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Theorem vdwlem7 15691

Description: Lemma for vdw 15698. (Contributed by Mario Carneiro, 12-Sep-2014.)

**Hypotheses**
Ref	Expression
vdwlem3.v	⊢ (𝜑 → 𝑉 ∈ ℕ)
vdwlem3.w	⊢ (𝜑 → 𝑊 ∈ ℕ)
vdwlem4.r	⊢ (𝜑 → 𝑅 ∈ Fin)
vdwlem4.h	⊢ (𝜑 → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
vdwlem4.f	⊢ 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))
vdwlem7.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
vdwlem7.g	⊢ (𝜑 → 𝐺:(1...𝑊)⟶𝑅)
vdwlem7.k	⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
vdwlem7.a	⊢ (𝜑 → 𝐴 ∈ ℕ)
vdwlem7.d	⊢ (𝜑 → 𝐷 ∈ ℕ)
vdwlem7.s	⊢ (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (^◡𝐹 “ {𝐺}))

**Assertion**
Ref	Expression
vdwlem7	⊢ (𝜑 → (⟨𝑀, 𝐾⟩ PolyAP 𝐺 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐺,𝑦 𝑥,𝐾,𝑦 𝜑,𝑥,𝑦 𝑥,𝑅,𝑦 𝑥,𝐻,𝑦 𝑥,𝑀,𝑦 𝑥,𝐷,𝑦 𝑥,𝑊,𝑦 𝑥,𝑉,𝑦 Allowed substitution hints: 𝐹(𝑥,𝑦)

Proof of Theorem vdwlem7 Dummy variables 𝑘 𝑎 𝑑 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6678	. . 3 ⊢ (1...𝑊) ∈ V
2		2nn0 11309	. . . 4 ⊢ 2 ∈ ℕ₀
3		vdwlem7.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
4		eluznn0 11757	. . . 4 ⊢ ((2 ∈ ℕ₀ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 𝐾 ∈ ℕ₀)
5	2, 3, 4	sylancr 695	. . 3 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
6		vdwlem7.g	. . 3 ⊢ (𝜑 → 𝐺:(1...𝑊)⟶𝑅)
7		vdwlem7.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
8		eqid 2622	. . 3 ⊢ (1...𝑀) = (1...𝑀)
9	1, 5, 6, 7, 8	vdwpc 15684	. 2 ⊢ (𝜑 → (⟨𝑀, 𝐾⟩ PolyAP 𝐺 ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀))(∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)))
10		vdwlem3.v	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ ℕ)
11	10	ad2antrr 762	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝑉 ∈ ℕ)
12		vdwlem3.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℕ)
13	12	ad2antrr 762	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝑊 ∈ ℕ)
14		vdwlem4.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Fin)
15	14	ad2antrr 762	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝑅 ∈ Fin)
16		vdwlem4.h	. . . . . 6 ⊢ (𝜑 → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
17	16	ad2antrr 762	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝐻:(1...(𝑊 · (2 · 𝑉)))⟶𝑅)
18		vdwlem4.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (1...𝑉) ↦ (𝑦 ∈ (1...𝑊) ↦ (𝐻‘(𝑦 + (𝑊 · ((𝑥 − 1) + 𝑉))))))
19	7	ad2antrr 762	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝑀 ∈ ℕ)
20	6	ad2antrr 762	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝐺:(1...𝑊)⟶𝑅)
21	3	ad2antrr 762	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝐾 ∈ (ℤ_≥‘2))
22		vdwlem7.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℕ)
23	22	ad2antrr 762	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝐴 ∈ ℕ)
24		vdwlem7.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℕ)
25	24	ad2antrr 762	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝐷 ∈ ℕ)
26		vdwlem7.s	. . . . . 6 ⊢ (𝜑 → (𝐴(AP‘𝐾)𝐷) ⊆ (^◡𝐹 “ {𝐺}))
27	26	ad2antrr 762	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → (𝐴(AP‘𝐾)𝐷) ⊆ (^◡𝐹 “ {𝐺}))
28		simplrl 800	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝑎 ∈ ℕ)
29		simplrr 801	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))
30		nnex 11026	. . . . . . 7 ⊢ ℕ ∈ V
31		ovex 6678	. . . . . . 7 ⊢ (1...𝑀) ∈ V
32	30, 31	elmap 7886	. . . . . 6 ⊢ (𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)) ↔ 𝑑:(1...𝑀)⟶ℕ)
33	29, 32	sylib 208	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → 𝑑:(1...𝑀)⟶ℕ)
34		simprl 794	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → ∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}))
35		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝑑‘𝑖) = (𝑑‘𝑘))
36	35	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑎 + (𝑑‘𝑖)) = (𝑎 + (𝑑‘𝑘)))
37	36, 35	oveq12d 6668	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) = ((𝑎 + (𝑑‘𝑘))(AP‘𝐾)(𝑑‘𝑘)))
38	36	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝐺‘(𝑎 + (𝑑‘𝑖))) = (𝐺‘(𝑎 + (𝑑‘𝑘))))
39	38	sneqd 4189	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → {(𝐺‘(𝑎 + (𝑑‘𝑖)))} = {(𝐺‘(𝑎 + (𝑑‘𝑘)))})
40	39	imaeq2d 5466	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) = (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑘)))}))
41	37, 40	sseq12d 3634	. . . . . . 7 ⊢ (𝑖 = 𝑘 → (((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ↔ ((𝑎 + (𝑑‘𝑘))(AP‘𝐾)(𝑑‘𝑘)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑘)))})))
42	41	cbvralv 3171	. . . . . 6 ⊢ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ↔ ∀𝑘 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑘))(AP‘𝐾)(𝑑‘𝑘)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑘)))}))
43	34, 42	sylib 208	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → ∀𝑘 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑘))(AP‘𝐾)(𝑑‘𝑘)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑘)))}))
44	38	cbvmptv 4750	. . . . 5 ⊢ (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖)))) = (𝑘 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑘))))
45		simprr 796	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)
46		eqid 2622	. . . . 5 ⊢ (𝑎 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1))) = (𝑎 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1)))
47		eqid 2622	. . . . 5 ⊢ (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (𝑑‘𝑗)) + (𝑊 · 𝐷))) = (𝑗 ∈ (1...(𝑀 + 1)) ↦ (if(𝑗 = (𝑀 + 1), 0, (𝑑‘𝑗)) + (𝑊 · 𝐷)))
48	11, 13, 15, 17, 18, 19, 20, 21, 23, 25, 27, 28, 33, 43, 44, 45, 46, 47	vdwlem6 15690	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) ∧ (∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀)) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺))
49	48	ex 450	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ℕ ∧ 𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀)))) → ((∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)))
50	49	rexlimdvva 3038	. 2 ⊢ (𝜑 → (∃𝑎 ∈ ℕ ∃𝑑 ∈ (ℕ ↑_𝑚 (1...𝑀))(∀𝑖 ∈ (1...𝑀)((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) ⊆ (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) ∧ (#‘ran (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖))))) = 𝑀) → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)))
51	9, 50	sylbid 230	1 ⊢ (𝜑 → (⟨𝑀, 𝐾⟩ PolyAP 𝐺 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ⊆ wss 3574 ifcif 4086 {csn 4177 ⟨cop 4183 class class class wbr 4653 ↦ cmpt 4729 ^◡ccnv 5113 ran crn 5115 “ cima 5117 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑_𝑚 cmap 7857 Fincfn 7955 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 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-oadd 7564 df-er 7742 df-map 7859 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-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: vdwlem9 15693

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