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Theorem vdwlem11 16863

Description: Lemma for vdw 16866. (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 14256	. . . . 5 ⊢ (𝑅 ∈ Fin → (♯‘𝑅) ∈ ℕ₀)
5	1, 4	syl 17	. . . 4 ⊢ (𝜑 → (♯‘𝑅) ∈ ℕ₀)
6		nn0p1nn 12452	. . . 4 ⊢ ((♯‘𝑅) ∈ ℕ₀ → ((♯‘𝑅) + 1) ∈ ℕ)
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ)
8	1, 2, 3, 7	vdwlem10 16862	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑_m (1...𝑛))(⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓))
9	1	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ Fin)
10		ovex 7390	. . . . . . 7 ⊢ (1...𝑛) ∈ V
11		elmapg 8778	. . . . . . 7 ⊢ ((𝑅 ∈ Fin ∧ (1...𝑛) ∈ V) → (𝑓 ∈ (𝑅 ↑_m (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅))
12	9, 10, 11	sylancl 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑓 ∈ (𝑅 ↑_m (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅))
13	12	biimpa 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑_m (1...𝑛))) → 𝑓:(1...𝑛)⟶𝑅)
14	5	nn0red 12474	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑅) ∈ ℝ)
15	14	ltp1d 12085	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑅) < ((♯‘𝑅) + 1))
16		peano2re 11328	. . . . . . . . . . . 12 ⊢ ((♯‘𝑅) ∈ ℝ → ((♯‘𝑅) + 1) ∈ ℝ)
17	14, 16	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘𝑅) + 1) ∈ ℝ)
18	14, 17	ltnled 11302	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝑅) < ((♯‘𝑅) + 1) ↔ ¬ ((♯‘𝑅) + 1) ≤ (♯‘𝑅)))
19	15, 18	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → ¬ ((♯‘𝑅) + 1) ≤ (♯‘𝑅))
20	19	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ ((♯‘𝑅) + 1) ≤ (♯‘𝑅))
21		eluz2nn 12809	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (ℤ_≥‘2) → 𝐾 ∈ ℕ)
22	2, 21	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ ℕ)
23	22	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝐾 ∈ ℕ)
24	23	nnnn0d 12473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝐾 ∈ ℕ₀)
25		simprr 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝑓:(1...𝑛)⟶𝑅)
26	7	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ((♯‘𝑅) + 1) ∈ ℕ)
27		eqid 2736	. . . . . . . . . 10 ⊢ (1...((♯‘𝑅) + 1)) = (1...((♯‘𝑅) + 1))
28	10, 24, 25, 26, 27	vdwpc 16852	. . . . . . . . 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 728	. . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . 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	25	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . 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)))})) → 𝑓:(1...𝑛)⟶𝑅)
32		simpr 485	. . . . . . . . . . . . . . . . . . . . . . 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)))}))
33		cnvimass 6033	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ⊆ dom 𝑓
34	32, 33	sstrdi 3956	. . . . . . . . . . . . . . . . . . . . . 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)
35	31, 34	fssdmd 6687	. . . . . . . . . . . . . . . . . . . . 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))
36	22	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → 𝐾 ∈ ℕ)
37		simplrl 775	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → 𝑎 ∈ ℕ)
38		simprr 771	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))))) → 𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))))
39		nnex 12159	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ℕ ∈ V
40		ovex 7390	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (1...((♯‘𝑅) + 1)) ∈ V
41	39, 40	elmap 8809	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))) ↔ 𝑑:(1...((♯‘𝑅) + 1))⟶ℕ)
42	38, 41	sylib 217	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))))) → 𝑑:(1...((♯‘𝑅) + 1))⟶ℕ)
43	42	ffvelcdmda 7035	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → (𝑑‘𝑖) ∈ ℕ)
44	37, 43	nnaddcld 12205	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → (𝑎 + (𝑑‘𝑖)) ∈ ℕ)
45		vdwapid1 16847	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐾 ∈ ℕ ∧ (𝑎 + (𝑑‘𝑖)) ∈ ℕ ∧ (𝑑‘𝑖) ∈ ℕ) → (𝑎 + (𝑑‘𝑖)) ∈ ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)))
46	36, 44, 43, 45	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . 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)))
47	46	adantr 481	. . . . . . . . . . . . . . . . . . . . 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)))
48	35, 47	sseldd 3945	. . . . . . . . . . . . . . . . . . . 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))
49	48	ex 413	. . . . . . . . . . . . . . . . . . 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)))
50		ffvelcdm 7032	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(1...𝑛)⟶𝑅 ∧ (𝑎 + (𝑑‘𝑖)) ∈ (1...𝑛)) → (𝑓‘(𝑎 + (𝑑‘𝑖))) ∈ 𝑅)
51	30, 49, 50	syl6an 682	. . . . . . . . . . . . . . . . . 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))
52	51	ralimdva 3164	. . . . . . . . . . . . . . . . 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))
53	52	imp 407	. . . . . . . . . . . . . . . 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)
54		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) = (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))
55	54	fmpt 7058	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑖 ∈ (1...((♯‘𝑅) + 1))(𝑓‘(𝑎 + (𝑑‘𝑖))) ∈ 𝑅 ↔ (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))):(1...((♯‘𝑅) + 1))⟶𝑅)
56	53, 55	sylib 217	. . . . . . . . . . . . . . 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)
57	56	frnd 6676	. . . . . . . . . . . . . 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)
58		ssdomg 8940	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Fin → (ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ⊆ 𝑅 → ran (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) ≼ 𝑅))
59	29, 57, 58	sylc 65	. . . . . . . . . . . . 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)
60	29, 57	ssfid 9211	. . . . . . . . . . . . . 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)
61		hashdom 14279	. . . . . . . . . . . . . 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))
62	60, 29, 61	syl2anc 584	. . . . . . . . . . . . 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))
63	59, 62	mpbird 256	. . . . . . . . . . . 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))
64		breq1 5108	. . . . . . . . . . . 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)))
65	63, 64	syl5ibcom 244	. . . . . . . . . . 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)))
66	65	expimpd 454	. . . . . . . . . 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)))
67	66	rexlimdvva 3205	. . . . . . . . 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)))
68	28, 67	sylbid 239	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 → ((♯‘𝑅) + 1) ≤ (♯‘𝑅)))
69	20, 68	mtod 197	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ ⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓)
70		biorf 935	. . . . . . 7 ⊢ (¬ ⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
71	69, 70	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
72	71	anassrs 468	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑓:(1...𝑛)⟶𝑅) → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
73	13, 72	syldan 591	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑_m (1...𝑛))) → ((𝐾 + 1) MonoAP 𝑓 ↔ (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 ∨ (𝐾 + 1) MonoAP 𝑓)))
74	73	ralbidva 3172	. . 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)))
75	74	rexbidva 3173	. 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)))
76	8, 75	mpbird 256	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 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 Vcvv 3445 ⊆ wss 3910 {csn 4586 ⟨cop 4592 class class class wbr 5105 ↦ cmpt 5188 ^◡ccnv 5632 dom cdm 5633 ran crn 5634 “ cima 5636 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ↑_m cmap 8765 ≼ cdom 8881 Fincfn 8883 ℝcr 11050 1c1 11052 + caddc 11054 < clt 11189 ≤ cle 11190 ℕcn 12153 2c2 12208 ℕ₀cn0 12413 ℤ_≥cuz 12763 ...cfz 13424 ♯chash 14230 APcvdwa 16837 MonoAP cvdwm 16838 PolyAP cvdwp 16839

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-oadd 8416 df-er 8648 df-map 8767 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-rp 12916 df-fz 13425 df-hash 14231 df-vdwap 16840 df-vdwmc 16841 df-vdwpc 16842

This theorem is referenced by: vdwlem13 16865

W3C validator