vdwlem11 - Metamath Proof Explorer

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Theorem vdwlem11 16969

Description: Lemma for vdw 16972. (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 14328	. . . . 5 ⊢ (𝑅 ∈ Fin → (♯‘𝑅) ∈ ℕ₀)
5	1, 4	syl 17	. . . 4 ⊢ (𝜑 → (♯‘𝑅) ∈ ℕ₀)
6		nn0p1nn 12488	. . . 4 ⊢ ((♯‘𝑅) ∈ ℕ₀ → ((♯‘𝑅) + 1) ∈ ℕ)
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → ((♯‘𝑅) + 1) ∈ ℕ)
8	1, 2, 3, 7	vdwlem10 16968	. 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 7423	. . . . . . 7 ⊢ (1...𝑛) ∈ V
11		elmapg 8815	. . . . . . 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 476	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ (𝑅 ↑_m (1...𝑛))) → 𝑓:(1...𝑛)⟶𝑅)
14	5	nn0red 12511	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝑅) ∈ ℝ)
15	14	ltp1d 12120	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑅) < ((♯‘𝑅) + 1))
16		peano2re 11354	. . . . . . . . . . . 12 ⊢ ((♯‘𝑅) ∈ ℝ → ((♯‘𝑅) + 1) ∈ ℝ)
17	14, 16	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘𝑅) + 1) ∈ ℝ)
18	14, 17	ltnled 11328	. . . . . . . . . 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 12854	. . . . . . . . . . . . 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 12510	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) → 𝐾 ∈ ℕ₀)
25		simprr 772	. . . . . . . . . 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 2730	. . . . . . . . . 10 ⊢ (1...((♯‘𝑅) + 1)) = (1...((♯‘𝑅) + 1))
28	10, 24, 25, 26, 27	vdwpc 16958	. . . . . . . . 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 730	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . 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 6056	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (^◡𝑓 “ {(𝑓‘(𝑎 + (𝑑‘𝑖)))}) ⊆ dom 𝑓
34	32, 33	sstrdi 3962	. . . . . . . . . . . . . . . . . . . . . 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 6709	. . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → 𝐾 ∈ ℕ)
37		simplrl 776	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))))) ∧ 𝑖 ∈ (1...((♯‘𝑅) + 1))) → 𝑎 ∈ ℕ)
38		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ ∧ 𝑓:(1...𝑛)⟶𝑅)) ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))))) → 𝑑 ∈ (ℕ ↑_m (1...((♯‘𝑅) + 1))))
39		nnex 12199	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ℕ ∈ V
40		ovex 7423	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (1...((♯‘𝑅) + 1)) ∈ V
41	39, 40	elmap 8847	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7059	. . . . . . . . . . . . . . . . . . . . . . . 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 12245	. . . . . . . . . . . . . . . . . . . . . . 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 16953	. . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . 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 3950	. . . . . . . . . . . . . . . . . . . 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 7056	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(1...𝑛)⟶𝑅 ∧ (𝑎 + (𝑑‘𝑖)) ∈ (1...𝑛)) → (𝑓‘(𝑎 + (𝑑‘𝑖))) ∈ 𝑅)
51	30, 49, 50	syl6an 684	. . . . . . . . . . . . . . . . . 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 3146	. . . . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖)))) = (𝑖 ∈ (1...((♯‘𝑅) + 1)) ↦ (𝑓‘(𝑎 + (𝑑‘𝑖))))
55	54	fmpt 7085	. . . . . . . . . . . . . . . 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 6699	. . . . . . . . . . . . . 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 8974	. . . . . . . . . . . . . 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 9219	. . . . . . . . . . . . . 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 14351	. . . . . . . . . . . . . 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 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 5113	. . . . . . . . . . . 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 936	. . . . . . 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 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 3155	. . 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 3156	. 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 847 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 Vcvv 3450 ⊆ wss 3917 {csn 4592 ⟨cop 4598 class class class wbr 5110 ↦ cmpt 5191 ^◡ccnv 5640 dom cdm 5641 ran crn 5642 “ cima 5644 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ↑_m cmap 8802 ≼ cdom 8919 Fincfn 8921 ℝcr 11074 1c1 11076 + caddc 11078 < clt 11215 ≤ cle 11216 ℕcn 12193 2c2 12248 ℕ₀cn0 12449 ℤ_≥cuz 12800 ...cfz 13475 ♯chash 14302 APcvdwa 16943 MonoAP cvdwm 16944 PolyAP cvdwp 16945

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-oadd 8441 df-er 8674 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-n0 12450 df-xnn0 12523 df-z 12537 df-uz 12801 df-rp 12959 df-fz 13476 df-hash 14303 df-vdwap 16946 df-vdwmc 16947 df-vdwpc 16948

This theorem is referenced by: vdwlem13 16971

W3C validator