poimirlem1 - Mathbox for Brendan Leahy

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Theorem poimirlem1 32580

Description: Lemma for poimir 32612- the vertices on either side of a skipped vertex differ in at least two dimensions. (Contributed by Brendan Leahy, 21-Aug-2020.)

**Hypotheses**
Ref	Expression
poimir.0	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimirlem2.1	⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))
poimirlem2.2	⊢ (𝜑 → 𝑇:(1...𝑁)⟶ℤ)
poimirlem2.3	⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁))
poimirlem1.4	⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1)))

**Assertion**
Ref	Expression
poimirlem1	⊢ (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛))

Distinct variable groups: 𝑗,𝑛,𝑦,𝜑 𝑗,𝐹,𝑛,𝑦 𝑗,𝑀,𝑛,𝑦 𝑗,𝑁,𝑛,𝑦 𝑇,𝑗,𝑛,𝑦 𝑈,𝑗,𝑛,𝑦

Proof of Theorem poimirlem1 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		poimirlem2.3	. . . . 5 ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁))
2		f1of 6050	. . . . 5 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)⟶(1...𝑁))
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝑈:(1...𝑁)⟶(1...𝑁))
4		poimir.0	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
5	4	nncnd 10913	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ)
6		npcan1 10334	. . . . . . . 8 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
7	5, 6	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
8	4	nnzd 11357	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
9		peano2zm 11297	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
10		uzid 11578	. . . . . . . 8 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)))
11		peano2uz 11617	. . . . . . . 8 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
12	8, 9, 10, 11	4syl 19	. . . . . . 7 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
13	7, 12	eqeltrrd 2689	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
14		fzss2 12252	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
15	13, 14	syl 17	. . . . 5 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
16		poimirlem1.4	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1)))
17	15, 16	sseldd 3569	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))
18	3, 17	ffvelrnd 6268	. . 3 ⊢ (𝜑 → (𝑈‘𝑀) ∈ (1...𝑁))
19		fzp1elp1 12264	. . . . . 6 ⊢ (𝑀 ∈ (1...(𝑁 − 1)) → (𝑀 + 1) ∈ (1...((𝑁 − 1) + 1)))
20	16, 19	syl 17	. . . . 5 ⊢ (𝜑 → (𝑀 + 1) ∈ (1...((𝑁 − 1) + 1)))
21	7	oveq2d 6565	. . . . 5 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
22	20, 21	eleqtrd 2690	. . . 4 ⊢ (𝜑 → (𝑀 + 1) ∈ (1...𝑁))
23	3, 22	ffvelrnd 6268	. . 3 ⊢ (𝜑 → (𝑈‘(𝑀 + 1)) ∈ (1...𝑁))
24		imassrn 5396	. . . . . . . . . 10 ⊢ (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ ran 𝑈
25		frn 5966	. . . . . . . . . . 11 ⊢ (𝑈:(1...𝑁)⟶(1...𝑁) → ran 𝑈 ⊆ (1...𝑁))
26	1, 2, 25	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑈 ⊆ (1...𝑁))
27	24, 26	syl5ss 3579	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (1...𝑁))
28	27	sselda 3568	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → 𝑛 ∈ (1...𝑁))
29		poimirlem2.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇:(1...𝑁)⟶ℤ)
30	29	ffvelrnda 6267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑇‘𝑛) ∈ ℤ)
31	30	zred 11358	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑇‘𝑛) ∈ ℝ)
32	31	ltp1d 10833	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑇‘𝑛) < ((𝑇‘𝑛) + 1))
33	31, 32	ltned 10052	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑇‘𝑛) ≠ ((𝑇‘𝑛) + 1))
34	28, 33	syldan 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (𝑇‘𝑛) ≠ ((𝑇‘𝑛) + 1))
35		poimirlem2.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))))
36		breq1 4586	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑀 − 1) → (𝑦 < 𝑀 ↔ (𝑀 − 1) < 𝑀))
37		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑀 − 1) → 𝑦 = (𝑀 − 1))
38	36, 37	ifbieq1d 4059	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑀 − 1) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < 𝑀, (𝑀 − 1), (𝑦 + 1)))
39		elfzelz 12213	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ∈ ℤ)
40	16, 39	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑀 ∈ ℤ)
41	40	zred 11358	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℝ)
42	41	ltm1d 10835	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀 − 1) < 𝑀)
43	42	iftrued 4044	. . . . . . . . . . . . . 14 ⊢ (𝜑 → if((𝑀 − 1) < 𝑀, (𝑀 − 1), (𝑦 + 1)) = (𝑀 − 1))
44	38, 43	sylan9eqr 2666	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = (𝑀 − 1))
45	44	csbeq1d 3506	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋(𝑀 − 1) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
46	8, 9	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
47		elfzm1b 12287	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (𝑀 ∈ (1...(𝑁 − 1)) ↔ (𝑀 − 1) ∈ (0...((𝑁 − 1) − 1))))
48	40, 46, 47	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀 ∈ (1...(𝑁 − 1)) ↔ (𝑀 − 1) ∈ (0...((𝑁 − 1) − 1))))
49	16, 48	mpbid 221	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 − 1) ∈ (0...((𝑁 − 1) − 1)))
50		oveq2 6557	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (𝑀 − 1) → (1...𝑗) = (1...(𝑀 − 1)))
51	50	imaeq2d 5385	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = (𝑀 − 1) → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...(𝑀 − 1))))
52	51	xpeq1d 5062	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = (𝑀 − 1) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑀 − 1))) × {1}))
53	52	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑀 − 1))) × {1}))
54		oveq1 6556	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 = (𝑀 − 1) → (𝑗 + 1) = ((𝑀 − 1) + 1))
55	40	zcnd 11359	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑀 ∈ ℂ)
56		npcan1 10334	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀)
57	55, 56	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀)
58	54, 57	sylan9eqr 2666	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (𝑗 + 1) = 𝑀)
59	58	oveq1d 6564	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((𝑗 + 1)...𝑁) = (𝑀...𝑁))
60	59	imaeq2d 5385	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ (𝑀...𝑁)))
61	60	xpeq1d 5062	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ (𝑀...𝑁)) × {0}))
62	53, 61	uneq12d 3730	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))
63	62	oveq2d 6565	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))))
64	49, 63	csbied 3526	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⦋(𝑀 − 1) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))))
65	64	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋(𝑀 − 1) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))))
66	45, 65	eqtrd 2644	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))))
67	46	zcnd 11359	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 − 1) ∈ ℂ)
68		npcan1 10334	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 − 1) ∈ ℂ → (((𝑁 − 1) − 1) + 1) = (𝑁 − 1))
69	67, 68	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝑁 − 1) − 1) + 1) = (𝑁 − 1))
70		peano2zm 11297	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 − 1) ∈ ℤ → ((𝑁 − 1) − 1) ∈ ℤ)
71		uzid 11578	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 − 1) − 1) ∈ ℤ → ((𝑁 − 1) − 1) ∈ (ℤ_≥‘((𝑁 − 1) − 1)))
72		peano2uz 11617	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 − 1) − 1) ∈ (ℤ_≥‘((𝑁 − 1) − 1)) → (((𝑁 − 1) − 1) + 1) ∈ (ℤ_≥‘((𝑁 − 1) − 1)))
73	46, 70, 71, 72	4syl 19	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝑁 − 1) − 1) + 1) ∈ (ℤ_≥‘((𝑁 − 1) − 1)))
74	69, 73	eqeltrrd 2689	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ_≥‘((𝑁 − 1) − 1)))
75		fzss2 12252	. . . . . . . . . . . . 13 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘((𝑁 − 1) − 1)) → (0...((𝑁 − 1) − 1)) ⊆ (0...(𝑁 − 1)))
76	74, 75	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0...((𝑁 − 1) − 1)) ⊆ (0...(𝑁 − 1)))
77	76, 49	sseldd 3569	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1)))
78		ovex 6577	. . . . . . . . . . . 12 ⊢ (𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))) ∈ V
79	78	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))) ∈ V)
80	35, 66, 77, 79	fvmptd 6197	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘(𝑀 − 1)) = (𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))))
81	80	fveq1d 6105	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))‘𝑛))
82	81	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))‘𝑛))
83		ffn 5958	. . . . . . . . . . . 12 ⊢ (𝑇:(1...𝑁)⟶ℤ → 𝑇 Fn (1...𝑁))
84	29, 83	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 Fn (1...𝑁))
85	84	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → 𝑇 Fn (1...𝑁))
86		1ex 9914	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ V
87		fnconstg 6006	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ V → ((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1))))
88	86, 87	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1)))
89		c0ex 9913	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
90		fnconstg 6006	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ V → ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁)))
91	89, 90	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁))
92	88, 91	pm3.2i 470	. . . . . . . . . . . . 13 ⊢ (((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1))) ∧ ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁)))
93		dff1o3 6056	. . . . . . . . . . . . . . . 16 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (𝑈:(1...𝑁)–onto→(1...𝑁) ∧ Fun ^◡𝑈))
94	93	simprbi 479	. . . . . . . . . . . . . . 15 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡𝑈)
95		imain 5888	. . . . . . . . . . . . . . 15 ⊢ (Fun ^◡𝑈 → (𝑈 “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))))
96	1, 94, 95	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑈 “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))))
97		fzdisj 12239	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 − 1) < 𝑀 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
98	42, 97	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅)
99	98	imaeq2d 5385	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = (𝑈 “ ∅))
100		ima0 5400	. . . . . . . . . . . . . . 15 ⊢ (𝑈 “ ∅) = ∅
101	99, 100	syl6eq 2660	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑈 “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ∅)
102	96, 101	eqtr3d 2646	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅)
103		fnun 5911	. . . . . . . . . . . . 13 ⊢ (((((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1))) ∧ ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁))) ∧ ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅) → (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))))
104	92, 102, 103	sylancr 694	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))))
105		elfzuz 12209	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ∈ (ℤ_≥‘1))
106	16, 105	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
107	57, 106	eqeltrd 2688	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑀 − 1) + 1) ∈ (ℤ_≥‘1))
108		peano2zm 11297	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
109		uzid 11578	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑀 − 1) ∈ ℤ → (𝑀 − 1) ∈ (ℤ_≥‘(𝑀 − 1)))
110		peano2uz 11617	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑀 − 1) ∈ (ℤ_≥‘(𝑀 − 1)) → ((𝑀 − 1) + 1) ∈ (ℤ_≥‘(𝑀 − 1)))
111	40, 108, 109, 110	4syl 19	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑀 − 1) + 1) ∈ (ℤ_≥‘(𝑀 − 1)))
112	57, 111	eqeltrrd 2689	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘(𝑀 − 1)))
113		peano2uz 11617	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 ∈ (ℤ_≥‘(𝑀 − 1)) → (𝑀 + 1) ∈ (ℤ_≥‘(𝑀 − 1)))
114		uzss 11584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑀 + 1) ∈ (ℤ_≥‘(𝑀 − 1)) → (ℤ_≥‘(𝑀 + 1)) ⊆ (ℤ_≥‘(𝑀 − 1)))
115	112, 113, 114	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℤ_≥‘(𝑀 + 1)) ⊆ (ℤ_≥‘(𝑀 − 1)))
116		elfzuz3 12210	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ (1...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ_≥‘𝑀))
117		eluzp1p1 11589	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘𝑀) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑀 + 1)))
118	16, 116, 117	3syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑀 + 1)))
119	7, 118	eqeltrrd 2689	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑀 + 1)))
120	115, 119	sseldd 3569	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑀 − 1)))
121		fzsplit2 12237	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑀 − 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘(𝑀 − 1))) → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
122	107, 120, 121	syl2anc 691	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)))
123	57	oveq1d 6564	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((𝑀 − 1) + 1)...𝑁) = (𝑀...𝑁))
124	123	uneq2d 3729	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
125	122, 124	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))
126	125	imaeq2d 5385	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))))
127		imaundi 5464	. . . . . . . . . . . . . . 15 ⊢ (𝑈 “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) = ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁)))
128	126, 127	syl6eq 2660	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑈 “ (1...𝑁)) = ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))))
129		f1ofo 6057	. . . . . . . . . . . . . . 15 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)–onto→(1...𝑁))
130		foima 6033	. . . . . . . . . . . . . . 15 ⊢ (𝑈:(1...𝑁)–onto→(1...𝑁) → (𝑈 “ (1...𝑁)) = (1...𝑁))
131	1, 129, 130	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑈 “ (1...𝑁)) = (1...𝑁))
132	128, 131	eqtr3d 2646	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))) = (1...𝑁))
133	132	fneq2d 5896	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))) ↔ (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn (1...𝑁)))
134	104, 133	mpbid 221	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))
135	134	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))
136		ovex 6577	. . . . . . . . . . 11 ⊢ (1...𝑁) ∈ V
137	136	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (1...𝑁) ∈ V)
138		inidm 3784	. . . . . . . . . 10 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
139		eqidd 2611	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → (𝑇‘𝑛) = (𝑇‘𝑛))
140	102	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅)
141		fzss2 12252	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑀 + 1)) → (𝑀...(𝑀 + 1)) ⊆ (𝑀...𝑁))
142		imass2 5420	. . . . . . . . . . . . . . 15 ⊢ ((𝑀...(𝑀 + 1)) ⊆ (𝑀...𝑁) → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (𝑈 “ (𝑀...𝑁)))
143	119, 141, 142	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (𝑈 “ (𝑀...𝑁)))
144	143	sselda 3568	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → 𝑛 ∈ (𝑈 “ (𝑀...𝑁)))
145		fvun2 6180	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1))) ∧ ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁)) ∧ (((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (𝑀...𝑁)))) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛))
146	88, 91, 145	mp3an12 1406	. . . . . . . . . . . . 13 ⊢ ((((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (𝑀...𝑁))) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛))
147	140, 144, 146	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛))
148	89	fvconst2 6374	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝑈 “ (𝑀...𝑁)) → (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛) = 0)
149	144, 148	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛) = 0)
150	147, 149	eqtrd 2644	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = 0)
151	150	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = 0)
152	85, 135, 137, 137, 138, 139, 151	ofval 6804	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))‘𝑛) = ((𝑇‘𝑛) + 0))
153	28, 152	mpdan 699	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))‘𝑛) = ((𝑇‘𝑛) + 0))
154	30	zcnd 11359	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑇‘𝑛) ∈ ℂ)
155	154	addid1d 10115	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇‘𝑛) + 0) = (𝑇‘𝑛))
156	28, 155	syldan 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑇‘𝑛) + 0) = (𝑇‘𝑛))
157	82, 153, 156	3eqtrd 2648	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹‘(𝑀 − 1))‘𝑛) = (𝑇‘𝑛))
158		breq1 4586	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑀 → (𝑦 < 𝑀 ↔ 𝑀 < 𝑀))
159		oveq1 6556	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑀 → (𝑦 + 1) = (𝑀 + 1))
160	158, 159	ifbieq2d 4061	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑀 → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = if(𝑀 < 𝑀, 𝑦, (𝑀 + 1)))
161	41	ltnrd 10050	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝑀 < 𝑀)
162	161	iffalsed 4047	. . . . . . . . . . . . . 14 ⊢ (𝜑 → if(𝑀 < 𝑀, 𝑦, (𝑀 + 1)) = (𝑀 + 1))
163	160, 162	sylan9eqr 2666	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 = 𝑀) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = (𝑀 + 1))
164	163	csbeq1d 3506	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋(𝑀 + 1) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))
165		ovex 6577	. . . . . . . . . . . . 13 ⊢ (𝑀 + 1) ∈ V
166		oveq2 6557	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = (𝑀 + 1) → (1...𝑗) = (1...(𝑀 + 1)))
167	166	imaeq2d 5385	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑀 + 1) → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...(𝑀 + 1))))
168	167	xpeq1d 5062	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑀 + 1) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑀 + 1))) × {1}))
169		oveq1 6556	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = (𝑀 + 1) → (𝑗 + 1) = ((𝑀 + 1) + 1))
170	169	oveq1d 6564	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = (𝑀 + 1) → ((𝑗 + 1)...𝑁) = (((𝑀 + 1) + 1)...𝑁))
171	170	imaeq2d 5385	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑀 + 1) → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ (((𝑀 + 1) + 1)...𝑁)))
172	171	xpeq1d 5062	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑀 + 1) → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))
173	168, 172	uneq12d 3730	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑀 + 1) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))
174	173	oveq2d 6565	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑀 + 1) → (𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))))
175	165, 174	csbie 3525	. . . . . . . . . . . 12 ⊢ ⦋(𝑀 + 1) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))
176	164, 175	syl6eq 2660	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘_𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))))
177		1eluzge0 11608	. . . . . . . . . . . . 13 ⊢ 1 ∈ (ℤ_≥‘0)
178		fzss1 12251	. . . . . . . . . . . . 13 ⊢ (1 ∈ (ℤ_≥‘0) → (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1)))
179	177, 178	ax-mp 5	. . . . . . . . . . . 12 ⊢ (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))
180	179, 16	sseldi 3566	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (0...(𝑁 − 1)))
181		ovex 6577	. . . . . . . . . . . 12 ⊢ (𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))) ∈ V
182	181	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))) ∈ V)
183	35, 176, 180, 182	fvmptd 6197	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘𝑀) = (𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))))
184	183	fveq1d 6105	. . . . . . . . 9 ⊢ (𝜑 → ((𝐹‘𝑀)‘𝑛) = ((𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))‘𝑛))
185	184	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹‘𝑀)‘𝑛) = ((𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))‘𝑛))
186		fnconstg 6006	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ V → ((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1))))
187	86, 186	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1)))
188		fnconstg 6006	. . . . . . . . . . . . . . 15 ⊢ (0 ∈ V → ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁)))
189	89, 188	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁))
190	187, 189	pm3.2i 470	. . . . . . . . . . . . 13 ⊢ (((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1))) ∧ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁)))
191		imain 5888	. . . . . . . . . . . . . . . 16 ⊢ (Fun ^◡𝑈 → (𝑈 “ ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))))
192	1, 94, 191	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 “ ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))))
193		peano2re 10088	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
194	41, 193	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ)
195	194	ltp1d 10833	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑀 + 1) < ((𝑀 + 1) + 1))
196		fzdisj 12239	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 + 1) < ((𝑀 + 1) + 1) → ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁)) = ∅)
197	195, 196	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁)) = ∅)
198	197	imaeq2d 5385	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 “ ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁))) = (𝑈 “ ∅))
199	192, 198	eqtr3d 2646	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = (𝑈 “ ∅))
200	199, 100	syl6eq 2660	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅)
201		fnun 5911	. . . . . . . . . . . . 13 ⊢ (((((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1))) ∧ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) ∧ ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅) → (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))))
202	190, 200, 201	sylancr 694	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))))
203		fzsplit 12238	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀 + 1) ∈ (1...𝑁) → (1...𝑁) = ((1...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
204	22, 203	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
205	204	imaeq2d 5385	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁))))
206		imaundi 5464	. . . . . . . . . . . . . . 15 ⊢ (𝑈 “ ((1...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁)))
207	205, 206	syl6eq 2660	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑈 “ (1...𝑁)) = ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))))
208	207, 131	eqtr3d 2646	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = (1...𝑁))
209	208	fneq2d 5896	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) ↔ (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁)))
210	202, 209	mpbid 221	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
211	210	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
212	200	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅)
213		fzss1 12251	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (ℤ_≥‘1) → (𝑀...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)))
214		imass2 5420	. . . . . . . . . . . . . . 15 ⊢ ((𝑀...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (𝑈 “ (1...(𝑀 + 1))))
215	106, 213, 214	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (𝑈 “ (1...(𝑀 + 1))))
216	215	sselda 3568	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → 𝑛 ∈ (𝑈 “ (1...(𝑀 + 1))))
217		fvun1 6179	. . . . . . . . . . . . . 14 ⊢ ((((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1))) ∧ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁)) ∧ (((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (1...(𝑀 + 1))))) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛))
218	187, 189, 217	mp3an12 1406	. . . . . . . . . . . . 13 ⊢ ((((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (1...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛))
219	212, 216, 218	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛))
220	86	fvconst2 6374	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (𝑈 “ (1...(𝑀 + 1))) → (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛) = 1)
221	216, 220	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛) = 1)
222	219, 221	eqtrd 2644	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 1)
223	222	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 1)
224	85, 211, 137, 137, 138, 139, 223	ofval 6804	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇‘𝑛) + 1))
225	28, 224	mpdan 699	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑇 ∘_𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇‘𝑛) + 1))
226	185, 225	eqtrd 2644	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹‘𝑀)‘𝑛) = ((𝑇‘𝑛) + 1))
227	34, 157, 226	3netr4d 2859	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛))
228	227	ralrimiva 2949	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛))
229		fzpr 12266	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)})
230	16, 39, 229	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)})
231	230	imaeq2d 5385	. . . . . . 7 ⊢ (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) = (𝑈 “ {𝑀, (𝑀 + 1)}))
232		f1ofn 6051	. . . . . . . . 9 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈 Fn (1...𝑁))
233	1, 232	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 Fn (1...𝑁))
234		fnimapr 6172	. . . . . . . 8 ⊢ ((𝑈 Fn (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ (𝑀 + 1) ∈ (1...𝑁)) → (𝑈 “ {𝑀, (𝑀 + 1)}) = {(𝑈‘𝑀), (𝑈‘(𝑀 + 1))})
235	233, 17, 22, 234	syl3anc 1318	. . . . . . 7 ⊢ (𝜑 → (𝑈 “ {𝑀, (𝑀 + 1)}) = {(𝑈‘𝑀), (𝑈‘(𝑀 + 1))})
236	231, 235	eqtrd 2644	. . . . . 6 ⊢ (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) = {(𝑈‘𝑀), (𝑈‘(𝑀 + 1))})
237	236	raleqdv 3121	. . . . 5 ⊢ (𝜑 → (∀𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ ∀𝑛 ∈ {(𝑈‘𝑀), (𝑈‘(𝑀 + 1))} ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛)))
238	228, 237	mpbid 221	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ {(𝑈‘𝑀), (𝑈‘(𝑀 + 1))} ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛))
239		fvex 6113	. . . . 5 ⊢ (𝑈‘𝑀) ∈ V
240		fvex 6113	. . . . 5 ⊢ (𝑈‘(𝑀 + 1)) ∈ V
241		fveq2 6103	. . . . . 6 ⊢ (𝑛 = (𝑈‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)))
242		fveq2 6103	. . . . . 6 ⊢ (𝑛 = (𝑈‘𝑀) → ((𝐹‘𝑀)‘𝑛) = ((𝐹‘𝑀)‘(𝑈‘𝑀)))
243	241, 242	neeq12d 2843	. . . . 5 ⊢ (𝑛 = (𝑈‘𝑀) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ ((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀))))
244		fveq2 6103	. . . . . 6 ⊢ (𝑛 = (𝑈‘(𝑀 + 1)) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))))
245		fveq2 6103	. . . . . 6 ⊢ (𝑛 = (𝑈‘(𝑀 + 1)) → ((𝐹‘𝑀)‘𝑛) = ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1))))
246	244, 245	neeq12d 2843	. . . . 5 ⊢ (𝑛 = (𝑈‘(𝑀 + 1)) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1)))))
247	239, 240, 243, 246	ralpr 4185	. . . 4 ⊢ (∀𝑛 ∈ {(𝑈‘𝑀), (𝑈‘(𝑀 + 1))} ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ (((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1)))))
248	238, 247	sylib 207	. . 3 ⊢ (𝜑 → (((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1)))))
249	41	ltp1d 10833	. . . . 5 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
250	41, 249	ltned 10052	. . . 4 ⊢ (𝜑 → 𝑀 ≠ (𝑀 + 1))
251		f1of1 6049	. . . . . . 7 ⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)–1-1→(1...𝑁))
252	1, 251	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈:(1...𝑁)–1-1→(1...𝑁))
253		f1veqaeq 6418	. . . . . 6 ⊢ ((𝑈:(1...𝑁)–1-1→(1...𝑁) ∧ (𝑀 ∈ (1...𝑁) ∧ (𝑀 + 1) ∈ (1...𝑁))) → ((𝑈‘𝑀) = (𝑈‘(𝑀 + 1)) → 𝑀 = (𝑀 + 1)))
254	252, 17, 22, 253	syl12anc 1316	. . . . 5 ⊢ (𝜑 → ((𝑈‘𝑀) = (𝑈‘(𝑀 + 1)) → 𝑀 = (𝑀 + 1)))
255	254	necon3d 2803	. . . 4 ⊢ (𝜑 → (𝑀 ≠ (𝑀 + 1) → (𝑈‘𝑀) ≠ (𝑈‘(𝑀 + 1))))
256	250, 255	mpd 15	. . 3 ⊢ (𝜑 → (𝑈‘𝑀) ≠ (𝑈‘(𝑀 + 1)))
257	243	anbi1d 737	. . . . 5 ⊢ (𝑛 = (𝑈‘𝑀) → ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ↔ (((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚))))
258		neeq1 2844	. . . . 5 ⊢ (𝑛 = (𝑈‘𝑀) → (𝑛 ≠ 𝑚 ↔ (𝑈‘𝑀) ≠ 𝑚))
259	257, 258	anbi12d 743	. . . 4 ⊢ (𝑛 = (𝑈‘𝑀) → (((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚) ↔ ((((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ (𝑈‘𝑀) ≠ 𝑚)))
260		fveq2 6103	. . . . . . 7 ⊢ (𝑚 = (𝑈‘(𝑀 + 1)) → ((𝐹‘(𝑀 − 1))‘𝑚) = ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))))
261		fveq2 6103	. . . . . . 7 ⊢ (𝑚 = (𝑈‘(𝑀 + 1)) → ((𝐹‘𝑀)‘𝑚) = ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1))))
262	260, 261	neeq12d 2843	. . . . . 6 ⊢ (𝑚 = (𝑈‘(𝑀 + 1)) → (((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚) ↔ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1)))))
263	262	anbi2d 736	. . . . 5 ⊢ (𝑚 = (𝑈‘(𝑀 + 1)) → ((((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ↔ (((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1))))))
264		neeq2 2845	. . . . 5 ⊢ (𝑚 = (𝑈‘(𝑀 + 1)) → ((𝑈‘𝑀) ≠ 𝑚 ↔ (𝑈‘𝑀) ≠ (𝑈‘(𝑀 + 1))))
265	263, 264	anbi12d 743	. . . 4 ⊢ (𝑚 = (𝑈‘(𝑀 + 1)) → (((((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ (𝑈‘𝑀) ≠ 𝑚) ↔ ((((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1)))) ∧ (𝑈‘𝑀) ≠ (𝑈‘(𝑀 + 1)))))
266	259, 265	rspc2ev 3295	. . 3 ⊢ (((𝑈‘𝑀) ∈ (1...𝑁) ∧ (𝑈‘(𝑀 + 1)) ∈ (1...𝑁) ∧ ((((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1)))) ∧ (𝑈‘𝑀) ≠ (𝑈‘(𝑀 + 1)))) → ∃𝑛 ∈ (1...𝑁)∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚))
267	18, 23, 248, 256, 266	syl112anc 1322	. 2 ⊢ (𝜑 → ∃𝑛 ∈ (1...𝑁)∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚))
268		dfrex2 2979	. . 3 ⊢ (∃𝑛 ∈ (1...𝑁)∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚) ↔ ¬ ∀𝑛 ∈ (1...𝑁) ¬ ∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚))
269		fveq2 6103	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘𝑚))
270		fveq2 6103	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑀)‘𝑛) = ((𝐹‘𝑀)‘𝑚))
271	269, 270	neeq12d 2843	. . . . 5 ⊢ (𝑛 = 𝑚 → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)))
272	271	rmo4 3366	. . . 4 ⊢ (∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)∀𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) → 𝑛 = 𝑚))
273		dfral2 2977	. . . . . 6 ⊢ (∀𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) → 𝑛 = 𝑚) ↔ ¬ ∃𝑚 ∈ (1...𝑁) ¬ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) → 𝑛 = 𝑚))
274		df-ne 2782	. . . . . . . . 9 ⊢ (𝑛 ≠ 𝑚 ↔ ¬ 𝑛 = 𝑚)
275	274	anbi2i 726	. . . . . . . 8 ⊢ (((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚) ↔ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ ¬ 𝑛 = 𝑚))
276		annim 440	. . . . . . . 8 ⊢ (((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ ¬ 𝑛 = 𝑚) ↔ ¬ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) → 𝑛 = 𝑚))
277	275, 276	bitri 263	. . . . . . 7 ⊢ (((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚) ↔ ¬ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) → 𝑛 = 𝑚))
278	277	rexbii 3023	. . . . . 6 ⊢ (∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚) ↔ ∃𝑚 ∈ (1...𝑁) ¬ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) → 𝑛 = 𝑚))
279	273, 278	xchbinxr 324	. . . . 5 ⊢ (∀𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) → 𝑛 = 𝑚) ↔ ¬ ∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚))
280	279	ralbii 2963	. . . 4 ⊢ (∀𝑛 ∈ (1...𝑁)∀𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) → 𝑛 = 𝑚) ↔ ∀𝑛 ∈ (1...𝑁) ¬ ∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚))
281	272, 280	bitri 263	. . 3 ⊢ (∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁) ¬ ∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚))
282	268, 281	xchbinxr 324	. 2 ⊢ (∃𝑛 ∈ (1...𝑁)∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚) ↔ ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛))
283	267, 282	sylib 207	1 ⊢ (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ∃*wrmo 2899 Vcvv 3173 ⦋csb 3499 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ifcif 4036 {csn 4125 {cpr 4127 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ^◡ccnv 5037 ran crn 5039 “ cima 5041 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 –1-1→wf1 5801 –onto→wfo 5802 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ∘_𝑓 cof 6793 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 − cmin 10145 ℕcn 10897 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197

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-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

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 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-iun 4457 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-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-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-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198

This theorem is referenced by: poimirlem8 32587 poimirlem18 32597 poimirlem21 32600 poimirlem22 32601

W3C validator