Theorem poimirlem28 32607

Description: Lemma for poimir 32612, a variant of Sperner's lemma. (Contributed by Brendan Leahy, 21-Aug-2020.)

Hypotheses

Ref	Expression
poimir.0	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimirlem28.1	⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
poimirlem28.2	⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
poimirlem28.3	⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛)
poimirlem28.4	⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))
poimirlem28.5	⊢ (𝜑 → 𝐾 ∈ ℕ)

Assertion

Ref	Expression
poimirlem28	⊢ (𝜑 → ∃𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)

Distinct variable groups:   𝑓,𝑖,𝑗,𝑛,𝑝,𝑠   𝜑,𝑗,𝑛   𝑗,𝑁,𝑛   𝜑,𝑖,𝑝,𝑠   𝐵,𝑓,𝑖,𝑗,𝑛,𝑠   𝑓,𝐾,𝑖,𝑗,𝑛,𝑝,𝑠   𝑓,𝑁,𝑖,𝑝,𝑠   𝐶,𝑖,𝑛,𝑝

Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑝)   𝐶(𝑓,𝑗,𝑠)

Proof of Theorem poimirlem28

Dummy variables 𝑘 𝑚 𝑞 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		poimir.0	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2	1	nnnn0d 11228	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
3	1	nnred 10912	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
4	3	leidd 10473	. . . . 5 ⊢ (𝜑 → 𝑁 ≤ 𝑁)
5	2, 2, 4	3jca 1235	. . . 4 ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 𝑁))
6		oveq2 6557	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → (1...𝑘) = (1...0))
7		fz10 12233	. . . . . . . . . . . . . . . 16 ⊢ (1...0) = ∅
8	6, 7	syl6eq 2660	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → (1...𝑘) = ∅)
9	8	oveq2d 6565	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 ∅))
10		fzofi 12635	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝐾) ∈ Fin
11		map0e 7781	. . . . . . . . . . . . . . . 16 ⊢ ((0..^𝐾) ∈ Fin → ((0..^𝐾) ↑𝑚 ∅) = 1𝑜)
12	10, 11	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((0..^𝐾) ↑𝑚 ∅) = 1𝑜
13		df1o2 7459	. . . . . . . . . . . . . . 15 ⊢ 1𝑜 = {∅}
14	12, 13	eqtri 2632	. . . . . . . . . . . . . 14 ⊢ ((0..^𝐾) ↑𝑚 ∅) = {∅}
15	9, 14	syl6eq 2660	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = {∅})
16		eqidd 2611	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → 𝑓 = 𝑓)
17	16, 8, 8	f1oeq123d 6046	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:∅–1-1-onto→∅))
18		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ ∅ = ∅
19		f1o00 6083	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:∅–1-1-onto→∅ ↔ (𝑓 = ∅ ∧ ∅ = ∅))
20	18, 19	mpbiran2 956	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:∅–1-1-onto→∅ ↔ 𝑓 = ∅)
21	17, 20	syl6bb 275	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓 = ∅))
22	21	abbidv 2728	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓 ∣ 𝑓 = ∅})
23		df-sn 4126	. . . . . . . . . . . . . 14 ⊢ {∅} = {𝑓 ∣ 𝑓 = ∅}
24	22, 23	syl6eqr 2662	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {∅})
25	15, 24	xpeq12d 5064	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = ({∅} × {∅}))
26		0ex 4718	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
27	26, 26	xpsn 6313	. . . . . . . . . . . 12 ⊢ ({∅} × {∅}) = {⟨∅, ∅⟩}
28	25, 27	syl6req 2661	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → {⟨∅, ∅⟩} = (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}))
29		elsni 4142	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → 𝑠 = ⟨∅, ∅⟩)
30	26, 26	op1std 7069	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 = ⟨∅, ∅⟩ → (1st ‘𝑠) = ∅)
31	29, 30	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → (1st ‘𝑠) = ∅)
32	31	oveq1d 6564	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ((1st ‘𝑠) ∘𝑓 + ∅) = (∅ ∘𝑓 + ∅))
33		f0 5999	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅:∅⟶∅
34		ffn 5958	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅:∅⟶∅ → ∅ Fn ∅)
35	33, 34	mp1i 13	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ∅ Fn ∅)
36	26	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ∅ ∈ V)
37		inidm 3784	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∩ ∅) = ∅
38		0fv 6137	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∅‘𝑛) = ∅
39	38	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 ∈ {⟨∅, ∅⟩} ∧ 𝑛 ∈ ∅) → (∅‘𝑛) = ∅)
40	35, 35, 36, 36, 37, 39, 39	offval 6802	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → (∅ ∘𝑓 + ∅) = (𝑛 ∈ ∅ ↦ (∅ + ∅)))
41		mpt0 5934	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ∅ ↦ (∅ + ∅)) = ∅
42	40, 41	syl6eq 2660	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → (∅ ∘𝑓 + ∅) = ∅)
43	32, 42	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ((1st ‘𝑠) ∘𝑓 + ∅) = ∅)
44	43	uneq1d 3728	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → (((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) = (∅ ∪ ((1...𝑁) × {0})))
45		uncom 3719	. . . . . . . . . . . . . . . 16 ⊢ (∅ ∪ ((1...𝑁) × {0})) = (((1...𝑁) × {0}) ∪ ∅)
46		un0 3919	. . . . . . . . . . . . . . . 16 ⊢ (((1...𝑁) × {0}) ∪ ∅) = ((1...𝑁) × {0})
47	45, 46	eqtri 2632	. . . . . . . . . . . . . . 15 ⊢ (∅ ∪ ((1...𝑁) × {0})) = ((1...𝑁) × {0})
48	44, 47	syl6req 2661	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ((1...𝑁) × {0}) = (((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})))
49	48	csbeq1d 3506	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)
50	49	eqeq2d 2620	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ {⟨∅, ∅⟩} → (0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 ↔ 0 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵))
51		oveq2 6557	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → (0...𝑘) = (0...0))
52		0z 11265	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
53		fzsn 12254	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ ℤ → (0...0) = {0})
54	52, 53	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (0...0) = {0}
55	51, 54	syl6eq 2660	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (0...𝑘) = {0})
56		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 0 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...0))
57	56	imaeq2d 5385	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 0 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...0)))
58	57	xpeq1d 5062	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 0 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))
59	58	uneq2d 3729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 0 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0})))
60	59	oveq2d 6565	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))))
61		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 0 → (𝑘 + 1) = (0 + 1))
62		0p1e1 11009	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 + 1) = 1
63	61, 62	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 0 → (𝑘 + 1) = 1)
64	63	oveq1d 6564	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 0 → ((𝑘 + 1)...𝑁) = (1...𝑁))
65	64	xpeq1d 5062	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 0 → (((𝑘 + 1)...𝑁) × {0}) = ((1...𝑁) × {0}))
66	60, 65	uneq12d 3730	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 0 → (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})))
67	66	csbeq1d 3506	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 0 → ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)
68	67	eqeq2d 2620	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 0 → (𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵))
69	55, 68	rexeqbidv 3130	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 0 → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ {0}𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵))
70		c0ex 9913	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ V
71		oveq2 6557	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 = 0 → (1...𝑗) = (1...0))
72	71, 7	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 0 → (1...𝑗) = ∅)
73	72	imaeq2d 5385	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 0 → ((2nd ‘𝑠) “ (1...𝑗)) = ((2nd ‘𝑠) “ ∅))
74		ima0 5400	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((2nd ‘𝑠) “ ∅) = ∅
75	73, 74	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 0 → ((2nd ‘𝑠) “ (1...𝑗)) = ∅)
76	75	xpeq1d 5062	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 0 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) = (∅ × {1}))
77		0xp 5122	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∅ × {1}) = ∅
78	76, 77	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 0 → (((2nd ‘𝑠) “ (1...𝑗)) × {1}) = ∅)
79		oveq1 6556	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 = 0 → (𝑗 + 1) = (0 + 1))
80	79, 62	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 = 0 → (𝑗 + 1) = 1)
81	80	oveq1d 6564	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 = 0 → ((𝑗 + 1)...0) = (1...0))
82	81, 7	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 0 → ((𝑗 + 1)...0) = ∅)
83	82	imaeq2d 5385	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 0 → ((2nd ‘𝑠) “ ((𝑗 + 1)...0)) = ((2nd ‘𝑠) “ ∅))
84	83, 74	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 = 0 → ((2nd ‘𝑠) “ ((𝑗 + 1)...0)) = ∅)
85	84	xpeq1d 5062	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 = 0 → (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}) = (∅ × {0}))
86		0xp 5122	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∅ × {0}) = ∅
87	85, 86	syl6eq 2660	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 = 0 → (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}) = ∅)
88	78, 87	uneq12d 3730	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 = 0 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0})) = (∅ ∪ ∅))
89		un0 3919	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∅ ∪ ∅) = ∅
90	88, 89	syl6eq 2660	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = 0 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0})) = ∅)
91	90	oveq2d 6565	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 0 → ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) = ((1st ‘𝑠) ∘𝑓 + ∅))
92	91	uneq1d 3728	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 0 → (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) = (((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})))
93	92	csbeq1d 3506	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 0 → ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)
94	93	eqeq2d 2620	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 0 → (𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵))
95	70, 94	rexsn 4170	. . . . . . . . . . . . . . 15 ⊢ (∃𝑗 ∈ {0}𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...0)) × {0}))) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)
96	69, 95	syl6bb 275	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵))
97	55, 96	raleqbidv 3129	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ {0}𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵))
98		eqeq1 2614	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 0 → (𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 0 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵))
99	70, 98	ralsn 4169	. . . . . . . . . . . . 13 ⊢ (∀𝑖 ∈ {0}𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 0 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵)
100	97, 99	syl6rbb 276	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → (0 = ⦋(((1st ‘𝑠) ∘𝑓 + ∅) ∪ ((1...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
101	50, 100	sylan9bbr 733	. . . . . . . . . . 11 ⊢ ((𝑘 = 0 ∧ 𝑠 ∈ {⟨∅, ∅⟩}) → (0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
102	28, 101	rabeqbidva 3169	. . . . . . . . . 10 ⊢ (𝑘 = 0 → {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})
103	102	eqcomd 2616	. . . . . . . . 9 ⊢ (𝑘 = 0 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})
104	103	fveq2d 6107	. . . . . . . 8 ⊢ (𝑘 = 0 → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (#‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))
105	104	breq2d 4595	. . . . . . 7 ⊢ (𝑘 = 0 → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (#‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})))
106	105	notbid 307	. . . . . 6 ⊢ (𝑘 = 0 → (¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ ¬ 2 ∥ (#‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})))
107	106	imbi2d 329	. . . . 5 ⊢ (𝑘 = 0 → ((𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ↔ (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))))
108		oveq2 6557	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → (1...𝑘) = (1...𝑚))
109	108	oveq2d 6565	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 (1...𝑚)))
110		eqidd 2611	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → 𝑓 = 𝑓)
111	110, 108, 108	f1oeq123d 6046	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)))
112	111	abbidv 2728	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)})
113	109, 112	xpeq12d 5064	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}))
114		oveq2 6557	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (0...𝑘) = (0...𝑚))
115		oveq2 6557	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑚 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...𝑚))
116	115	imaeq2d 5385	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑚 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)))
117	116	xpeq1d 5062	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑚 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))
118	117	uneq2d 3729	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0})))
119	118	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))))
120		oveq1 6556	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑚 → (𝑘 + 1) = (𝑚 + 1))
121	120	oveq1d 6564	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → ((𝑘 + 1)...𝑁) = ((𝑚 + 1)...𝑁))
122	121	xpeq1d 5062	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → (((𝑘 + 1)...𝑁) × {0}) = (((𝑚 + 1)...𝑁) × {0}))
123	119, 122	uneq12d 3730	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})))
124	123	csbeq1d 3506	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
125	124	eqeq2d 2620	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → (𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
126	114, 125	rexeqbidv 3130	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
127	114, 126	raleqbidv 3129	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
128	113, 127	rabeqbidv 3168	. . . . . . . . 9 ⊢ (𝑘 = 𝑚 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})
129	128	fveq2d 6107	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))
130	129	breq2d 4595	. . . . . . 7 ⊢ (𝑘 = 𝑚 → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
131	130	notbid 307	. . . . . 6 ⊢ (𝑘 = 𝑚 → (¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
132	131	imbi2d 329	. . . . 5 ⊢ (𝑘 = 𝑚 → ((𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ↔ (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))))
133		oveq2 6557	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑚 + 1) → (1...𝑘) = (1...(𝑚 + 1)))
134	133	oveq2d 6565	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑚 + 1) → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))))
135		eqidd 2611	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑚 + 1) → 𝑓 = 𝑓)
136	135, 133, 133	f1oeq123d 6046	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑚 + 1) → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))))
137	136	abbidv 2728	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑚 + 1) → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))})
138	134, 137	xpeq12d 5064	. . . . . . . . . 10 ⊢ (𝑘 = (𝑚 + 1) → (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}))
139		oveq2 6557	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑚 + 1) → (0...𝑘) = (0...(𝑚 + 1)))
140		oveq2 6557	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = (𝑚 + 1) → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...(𝑚 + 1)))
141	140	imaeq2d 5385	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = (𝑚 + 1) → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))))
142	141	xpeq1d 5062	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑚 + 1) → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))
143	142	uneq2d 3729	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑚 + 1) → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0})))
144	143	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑚 + 1) → ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))))
145		oveq1 6556	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = (𝑚 + 1) → (𝑘 + 1) = ((𝑚 + 1) + 1))
146	145	oveq1d 6564	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑚 + 1) → ((𝑘 + 1)...𝑁) = (((𝑚 + 1) + 1)...𝑁))
147	146	xpeq1d 5062	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑚 + 1) → (((𝑘 + 1)...𝑁) × {0}) = ((((𝑚 + 1) + 1)...𝑁) × {0}))
148	144, 147	uneq12d 3730	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑚 + 1) → (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})))
149	148	csbeq1d 3506	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑚 + 1) → ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
150	149	eqeq2d 2620	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑚 + 1) → (𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
151	139, 150	rexeqbidv 3130	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑚 + 1) → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
152	139, 151	raleqbidv 3129	. . . . . . . . . 10 ⊢ (𝑘 = (𝑚 + 1) → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
153	138, 152	rabeqbidv 3168	. . . . . . . . 9 ⊢ (𝑘 = (𝑚 + 1) → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})
154	153	fveq2d 6107	. . . . . . . 8 ⊢ (𝑘 = (𝑚 + 1) → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))
155	154	breq2d 4595	. . . . . . 7 ⊢ (𝑘 = (𝑚 + 1) → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
156	155	notbid 307	. . . . . 6 ⊢ (𝑘 = (𝑚 + 1) → (¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
157	156	imbi2d 329	. . . . 5 ⊢ (𝑘 = (𝑚 + 1) → ((𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ↔ (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))))
158		oveq2 6557	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑁 → (1...𝑘) = (1...𝑁))
159	158	oveq2d 6565	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑁 → ((0..^𝐾) ↑𝑚 (1...𝑘)) = ((0..^𝐾) ↑𝑚 (1...𝑁)))
160		eqidd 2611	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑁 → 𝑓 = 𝑓)
161	160, 158, 158	f1oeq123d 6046	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑁 → (𝑓:(1...𝑘)–1-1-onto→(1...𝑘) ↔ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)))
162	161	abbidv 2728	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑁 → {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)} = {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
163	159, 162	xpeq12d 5064	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) = (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
164		oveq2 6557	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑁 → (0...𝑘) = (0...𝑁))
165		oveq2 6557	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑁 → ((𝑗 + 1)...𝑘) = ((𝑗 + 1)...𝑁))
166	165	imaeq2d 5385	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑁 → ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) = ((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)))
167	166	xpeq1d 5062	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑁 → (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}) = (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))
168	167	uneq2d 3729	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑁 → ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0})) = ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
169	168	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑁 → ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))))
170		oveq1 6556	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑁 → (𝑘 + 1) = (𝑁 + 1))
171	170	oveq1d 6564	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑁 → ((𝑘 + 1)...𝑁) = ((𝑁 + 1)...𝑁))
172	171	xpeq1d 5062	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑁 → (((𝑘 + 1)...𝑁) × {0}) = (((𝑁 + 1)...𝑁) × {0}))
173	169, 172	uneq12d 3730	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑁 → (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) = (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})))
174	173	csbeq1d 3506	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑁 → ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
175	174	eqeq2d 2620	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑁 → (𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
176	164, 175	rexeqbidv 3130	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑁 → (∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
177	164, 176	raleqbidv 3129	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → (∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
178	163, 177	rabeqbidv 3168	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})
179	178	fveq2d 6107	. . . . . . . 8 ⊢ (𝑘 = 𝑁 → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))
180	179	breq2d 4595	. . . . . . 7 ⊢ (𝑘 = 𝑁 → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
181	180	notbid 307	. . . . . 6 ⊢ (𝑘 = 𝑁 → (¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
182	181	imbi2d 329	. . . . 5 ⊢ (𝑘 = 𝑁 → ((𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑘)) × {𝑓 ∣ 𝑓:(1...𝑘)–1-1-onto→(1...𝑘)}) ∣ ∀𝑖 ∈ (0...𝑘)∃𝑗 ∈ (0...𝑘)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑘)) × {0}))) ∪ (((𝑘 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ↔ (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))))
183		n2dvds1 14942	. . . . . . 7 ⊢ ¬ 2 ∥ 1
184		opex 4859	. . . . . . . . . 10 ⊢ ⟨∅, ∅⟩ ∈ V
185		hashsng 13020	. . . . . . . . . 10 ⊢ (⟨∅, ∅⟩ ∈ V → (#‘{⟨∅, ∅⟩}) = 1)
186	184, 185	ax-mp 5	. . . . . . . . 9 ⊢ (#‘{⟨∅, ∅⟩}) = 1
187		nnuz 11599	. . . . . . . . . . . . . . . . 17 ⊢ ℕ = (ℤ≥‘1)
188	1, 187	syl6eleq 2698	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1))
189		eluzfz1 12219	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁))
190	188, 189	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ (1...𝑁))
191		poimirlem28.5	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ ℕ)
192	191	nnnn0d 11228	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
193		0elfz 12305	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ ℕ0 → 0 ∈ (0...𝐾))
194		fconst6g 6007	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0...𝐾) → ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾))
195	192, 193, 194	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾))
196	70	fvconst2 6374	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ (1...𝑁) → (((1...𝑁) × {0})‘1) = 0)
197	190, 196	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((1...𝑁) × {0})‘1) = 0)
198	190, 195, 197	3jca 1235	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0))
199		nfv 1830	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝(𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0))
200		nfcsb1v 3515	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑝⦋((1...𝑁) × {0}) / 𝑝⦌𝐵
201	200	nfeq1 2764	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0
202	199, 201	nfim 1813	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑝((𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)) → ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0)
203		ovex 6577	. . . . . . . . . . . . . . . 16 ⊢ (1...𝑁) ∈ V
204		snex 4835	. . . . . . . . . . . . . . . 16 ⊢ {0} ∈ V
205	203, 204	xpex 6860	. . . . . . . . . . . . . . 15 ⊢ ((1...𝑁) × {0}) ∈ V
206		feq1 5939	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ((1...𝑁) × {0}) → (𝑝:(1...𝑁)⟶(0...𝐾) ↔ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾)))
207		fveq1 6102	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = ((1...𝑁) × {0}) → (𝑝‘1) = (((1...𝑁) × {0})‘1))
208	207	eqeq1d 2612	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = ((1...𝑁) × {0}) → ((𝑝‘1) = 0 ↔ (((1...𝑁) × {0})‘1) = 0))
209	206, 208	3anbi23d 1394	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ((1...𝑁) × {0}) → ((1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0) ↔ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)))
210	209	anbi2d 736	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = ((1...𝑁) × {0}) → ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) ↔ (𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0))))
211		csbeq1a 3508	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = ((1...𝑁) × {0}) → 𝐵 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵)
212	211	eqeq1d 2612	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 = ((1...𝑁) × {0}) → (𝐵 = 0 ↔ ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0))
213	210, 212	imbi12d 333	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = ((1...𝑁) × {0}) → (((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 = 0) ↔ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)) → ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0)))
214		1ex 9914	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
215		eleq1 2676	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → (𝑛 ∈ (1...𝑁) ↔ 1 ∈ (1...𝑁)))
216		fveq2 6103	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 1 → (𝑝‘𝑛) = (𝑝‘1))
217	216	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 1 → ((𝑝‘𝑛) = 0 ↔ (𝑝‘1) = 0))
218	215, 217	3anbi13d 1393	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 1 → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0) ↔ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)))
219	218	anbi2d 736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) ↔ (𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0))))
220		breq2 4587	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → (𝐵 < 𝑛 ↔ 𝐵 < 1))
221	219, 220	imbi12d 333	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) ↔ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 < 1)))
222		poimirlem28.3	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛)
223	214, 221, 222	vtocl 3232	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 < 1)
224		poimirlem28.2	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
225		elfznn0 12302	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ (0...𝑁) → 𝐵 ∈ ℕ0)
226		nn0lt10b 11316	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ ℕ0 → (𝐵 < 1 ↔ 𝐵 = 0))
227	224, 225, 226	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝐵 < 1 ↔ 𝐵 = 0))
228	227	3ad2antr2 1220	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → (𝐵 < 1 ↔ 𝐵 = 0))
229	223, 228	mpbid 221	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘1) = 0)) → 𝐵 = 0)
230	202, 205, 213, 229	vtoclf 3231	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (1 ∈ (1...𝑁) ∧ ((1...𝑁) × {0}):(1...𝑁)⟶(0...𝐾) ∧ (((1...𝑁) × {0})‘1) = 0)) → ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0)
231	198, 230	mpdan 699	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵 = 0)
232	231	eqcomd 2616	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵)
233	232	ralrimivw 2950	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑠 ∈ {⟨∅, ∅⟩}0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵)
234		rabid2 3096	. . . . . . . . . . 11 ⊢ ({⟨∅, ∅⟩} = {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵} ↔ ∀𝑠 ∈ {⟨∅, ∅⟩}0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵)
235	233, 234	sylibr 223	. . . . . . . . . 10 ⊢ (𝜑 → {⟨∅, ∅⟩} = {𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})
236	235	fveq2d 6107	. . . . . . . . 9 ⊢ (𝜑 → (#‘{⟨∅, ∅⟩}) = (#‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))
237	186, 236	syl5eqr 2658	. . . . . . . 8 ⊢ (𝜑 → 1 = (#‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))
238	237	breq2d 4595	. . . . . . 7 ⊢ (𝜑 → (2 ∥ 1 ↔ 2 ∥ (#‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})))
239	183, 238	mtbii 315	. . . . . 6 ⊢ (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵}))
240	239	a1i 11	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ {⟨∅, ∅⟩} ∣ 0 = ⦋((1...𝑁) × {0}) / 𝑝⦌𝐵})))
241		2z 11286	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
242		fzfi 12633	. . . . . . . . . . . . . . . . 17 ⊢ (1...(𝑚 + 1)) ∈ Fin
243		mapfi 8145	. . . . . . . . . . . . . . . . 17 ⊢ (((0..^𝐾) ∈ Fin ∧ (1...(𝑚 + 1)) ∈ Fin) → ((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin)
244	10, 242, 243	mp2an 704	. . . . . . . . . . . . . . . 16 ⊢ ((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin
245		ovex 6577	. . . . . . . . . . . . . . . . . . 19 ⊢ (1...(𝑚 + 1)) ∈ V
246	245, 245	mapval 7756	. . . . . . . . . . . . . . . . . 18 ⊢ ((1...(𝑚 + 1)) ↑𝑚 (1...(𝑚 + 1))) = {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}
247		mapfi 8145	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1...(𝑚 + 1)) ∈ Fin ∧ (1...(𝑚 + 1)) ∈ Fin) → ((1...(𝑚 + 1)) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin)
248	242, 242, 247	mp2an 704	. . . . . . . . . . . . . . . . . 18 ⊢ ((1...(𝑚 + 1)) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin
249	246, 248	eqeltrri 2685	. . . . . . . . . . . . . . . . 17 ⊢ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} ∈ Fin
250		f1of 6050	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1)) → 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1)))
251	250	ss2abi 3637	. . . . . . . . . . . . . . . . 17 ⊢ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ⊆ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}
252		ssfi 8065	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))} ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ⊆ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))⟶(1...(𝑚 + 1))}) → {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ∈ Fin)
253	249, 251, 252	mp2an 704	. . . . . . . . . . . . . . . 16 ⊢ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ∈ Fin
254		xpfi 8116	. . . . . . . . . . . . . . . 16 ⊢ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))} ∈ Fin) → (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin)
255	244, 253, 254	mp2an 704	. . . . . . . . . . . . . . 15 ⊢ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin
256		rabfi 8070	. . . . . . . . . . . . . . 15 ⊢ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin)
257		hashcl 13009	. . . . . . . . . . . . . . 15 ⊢ ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℕ0)
258	255, 256, 257	mp2b 10	. . . . . . . . . . . . . 14 ⊢ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℕ0
259	258	nn0zi 11279	. . . . . . . . . . . . 13 ⊢ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ
260		rabfi 8070	. . . . . . . . . . . . . . 15 ⊢ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin)
261		hashcl 13009	. . . . . . . . . . . . . . 15 ⊢ ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℕ0)
262	255, 260, 261	mp2b 10	. . . . . . . . . . . . . 14 ⊢ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℕ0
263	262	nn0zi 11279	. . . . . . . . . . . . 13 ⊢ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℤ
264	241, 259, 263	3pm3.2i 1232	. . . . . . . . . . . 12 ⊢ (2 ∈ ℤ ∧ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ ∧ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℤ)
265		nn0p1nn 11209	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ)
266	265	ad2antrl 760	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (𝑚 + 1) ∈ ℕ)
267		uneq1 3722	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) = (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})))
268	267	csbeq1d 3506	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
269	70	fconst 6004	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0}
270	269	jctr 563	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) → (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0}))
271	265	nnred 10912	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℝ)
272	271	ltp1d 10833	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) < ((𝑚 + 1) + 1))
273		fzdisj 12239	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑚 + 1) < ((𝑚 + 1) + 1) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
274	272, 273	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ℕ0 → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
275		fun 5979	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}):(((𝑚 + 1) + 1)...𝑁)⟶{0}) ∧ ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}))
276	270, 274, 275	syl2anr 494	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}))
277	276	adantlr 747	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}))
278	277	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}))
279	265	peano2nnd 10914	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ ℕ)
280	279, 187	syl6eleq 2698	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ (ℤ≥‘1))
281	280	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((𝑚 + 1) + 1) ∈ (ℤ≥‘1))
282		nn0z 11277	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℤ)
283	1	nnzd 11357	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑁 ∈ ℤ)
284		zltp1le 11304	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑚 < 𝑁 ↔ (𝑚 + 1) ≤ 𝑁))
285	282, 283, 284	syl2anr 494	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑚 < 𝑁 ↔ (𝑚 + 1) ≤ 𝑁))
286	285	biimpa 500	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) → (𝑚 + 1) ≤ 𝑁)
287	286	anasss 677	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (𝑚 + 1) ≤ 𝑁)
288	282	peano2zd 11361	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℤ)
289	288	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) → (𝑚 + 1) ∈ ℤ)
290		eluz 11577	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑚 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘(𝑚 + 1)) ↔ (𝑚 + 1) ≤ 𝑁))
291	289, 283, 290	syl2anr 494	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (𝑁 ∈ (ℤ≥‘(𝑚 + 1)) ↔ (𝑚 + 1) ≤ 𝑁))
292	287, 291	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑁 ∈ (ℤ≥‘(𝑚 + 1)))
293		fzsplit2 12237	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑚 + 1) + 1) ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑚 + 1))) → (1...𝑁) = ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)))
294	281, 292, 293	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (1...𝑁) = ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)))
295	294	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁)) = (1...𝑁))
296	192, 193	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 0 ∈ (0...𝐾))
297	296	snssd 4281	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → {0} ⊆ (0...𝐾))
298		ssequn2 3748	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ({0} ⊆ (0...𝐾) ↔ ((0...𝐾) ∪ {0}) = (0...𝐾))
299	297, 298	sylib 207	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((0...𝐾) ∪ {0}) = (0...𝐾))
300	299	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((0...𝐾) ∪ {0}) = (0...𝐾))
301	295, 300	feq23d 5953	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)))
302	301	adantrr 749	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):((1...(𝑚 + 1)) ∪ (((𝑚 + 1) + 1)...𝑁))⟶((0...𝐾) ∪ {0}) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)))
303	278, 302	mpbid 221	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
304		nfv 1830	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑝(𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
305		nfcsb1v 3515	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵
306	305	nfel1 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)
307	304, 306	nfim 1813	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑝((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))
308		vex 3176	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑞 ∈ V
309		ovex 6577	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑚 + 1) + 1)...𝑁) ∈ V
310	309, 204	xpex 6860	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑚 + 1) + 1)...𝑁) × {0}) ∈ V
311	308, 310	unex 6854	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) ∈ V
312		feq1 5939	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝:(1...𝑁)⟶(0...𝐾) ↔ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)))
313	312	anbi2d 736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) ↔ (𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))))
314		csbeq1a 3508	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → 𝐵 = ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵)
315	314	eleq1d 2672	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 ∈ (0...𝑁) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁)))
316	313, 315	imbi12d 333	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁)) ↔ ((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))))
317	307, 311, 316, 224	vtoclf 3231	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))
318	303, 317	syldan 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))
319	318	anassrs 678	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))
320		elfznn0 12302	. . . . . . . . . . . . . . . . 17 ⊢ (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℕ0)
321	319, 320	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℕ0)
322	265	nnnn0d 11228	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
323	322	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) → (𝑚 + 1) ∈ ℕ0)
324	323	ad2antlr 759	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑚 + 1) ∈ ℕ0)
325		leloe 10003	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑚 + 1) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑚 + 1) ≤ 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
326	271, 3, 325	syl2anr 494	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) ≤ 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
327	285, 326	bitrd 267	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑚 < 𝑁 ↔ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
328	327	biimpd 218	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑚 < 𝑁 → ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
329	328	imdistani 722	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) → ((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
330	329	anasss 677	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → ((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)))
331		simplll 794	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝜑)
332	279	nnge1d 10940	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ ℕ0 → 1 ≤ ((𝑚 + 1) + 1))
333	332	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 1 ≤ ((𝑚 + 1) + 1))
334		zltp1le 11304	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑚 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑚 + 1) < 𝑁 ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
335	288, 283, 334	syl2anr 494	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((𝑚 + 1) < 𝑁 ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
336	335	biimpa 500	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ≤ 𝑁)
337	288	peano2zd 11361	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) + 1) ∈ ℤ)
338		1z 11284	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 1 ∈ ℤ
339		elfz 12203	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑚 + 1) + 1) ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
340	338, 339	mp3an2 1404	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑚 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
341	337, 283, 340	syl2anr 494	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
342	341	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (((𝑚 + 1) + 1) ∈ (1...𝑁) ↔ (1 ≤ ((𝑚 + 1) + 1) ∧ ((𝑚 + 1) + 1) ≤ 𝑁)))
343	333, 336, 342	mpbir2and 959	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (1...𝑁))
344	343	adantlr 747	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (1...𝑁))
345		nn0re 11178	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℝ)
346	345	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 ∈ ℝ)
347	271	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) ∈ ℝ)
348	3	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑁 ∈ ℝ)
349	345	ltp1d 10833	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ ℕ0 → 𝑚 < (𝑚 + 1))
350	349	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < (𝑚 + 1))
351		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) < 𝑁)
352	346, 347, 348, 350, 351	lttrd 10077	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < 𝑁)
353	352	adantlr 747	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝑚 < 𝑁)
354		anass 679	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ↔ (𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)))
355	303	anassrs 678	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
356	354, 355	sylanb 488	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
357	356	an32s 842	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
358	353, 357	syldan 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
359		ffn 5958	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑞:(1...(𝑚 + 1))⟶(0...𝐾) → 𝑞 Fn (1...(𝑚 + 1)))
360	359	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → 𝑞 Fn (1...(𝑚 + 1)))
361	274	ad3antlr 763	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
362		eluz 11577	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑚 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
363	337, 283, 362	syl2anr 494	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑁 ∈ (ℤ≥‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
364	363	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → (𝑁 ∈ (ℤ≥‘((𝑚 + 1) + 1)) ↔ ((𝑚 + 1) + 1) ≤ 𝑁))
365	336, 364	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → 𝑁 ∈ (ℤ≥‘((𝑚 + 1) + 1)))
366		eluzfz1 12219	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ (ℤ≥‘((𝑚 + 1) + 1)) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))
367	365, 366	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))
368	367	adantlr 747	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))
369		fnconstg 6006	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 ∈ V → ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁))
370	70, 369	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁)
371		fvun2 6180	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑞 Fn (1...(𝑚 + 1)) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)))
372	370, 371	mp3an2 1404	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑞 Fn (1...(𝑚 + 1)) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ ((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)))
373	360, 361, 368, 372	syl12anc 1316	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)))
374	70	fvconst2 6374	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑚 + 1) + 1) ∈ (((𝑚 + 1) + 1)...𝑁) → (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)) = 0)
375	368, 374	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (((((𝑚 + 1) + 1)...𝑁) × {0})‘((𝑚 + 1) + 1)) = 0)
376	373, 375	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)
377		nfv 1830	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑝(𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0))
378		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑝 <
379		nfcv 2751	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑝((𝑚 + 1) + 1)
380	305, 378, 379	nfbr 4629	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1)
381	377, 380	nfim 1813	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑝((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1))
382		fveq1 6102	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝‘((𝑚 + 1) + 1)) = ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)))
383	382	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝‘((𝑚 + 1) + 1)) = 0 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0))
384	312, 383	3anbi23d 1394	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0) ↔ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)))
385	384	anbi2d 736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) ↔ (𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0))))
386	314	breq1d 4593	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 < ((𝑚 + 1) + 1) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1)))
387	385, 386	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) → 𝐵 < ((𝑚 + 1) + 1)) ↔ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1))))
388		ovex 6577	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 + 1) + 1) ∈ V
389		eleq1 2676	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = ((𝑚 + 1) + 1) → (𝑛 ∈ (1...𝑁) ↔ ((𝑚 + 1) + 1) ∈ (1...𝑁)))
390		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = ((𝑚 + 1) + 1) → (𝑝‘𝑛) = (𝑝‘((𝑚 + 1) + 1)))
391	390	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = ((𝑚 + 1) + 1) → ((𝑝‘𝑛) = 0 ↔ (𝑝‘((𝑚 + 1) + 1)) = 0))
392	389, 391	3anbi13d 1393	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = ((𝑚 + 1) + 1) → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0) ↔ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)))
393	392	anbi2d 736	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = ((𝑚 + 1) + 1) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) ↔ (𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0))))
394		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = ((𝑚 + 1) + 1) → (𝐵 < 𝑛 ↔ 𝐵 < ((𝑚 + 1) + 1)))
395	393, 394	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = ((𝑚 + 1) + 1) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) ↔ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) → 𝐵 < ((𝑚 + 1) + 1))))
396	388, 395, 222	vtocl 3232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘((𝑚 + 1) + 1)) = 0)) → 𝐵 < ((𝑚 + 1) + 1))
397	381, 311, 387, 396	vtoclf 3231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (((𝑚 + 1) + 1) ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘((𝑚 + 1) + 1)) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1))
398	331, 344, 358, 376, 397	syl13anc 1320	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1))
399	354, 319	sylanb 488	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑚 < 𝑁) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))
400	399	an32s 842	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁))
401		elfzelz 12213	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℤ)
402	400, 401	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℤ)
403	353, 402	syldan 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℤ)
404	288	ad3antlr 763	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (𝑚 + 1) ∈ ℤ)
405		zleltp1 11305	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℤ ∧ (𝑚 + 1) ∈ ℤ) → (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1)))
406	403, 404, 405	syl2anc 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < ((𝑚 + 1) + 1)))
407	398, 406	mpbird 246	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1))
408	349	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → 𝑚 < (𝑚 + 1))
409		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 + 1) = 𝑁 → (𝑚 < (𝑚 + 1) ↔ 𝑚 < 𝑁))
410	409	biimpac 502	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 < (𝑚 + 1) ∧ (𝑚 + 1) = 𝑁) → 𝑚 < 𝑁)
411	408, 410	sylan 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → 𝑚 < 𝑁)
412		elfzle2 12216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ 𝑁)
413	400, 412	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ 𝑚 < 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ 𝑁)
414	411, 413	syldan 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ 𝑁)
415		simpr 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → (𝑚 + 1) = 𝑁)
416	414, 415	breqtrrd 4611	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ (𝑚 + 1) = 𝑁) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1))
417	407, 416	jaodan 822	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1))
418	417	an32s 842	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑚 + 1) < 𝑁 ∨ (𝑚 + 1) = 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1))
419	330, 418	sylan 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1))
420		elfz2nn0 12300	. . . . . . . . . . . . . . . 16 ⊢ (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...(𝑚 + 1)) ↔ (⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ ℕ0 ∧ (𝑚 + 1) ∈ ℕ0 ∧ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≤ (𝑚 + 1)))
421	321, 324, 419, 420	syl3anbrc 1239	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∈ (0...(𝑚 + 1)))
422		fzss2 12252	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ (ℤ≥‘(𝑚 + 1)) → (1...(𝑚 + 1)) ⊆ (1...𝑁))
423	292, 422	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (1...(𝑚 + 1)) ⊆ (1...𝑁))
424	423	sselda 3568	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ 𝑛 ∈ (1...(𝑚 + 1))) → 𝑛 ∈ (1...𝑁))
425	424	3ad2antr1 1219	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → 𝑛 ∈ (1...𝑁))
426	355	3ad2antr2 1220	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
427	359	ad2antll 761	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → 𝑞 Fn (1...(𝑚 + 1)))
428	274	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅)
429		simprl 790	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → 𝑛 ∈ (1...(𝑚 + 1)))
430		fvun1 6179	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑞 Fn (1...(𝑚 + 1)) ∧ ((((𝑚 + 1) + 1)...𝑁) × {0}) Fn (((𝑚 + 1) + 1)...𝑁) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛))
431	370, 430	mp3an2 1404	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑞 Fn (1...(𝑚 + 1)) ∧ (((1...(𝑚 + 1)) ∩ (((𝑚 + 1) + 1)...𝑁)) = ∅ ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛))
432	427, 428, 429, 431	syl12anc 1316	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛))
433	432	adantlrr 753	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛))
434	433	3adantr3 1215	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛))
435		simpr3 1062	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → (𝑞‘𝑛) = 0)
436	434, 435	eqtrd 2644	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)
437	425, 426, 436	3jca 1235	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))
438		nfv 1830	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑝(𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))
439		nfcv 2751	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑝𝑛
440	305, 378, 439	nfbr 4629	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛
441	438, 440	nfim 1813	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑝((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛)
442		fveq1 6102	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝑝‘𝑛) = ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛))
443	442	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝‘𝑛) = 0 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))
444	312, 443	3anbi23d 1394	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)))
445	444	anbi2d 736	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) ↔ (𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0))))
446	314	breq1d 4593	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 < 𝑛 ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛))
447	445, 446	imbi12d 333	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛) ↔ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛)))
448	441, 311, 447, 222	vtoclf 3231	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛)
449	448	adantlr 747	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛)
450	437, 449	syldan 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 0)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 < 𝑛)
451		simp1 1054	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾) → 𝑛 ∈ (1...(𝑚 + 1)))
452	424	anasss 677	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ 𝑛 ∈ (1...(𝑚 + 1)))) → 𝑛 ∈ (1...𝑁))
453	451, 452	sylanr2 683	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → 𝑛 ∈ (1...𝑁))
454		simp2 1055	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾) → 𝑞:(1...(𝑚 + 1))⟶(0...𝐾))
455	454, 303	sylanr2 683	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾))
456	432	3adantr3 1215	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = (𝑞‘𝑛))
457		simpr3 1062	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾)) → (𝑞‘𝑛) = 𝐾)
458	456, 457	eqtrd 2644	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾)) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)
459	458	anasss 677	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)
460	459	adantrlr 755	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)
461	453, 455, 460	3jca 1235	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))
462		nfv 1830	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑝(𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))
463		nfcv 2751	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑝(𝑛 − 1)
464	305, 463	nfne 2882	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑝⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1)
465	462, 464	nfim 1813	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑝((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1))
466	442	eqeq1d 2612	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑝‘𝑛) = 𝐾 ↔ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))
467	312, 466	3anbi23d 1394	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾) ↔ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)))
468	467	anbi2d 736	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) ↔ (𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾))))
469	314	neeq1d 2841	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (𝐵 ≠ (𝑛 − 1) ↔ ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1)))
470	468, 469	imbi12d 333	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 = (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) → (((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1)) ↔ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1))))
471		poimirlem28.4	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))
472	465, 311, 470, 471	vtoclf 3231	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ (𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})):(1...𝑁)⟶(0...𝐾) ∧ ((𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0}))‘𝑛) = 𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1))
473	461, 472	syldan 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾))) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1))
474	473	anassrs 678	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) ∧ (𝑛 ∈ (1...(𝑚 + 1)) ∧ 𝑞:(1...(𝑚 + 1))⟶(0...𝐾) ∧ (𝑞‘𝑛) = 𝐾)) → ⦋(𝑞 ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ≠ (𝑛 − 1))
475	266, 268, 421, 450, 474	poimirlem27 32606	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})))
476	266, 268, 421	poimirlem26 32605	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
477		fzfi 12633	. . . . . . . . . . . . . . . . . . 19 ⊢ (0...(𝑚 + 1)) ∈ Fin
478		xpfi 8116	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin ∧ (0...(𝑚 + 1)) ∈ Fin) → ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin)
479	255, 477, 478	mp2an 704	. . . . . . . . . . . . . . . . . 18 ⊢ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin
480		rabfi 8070	. . . . . . . . . . . . . . . . . 18 ⊢ (((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin)
481		hashcl 13009	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin → (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℕ0)
482	479, 480, 481	mp2b 10	. . . . . . . . . . . . . . . . 17 ⊢ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℕ0
483	482	nn0zi 11279	. . . . . . . . . . . . . . . 16 ⊢ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ
484		zsubcl 11296	. . . . . . . . . . . . . . . 16 ⊢ (((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ ∧ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℤ) → ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∈ ℤ)
485	483, 263, 484	mp2an 704	. . . . . . . . . . . . . . 15 ⊢ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∈ ℤ
486		zsubcl 11296	. . . . . . . . . . . . . . . 16 ⊢ (((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ ∧ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ) → ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ∈ ℤ)
487	483, 259, 486	mp2an 704	. . . . . . . . . . . . . . 15 ⊢ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ∈ ℤ
488		dvds2sub 14854	. . . . . . . . . . . . . . 15 ⊢ ((2 ∈ ℤ ∧ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∈ ℤ ∧ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) ∈ ℤ) → ((2 ∥ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∧ 2 ∥ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) → 2 ∥ (((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))))
489	241, 485, 487, 488	mp3an 1416	. . . . . . . . . . . . . 14 ⊢ ((2 ∥ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) ∧ 2 ∥ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) → 2 ∥ (((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))))
490	475, 476, 489	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ (((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))))
491	482	nn0cni 11181	. . . . . . . . . . . . . 14 ⊢ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℂ
492	262	nn0cni 11181	. . . . . . . . . . . . . 14 ⊢ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℂ
493	258	nn0cni 11181	. . . . . . . . . . . . . 14 ⊢ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℂ
494		nnncan1 10196	. . . . . . . . . . . . . 14 ⊢ (((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℂ ∧ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℂ ∧ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℂ) → (((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) = ((#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})))
495	491, 492, 493, 494	mp3an 1416	. . . . . . . . . . . . 13 ⊢ (((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})) − ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) × (0...(𝑚 + 1))) ∣ ∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ ((0...(𝑚 + 1)) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))) = ((#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))
496	490, 495	syl6breq 4624	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 2 ∥ ((#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})))
497		dvdssub2 14861	. . . . . . . . . . . 12 ⊢ (((2 ∈ ℤ ∧ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ∈ ℤ ∧ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ∈ ℤ) ∧ 2 ∥ ((#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))) → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})))
498	264, 496, 497	sylancr 694	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})))
499		nn0cn 11179	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ)
500		pncan1 10333	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℂ → ((𝑚 + 1) − 1) = 𝑚)
501	499, 500	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ ℕ0 → ((𝑚 + 1) − 1) = 𝑚)
502	501	oveq2d 6565	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕ0 → (0...((𝑚 + 1) − 1)) = (0...𝑚))
503	502	rexeqdv 3122	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ ℕ0 → (∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
504	502, 503	raleqbidv 3129	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ ℕ0 → (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵))
505	504	3anbi1d 1395	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ0 → ((∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1)) ↔ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))))
506	505	rabbidv 3164	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ0 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})
507	506	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ0 → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))
508	507	ad2antrl 760	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))
509	1	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑁 ∈ ℕ)
510	191	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝐾 ∈ ℕ)
511		simprl 790	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑚 ∈ ℕ0)
512		simprr 792	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → 𝑚 < 𝑁)
513	509, 510, 511, 512	poimirlem4 32583	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})
514		fzfi 12633	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑚) ∈ Fin
515		mapfi 8145	. . . . . . . . . . . . . . . . . 18 ⊢ (((0..^𝐾) ∈ Fin ∧ (1...𝑚) ∈ Fin) → ((0..^𝐾) ↑𝑚 (1...𝑚)) ∈ Fin)
516	10, 514, 515	mp2an 704	. . . . . . . . . . . . . . . . 17 ⊢ ((0..^𝐾) ↑𝑚 (1...𝑚)) ∈ Fin
517		ovex 6577	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1...𝑚) ∈ V
518	517, 517	mapval 7756	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1...𝑚) ↑𝑚 (1...𝑚)) = {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)}
519		mapfi 8145	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1...𝑚) ∈ Fin ∧ (1...𝑚) ∈ Fin) → ((1...𝑚) ↑𝑚 (1...𝑚)) ∈ Fin)
520	514, 514, 519	mp2an 704	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1...𝑚) ↑𝑚 (1...𝑚)) ∈ Fin
521	518, 520	eqeltrri 2685	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)} ∈ Fin
522		f1of 6050	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:(1...𝑚)–1-1-onto→(1...𝑚) → 𝑓:(1...𝑚)⟶(1...𝑚))
523	522	ss2abi 3637	. . . . . . . . . . . . . . . . . 18 ⊢ {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ⊆ {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)}
524		ssfi 8065	. . . . . . . . . . . . . . . . . 18 ⊢ (({𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)} ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ⊆ {𝑓 ∣ 𝑓:(1...𝑚)⟶(1...𝑚)}) → {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ∈ Fin)
525	521, 523, 524	mp2an 704	. . . . . . . . . . . . . . . . 17 ⊢ {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ∈ Fin
526		xpfi 8116	. . . . . . . . . . . . . . . . 17 ⊢ ((((0..^𝐾) ↑𝑚 (1...𝑚)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)} ∈ Fin) → (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∈ Fin)
527	516, 525, 526	mp2an 704	. . . . . . . . . . . . . . . 16 ⊢ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∈ Fin
528		rabfi 8070	. . . . . . . . . . . . . . . 16 ⊢ ((((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin)
529	527, 528	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin
530		rabfi 8070	. . . . . . . . . . . . . . . 16 ⊢ ((((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin)
531	255, 530	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin
532		hashen 12997	. . . . . . . . . . . . . . 15 ⊢ (({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ∈ Fin ∧ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))} ∈ Fin) → ((#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ↔ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))
533	529, 531, 532	mp2an 704	. . . . . . . . . . . . . 14 ⊢ ((#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ↔ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))})
534	513, 533	sylibr 223	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}))
535	508, 534	eqtr4d 2647	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))
536	535	breq2d 4595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ (∀𝑖 ∈ (0...((𝑚 + 1) − 1))∃𝑗 ∈ (0...((𝑚 + 1) − 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ∧ ((1st ‘𝑠)‘(𝑚 + 1)) = 0 ∧ ((2nd ‘𝑠)‘(𝑚 + 1)) = (𝑚 + 1))}) ↔ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
537	498, 536	bitrd 267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
538	537	biimpd 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) → 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
539	538	con3d 147	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁)) → (¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
540	539	expcom 450	. . . . . . 7 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) → (𝜑 → (¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))))
541	540	a2d 29	. . . . . 6 ⊢ ((𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) → ((𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) → (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))))
542	541	3adant1 1072	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑚 ∈ ℕ0 ∧ 𝑚 < 𝑁) → ((𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑚)) × {𝑓 ∣ 𝑓:(1...𝑚)–1-1-onto→(1...𝑚)}) ∣ ∀𝑖 ∈ (0...𝑚)∃𝑗 ∈ (0...𝑚)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑚)) × {0}))) ∪ (((𝑚 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})) → (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑚 + 1))) × {𝑓 ∣ 𝑓:(1...(𝑚 + 1))–1-1-onto→(1...(𝑚 + 1))}) ∣ ∀𝑖 ∈ (0...(𝑚 + 1))∃𝑗 ∈ (0...(𝑚 + 1))𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...(𝑚 + 1))) × {0}))) ∪ ((((𝑚 + 1) + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))))
543	107, 132, 157, 182, 240, 542	fnn0ind 11352	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 𝑁) → (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
544	5, 543	mpcom 37	. . 3 ⊢ (𝜑 → ¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}))
545		dvds0 14835	. . . . . . . 8 ⊢ (2 ∈ ℤ → 2 ∥ 0)
546	241, 545	ax-mp 5	. . . . . . 7 ⊢ 2 ∥ 0
547		hash0 13019	. . . . . . 7 ⊢ (#‘∅) = 0
548	546, 547	breqtrri 4610	. . . . . 6 ⊢ 2 ∥ (#‘∅)
549		fveq2 6103	. . . . . 6 ⊢ ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}) = (#‘∅))
550	548, 549	syl5breqr 4621	. . . . 5 ⊢ ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
551	3	ltp1d 10833	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 < (𝑁 + 1))
552	283	peano2zd 11361	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
553		fzn 12228	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
554	552, 283, 553	syl2anc 691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁 < (𝑁 + 1) ↔ ((𝑁 + 1)...𝑁) = ∅))
555	551, 554	mpbid 221	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑁 + 1)...𝑁) = ∅)
556	555	xpeq1d 5062	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝑁 + 1)...𝑁) × {0}) = (∅ × {0}))
557	556, 86	syl6eq 2660	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((𝑁 + 1)...𝑁) × {0}) = ∅)
558	557	uneq2d 3729	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) = (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ ∅))
559		un0 3919	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ ∅) = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0})))
560	558, 559	syl6eq 2660	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))))
561	560	csbeq1d 3506	. . . . . . . . . . . 12 ⊢ (𝜑 → ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = ⦋((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵)
562		ovex 6577	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
563		poimirlem28.1	. . . . . . . . . . . . 13 ⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
564	562, 563	csbie 3525	. . . . . . . . . . . 12 ⊢ ⦋((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) / 𝑝⦌𝐵 = 𝐶
565	561, 564	syl6eq 2660	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 = 𝐶)
566	565	eqeq2d 2620	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ 𝑖 = 𝐶))
567	566	rexbidv 3034	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
568	567	ralbidv 2969	. . . . . . . 8 ⊢ (𝜑 → (∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵 ↔ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶))
569	568	rabbidv 3164	. . . . . . 7 ⊢ (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵} = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})
570	569	fveq2d 6107	. . . . . 6 ⊢ (𝜑 → (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶}))
571	570	breq2d 4595	. . . . 5 ⊢ (𝜑 → (2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) ↔ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶})))
572	550, 571	syl5ibr 235	. . . 4 ⊢ (𝜑 → ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} = ∅ → 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵})))
573	572	necon3bd 2796	. . 3 ⊢ (𝜑 → (¬ 2 ∥ (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = ⦋(((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) ∪ (((𝑁 + 1)...𝑁) × {0})) / 𝑝⦌𝐵}) → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅))
574	544, 573	mpd 15	. 2 ⊢ (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅)
575		rabn0 3912	. 2 ⊢ ({𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ ∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶} ≠ ∅ ↔ ∃𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)
576	574, 575	sylib 207	1 ⊢ (𝜑 → ∃𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})∀𝑖 ∈ (0...𝑁)∃𝑗 ∈ (0...𝑁)𝑖 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  ⦋csb 3499   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036   “ cima 5041   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793  1^st c1st 7057  2^nd c2nd 7058  1_𝑜c1o 7440   ↑_𝑚 cmap 7744   ≈ cen 7838  Fincfn 7841  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  2c2 10947  ℕ₀cn0 11169  ℤcz 11254  ℤ_≥cuz 11563  ...cfz 12197  ..^cfzo 12334  #chash 12979   ∥ cdvds 14821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-dvds 14822

This theorem is referenced by:  poimirlem32  32611