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Theorem vdwlem11 16956

Description: Lemma for vdw 16959. (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 14312	. . . . 5 ⊢ (𝑅 ∈ Fin → (♯‘𝑅) ∈ ℕ₀)
5	1, 4	syl 17	. . . 4 ⊢ (𝜑 → (♯‘𝑅) ∈ ℕ₀)
6		nn0p1nn 12470	. . . 4 ⊢ ((♯‘𝑅) ∈ ℕ₀ → ((♯‘𝑅) + 1) ∈ ℕ)
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ)
8	1, 2, 3, 7	vdwlem10 16955	. 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 7394	. . . . . . 7 ⊢ (1...𝑛) ∈ V
11		elmapg 8780	. . . . . . 7 ⊢ ((𝑅 ∈ Fin ∧ (1...𝑛) ∈ V) → (𝑓 ∈ (𝑅 ↑_m (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅))
12	9, 10, 11	sylancl 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑓 ∈ (𝑅 ↑_m (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅))
13	12	biimpa 476	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑_m (1...𝑛))) → 𝑓:(1...𝑛)⟶𝑅)
14	5	nn0red 12493	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑅) ∈ ℝ)
15	14	ltp1d 12080	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑅) < ((♯‘𝑅) + 1))
16		peano2re 11313	. . . . . . . . . . . 12 ⊢ ((♯‘𝑅) ∈ ℝ → ((♯‘𝑅) + 1) ∈ ℝ)
17	14, 16	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘𝑅) + 1) ∈ ℝ)
18	14, 17	ltnled 11287	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝑅) < ((♯‘𝑅) + 1) ↔ ¬ ((♯‘𝑅) + 1) ≤ (♯‘𝑅)))
19	15, 18	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → ¬ ((♯‘𝑅) + 1) ≤ (♯‘𝑅))
20	19	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ ((♯‘𝑅) + 1) ≤ (♯‘𝑅))
21		eluz2nn 12832	. . . . . . . . . . . . 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 12492	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝐾 ∈ ℕ₀)
25		simprr 773	. . . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (1...((♯‘𝑅) + 1)) = (1...((♯‘𝑅) + 1))
28	10, 24, 25, 26, 27	vdwpc 16945	. . . . . . . . 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 731	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . 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 6042	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ⊆ dom 𝑓
34	32, 33	sstrdi 3935	. . . . . . . . . . . . . . . . . . . . . 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 6681	. . . . . . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → 𝐾 ∈ ℕ)
37		simplrl 777	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → 𝑎 ∈ ℕ)
38		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))))) → 𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))))
39		nnex 12174	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ℕ ∈ V
40		ovex 7394	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (1...((♯‘𝑅) + 1)) ∈ V
41	39, 40	elmap 8813	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))) ↔ 𝑑:(1...((♯‘𝑅) + 1))⟶ℕ)
42	38, 41	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))))) → 𝑑:(1...((♯‘𝑅) + 1))⟶ℕ)
43	42	ffvelcdmda 7031	. . . . . . . . . . . . . . . . . . . . . . . 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 12223	. . . . . . . . . . . . . . . . . . . . . . 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 16940	. . . . . . . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . . . . . . 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 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)))
48	35, 47	sseldd 3923	. . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . 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 7028	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(1...𝑛)⟶𝑅 ∧ (𝑎 + (𝑑‘𝑖)) ∈ (1...𝑛)) → (𝑓‘(𝑎 + (𝑑‘𝑖))) ∈ 𝑅)
51	30, 49, 50	syl6an 685	. . . . . . . . . . . . . . . . . 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 3150	. . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) = (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))
55	54	fmpt 7057	. . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . 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 6671	. . . . . . . . . . . . . 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 8941	. . . . . . . . . . . . . 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 9173	. . . . . . . . . . . . . 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 14335	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 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 257	. . . . . . . . . . . 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 5089	. . . . . . . . . . . 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 245	. . . . . . . . . . 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 453	. . . . . . . . . 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 3195	. . . . . . . . 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 240	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → (⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 → ((♯‘𝑅) + 1) ≤ (♯‘𝑅)))
69	20, 68	mtod 198	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ ⟨((♯‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓)
70		biorf 937	. . . . . . 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 467	. . . . 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 592	. . . 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 3159	. . 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 3160	. 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 257	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 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 Vcvv 3430 ⊆ wss 3890 {csn 4568 ⟨cop 4574 class class class wbr 5086 ↦ cmpt 5167 ^◡ccnv 5624 dom cdm 5625 ran crn 5626 “ cima 5628 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 ↑_m cmap 8767 ≼ cdom 8885 Fincfn 8887 ℝcr 11031 1c1 11033 + caddc 11035 < clt 11173 ≤ cle 11174 ℕcn 12168 2c2 12230 ℕ₀cn0 12431 ℤ_≥cuz 12782 ...cfz 13455 ♯chash 14286 APcvdwa 16930 MonoAP cvdwm 16931 PolyAP cvdwp 16932

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-oadd 8403 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-dju 9819 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-n0 12432 df-xnn0 12505 df-z 12519 df-uz 12783 df-rp 12937 df-fz 13456 df-hash 14287 df-vdwap 16933 df-vdwmc 16934 df-vdwpc 16935

This theorem is referenced by: vdwlem13 16958

W3C validator