vdwlem11 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > vdwlem11		Structured version Visualization version GIF version

Theorem vdwlem11 15695

Description: Lemma for vdw 15698. (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 13147	. . . . 5 ⊢ (𝑅 ∈ Fin → (#‘𝑅) ∈ ℕ₀)
5	1, 4	syl 17	. . . 4 ⊢ (𝜑 → (#‘𝑅) ∈ ℕ₀)
6		nn0p1nn 11332	. . . 4 ⊢ ((#‘𝑅) ∈ ℕ₀ → ((#‘𝑅) + 1) ∈ ℕ)
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → ((#‘𝑅) + 1) ∈ ℕ)
8	1, 2, 3, 7	vdwlem10 15694	. 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 6678	. . . . . . 7 ⊢ (1...𝑛) ∈ V
11		elmapg 7870	. . . . . . 7 ⊢ ((𝑅 ∈ Fin ∧ (1...𝑛) ∈ V) → (𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅))
12	9, 10, 11	sylancl 694	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛)) ↔ 𝑓:(1...𝑛)⟶𝑅))
13	12	biimpa 501	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑_𝑚 (1...𝑛))) → 𝑓:(1...𝑛)⟶𝑅)
14	5	nn0red 11352	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝑅) ∈ ℝ)
15	14	ltp1d 10954	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝑅) < ((#‘𝑅) + 1))
16		peano2re 10209	. . . . . . . . . . . 12 ⊢ ((#‘𝑅) ∈ ℝ → ((#‘𝑅) + 1) ∈ ℝ)
17	14, 16	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((#‘𝑅) + 1) ∈ ℝ)
18	14, 17	ltnled 10184	. . . . . . . . . 10 ⊢ (𝜑 → ((#‘𝑅) < ((#‘𝑅) + 1) ↔ ¬ ((#‘𝑅) + 1) ≤ (#‘𝑅)))
19	15, 18	mpbid 222	. . . . . . . . 9 ⊢ (𝜑 → ¬ ((#‘𝑅) + 1) ≤ (#‘𝑅))
20	19	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ ((#‘𝑅) + 1) ≤ (#‘𝑅))
21		eluz2nn 11726	. . . . . . . . . . . . 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 11351	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝐾 ∈ ℕ₀)
25		simprr 796	. . . . . . . . . 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 2622	. . . . . . . . . 10 ⊢ (1...((#‘𝑅) + 1)) = (1...((#‘𝑅) + 1))
28	10, 24, 25, 26, 27	vdwpc 15684	. . . . . . . . 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 766	. . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . . . . . . . . . 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 5485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ⊆ dom 𝑓
33	31, 32	syl6ss 3615	. . . . . . . . . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . . . . . . . . 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 6051	. . . . . . . . . . . . . . . . . . . . . . 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 3641	. . . . . . . . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) → 𝐾 ∈ ℕ)
39		simplrl 800	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((#‘𝑅) + 1))) → 𝑎 ∈ ℕ)
40		simprr 796	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) → 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))
41		nnex 11026	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ℕ ∈ V
42		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (1...((#‘𝑅) + 1)) ∈ V
43	41, 42	elmap 7886	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))) ↔ 𝑑:(1...((#‘𝑅) + 1))⟶ℕ)
44	40, 43	sylib 208	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...((#‘𝑅) + 1))))) → 𝑑:(1...((#‘𝑅) + 1))⟶ℕ)
45	44	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . . . . 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 11067	. . . . . . . . . . . . . . . . . . . . . . 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 15679	. . . . . . . . . . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . . . . . . . . . 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 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)))
50	37, 49	sseldd 3604	. . . . . . . . . . . . . . . . . . . 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 450	. . . . . . . . . . . . . . . . . . 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 6357	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(1...𝑛)⟶𝑅 ∧ (𝑎 + (𝑑‘𝑖)) ∈ (1...𝑛)) → (𝑓‘(𝑎 + (𝑑‘𝑖))) ∈ 𝑅)
53	30, 51, 52	syl6an 568	. . . . . . . . . . . . . . . . . 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 2962	. . . . . . . . . . . . . . . . 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 445	. . . . . . . . . . . . . . . 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 2622	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) = (𝑖 ∈ (1...((#‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))
57	56	fmpt 6381	. . . . . . . . . . . . . . . 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 208	. . . . . . . . . . . . . . 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 6053	. . . . . . . . . . . . . . 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 8001	. . . . . . . . . . . . . 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 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)
63		ssfi 8180	. . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . 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 13168	. . . . . . . . . . . . . 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 693	. . . . . . . . . . . . 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 247	. . . . . . . . . . . 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 4656	. . . . . . . . . . . 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 235	. . . . . . . . . . 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 629	. . . . . . . . . 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 3038	. . . . . . . . 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 230	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → (⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓 → ((#‘𝑅) + 1) ≤ (#‘𝑅)))
73	20, 72	mtod 189	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → ¬ ⟨((#‘𝑅) + 1), 𝐾⟩ PolyAP 𝑓)
74		biorf 420	. . . . . . 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 680	. . . . 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 487	. . . 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 2985	. . 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 3049	. 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 247	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 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ⊆ wss 3574 {csn 4177 ⟨cop 4183 class class class wbr 4653 ↦ cmpt 4729 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 “ cima 5117 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑_𝑚 cmap 7857 ≼ cdom 7953 Fincfn 7955 ℝcr 9935 1c1 9937 + caddc 9939 < clt 10074 ≤ cle 10075 ℕcn 11020 2c2 11070 ℕ₀cn0 11292 ℤ_≥cuz 11687 ...cfz 12326 #chash 13117 APcvdwa 15669 MonoAP cvdwm 15670 PolyAP cvdwp 15671

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-hash 13118 df-vdwap 15672 df-vdwmc 15673 df-vdwpc 15674

This theorem is referenced by: vdwlem13 15697

W3C validator