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Theorem vdwlem11 15533

Description: Lemma for vdw 15536. (Contributed by Mario Carneiro, 18-Aug-2014.)

**Hypotheses**
Ref	Expression
vdw.r	⊢ (𝜑 → 𝑅 ∈ Fin)
vdwlem9.k	⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
vdwlem9.s	⊢ (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓)

**Assertion**
Ref	Expression
vdwlem11	⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(𝐾 + 1) MonoAP 𝑓)

Distinct variable groups: 𝜑,𝑛,𝑓 𝑓,𝑠,𝐾,𝑛 𝑅,𝑓,𝑛,𝑠 𝜑,𝑓 Allowed substitution hint: 𝜑(𝑠)

Proof of Theorem vdwlem11 Dummy variables 𝑎 𝑑 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vdw.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Fin)
2		vdwlem9.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
3		vdwlem9.s	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑_𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
4		hashcl 13009	. . . . 5 ⊢ (𝑅 ∈ Fin → (#‘𝑅) ∈ ℕ₀)
5	1, 4	syl 17	. . . 4 ⊢ (𝜑 → (#‘𝑅) ∈ ℕ₀)
6		nn0p1nn 11209	. . . 4 ⊢ ((#‘𝑅) ∈ ℕ₀ → ((#‘𝑅) + 1) ∈ ℕ)
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → ((#‘𝑅) + 1) ∈ ℕ)
8	1, 2, 3, 7	vdwlem10 15532	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
9	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ Fin)
10		ovex 6577	. . . . . . 7 ⊢ (1...𝑛) ∈ V
11		elmapg 7757	. . . . . . 7 ⊢ ((𝑅 ∈ Fin ∧ (1...𝑛) ∈ V) → (𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅))
12	9, 10, 11	sylancl 693	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅))
13	12	biimpa 500	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))) → 𝑓:(1...𝑛)⟶𝑅)
14	5	nn0red 11229	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝑅) ∈ ℝ)
15	14	ltp1d 10833	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝑅) < ((#‘𝑅) + 1))
16		peano2re 10088	. . . . . . . . . . . 12 ⊢ ((#‘𝑅) ∈ ℝ → ((#‘𝑅) + 1) ∈ ℝ)
17	14, 16	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((#‘𝑅) + 1) ∈ ℝ)
18	14, 17	ltnled 10063	. . . . . . . . . 10 ⊢ (𝜑 → ((#‘𝑅) < ((#‘𝑅) + 1) ↔ ¬ ((#‘𝑅) + 1) ≤ (#‘𝑅)))
19	15, 18	mpbid 221	. . . . . . . . 9 ⊢ (𝜑 → ¬ ((#‘𝑅) + 1) ≤ (#‘𝑅))
20	19	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ ((#‘𝑅) + 1) ≤ (#‘𝑅))
21		eluz2nn 11602	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (ℤ_≥‘2) → 𝐾 ∈ ℕ)
22	2, 21	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ ℕ)
23	22	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝐾 ∈ ℕ)
24	23	nnnn0d 11228	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝐾 ∈ ℕ₀)
25		simprr 792	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝑓:(1...𝑛)⟶𝑅)
26	7	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ((#‘𝑅) + 1) ∈ ℕ)
27		eqid 2610	. . . . . . . . . 10 ⊢ (1...((#‘𝑅) + 1)) = (1...((#‘𝑅) + 1))
28	10, 24, 25, 26, 27	vdwpc 15522	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → (⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1)))(∀𝑖 ∈ (1...((#‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = ((#‘𝑅) + 1))))
29	1	ad3antrrr 762	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((#‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → 𝑅 ∈ Fin)
30	25	ad2antrr 758	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) → 𝑓:(1...𝑛)⟶𝑅)
31		simpr 476	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) ∧ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}))
32		cnvimass 5404	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ⊆ dom 𝑓
33	31, 32	syl6ss 3580	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) ∧ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ dom 𝑓)
34	25	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) ∧ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → 𝑓:(1...𝑛)⟶𝑅)
35		fdm 5964	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓:(1...𝑛)⟶𝑅 → dom 𝑓 = (1...𝑛))
36	34, 35	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) ∧ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → dom 𝑓 = (1...𝑛))
37	33, 36	sseqtrd 3604	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) ∧ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (1...𝑛))
38	22	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) → 𝐾 ∈ ℕ)
39		simplrl 796	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) → 𝑎 ∈ ℕ)
40		simprr 792	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) → 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))
41		nnex 10903	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ℕ ∈ V
42		ovex 6577	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (1...((#‘𝑅) + 1)) ∈ V
43	41, 42	elmap 7772	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))) ↔ 𝑑:(1...((#‘𝑅) + 1))⟶ℕ)
44	40, 43	sylib 207	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) → 𝑑:(1...((#‘𝑅) + 1))⟶ℕ)
45	44	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) → (𝑑‘𝑖) ∈ ℕ)
46	39, 45	nnaddcld 10944	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) → (𝑎 + (𝑑‘𝑖)) ∈ ℕ)
47		vdwapid1 15517	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐾 ∈ ℕ ∧ (𝑎 + (𝑑‘𝑖)) ∈ ℕ ∧ (𝑑‘𝑖) ∈ ℕ) → (𝑎 + (𝑑‘𝑖)) ∈ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)))
48	38, 46, 45, 47	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) → (𝑎 + (𝑑‘𝑖)) ∈ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)))
49	48	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) ∧ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → (𝑎 + (𝑑‘𝑖)) ∈ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)))
50	37, 49	sseldd 3569	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) ∧ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → (𝑎 + (𝑑‘𝑖)) ∈ (1...𝑛))
51	50	ex 449	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) → (((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) → (𝑎 + (𝑑‘𝑖)) ∈ (1...𝑛)))
52		ffvelrn 6265	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(1...𝑛)⟶𝑅 ∧ (𝑎 + (𝑑‘𝑖)) ∈ (1...𝑛)) → (𝑓‘(𝑎 + (𝑑‘𝑖))) ∈ 𝑅)
53	30, 51, 52	syl6an 566	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) → (((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) → (𝑓‘(𝑎 + (𝑑‘𝑖))) ∈ 𝑅))
54	53	ralimdva 2945	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) → (∀𝑖 ∈ (1...((#‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) → ∀𝑖 ∈ (1...((#‘𝑅) + 1))(𝑓‘(𝑎 + (𝑑‘𝑖))) ∈ 𝑅))
55	54	imp 444	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((#‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ∀𝑖 ∈ (1...((#‘𝑅) + 1))(𝑓‘(𝑎 + (𝑑‘𝑖))) ∈ 𝑅)
56		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) = (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))
57	56	fmpt 6289	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑖 ∈ (1...((#‘𝑅) + 1))(𝑓‘(𝑎 + (𝑑‘𝑖))) ∈ 𝑅 ↔ (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))):(1...((#‘𝑅) + 1))⟶𝑅)
58	55, 57	sylib 207	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((#‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))):(1...((#‘𝑅) + 1))⟶𝑅)
59		frn 5966	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))):(1...((#‘𝑅) + 1))⟶𝑅 → ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ⊆ 𝑅)
60	58, 59	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((#‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ⊆ 𝑅)
61		ssdomg 7887	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Fin → (ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ⊆ 𝑅 → ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ≼ 𝑅))
62	29, 60, 61	sylc 63	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((#‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ≼ 𝑅)
63		ssfi 8065	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Fin ∧ ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ⊆ 𝑅) → ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ∈ Fin)
64	29, 60, 63	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((#‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ∈ Fin)
65		hashdom 13029	. . . . . . . . . . . . . 14 ⊢ ((ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ∈ Fin ∧ 𝑅 ∈ Fin) → ((#‘ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) ≤ (#‘𝑅) ↔ ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ≼ 𝑅))
66	64, 29, 65	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((#‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ((#‘ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) ≤ (#‘𝑅) ↔ ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ≼ 𝑅))
67	62, 66	mpbird 246	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((#‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → (#‘ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) ≤ (#‘𝑅))
68		breq1 4586	. . . . . . . . . . . 12 ⊢ ((#‘ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = ((#‘𝑅) + 1) → ((#‘ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) ≤ (#‘𝑅) ↔ ((#‘𝑅) + 1) ≤ (#‘𝑅)))
69	67, 68	syl5ibcom 234	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ ∀𝑖 ∈ (1...((#‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))})) → ((#‘ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = ((#‘𝑅) + 1) → ((#‘𝑅) + 1) ≤ (#‘𝑅)))
70	69	expimpd 627	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) → ((∀𝑖 ∈ (1...((#‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = ((#‘𝑅) + 1)) → ((#‘𝑅) + 1) ≤ (#‘𝑅)))
71	70	rexlimdvva 3020	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1)))(∀𝑖 ∈ (1...((#‘𝑅) + 1))((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))) = ((#‘𝑅) + 1)) → ((#‘𝑅) + 1) ≤ (#‘𝑅)))
72	28, 71	sylbid 229	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → (⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 → ((#‘𝑅) + 1) ≤ (#‘𝑅)))
73	20, 72	mtod 188	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ ⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓)
74		biorf 419	. . . . . . 7 ⊢ (¬ ⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
75	73, 74	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
76	75	anassrs 678	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑓:(1...𝑛)⟶𝑅) → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
77	13, 76	syldan 486	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))) → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
78	77	ralbidva 2968	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(𝐾 + 1) MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
79	78	rexbidva 3031	. 2 ⊢ (𝜑 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(𝐾 + 1) MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
80	8, 79	mpbird 246	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))(𝐾 + 1) MonoAP 𝑓)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 {csn 4125 ⟨cop 4131 class class class wbr 4583 ↦ cmpt 4643 ^◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑_𝑚 cmap 7744 ≼ cdom 7839 Fincfn 7841 ℝcr 9814 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 ℕcn 10897 2c2 10947 ℕ₀cn0 11169 ℤ_≥cuz 11563 ...cfz 12197 #chash 12979 APcvdwa 15507 MonoAP cvdwm 15508 PolyAP cvdwp 15509

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-int 4411 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-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-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-nn 10898 df-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-hash 12980 df-vdwap 15510 df-vdwmc 15511 df-vdwpc 15512

This theorem is referenced by: vdwlem13 15535

W3C validator