vdwlem7 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > vdwlem7		Structured version Visualization version GIF version

Theorem vdwlem7 17071

Description: Lemma for vdw 17078. (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 7452	. . 3 ⊢ (1...𝑊) ∈ V
2		2nn0 12538	. . . 4 ⊢ 2 ∈ ℕ₀
3		vdwlem7.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
4		eluznn0 12959	. . . 4 ⊢ ((2 ∈ ℕ₀ ∧ 𝐾 ∈ (ℤ_≥‘2)) → 𝐾 ∈ ℕ₀)
5	2, 3, 4	sylancr 599	. . 3 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
6		vdwlem7.g	. . 3 ⊢ (𝜑 → 𝐺:(1...𝑊)⟶𝑅)
7		vdwlem7.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℕ)
8		eqid 2765	. . 3 ⊢ (1...𝑀) = (1...𝑀)
9	1, 5, 6, 7, 8	vdwpc 17064	. 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 739	. . . . 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 739	. . . . 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 739	. . . . 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 739	. . . . 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 739	. . . . 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 739	. . . . 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 739	. . . . 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 739	. . . . 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 739	. . . . 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 739	. . . . 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 789	. . . . 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 790	. . . . . 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 12256	. . . . . . 7 ⊢ ℕ ∈ V
31		ovex 7452	. . . . . . 7 ⊢ (1...𝑀) ∈ V
32	30, 31	elmap 8875	. . . . . 6 ⊢ (𝑑 ∈ (ℕ ↑_m (1...𝑀)) ↔ 𝑑:(1...𝑀)⟶ℕ)
33	29, 32	sylib 221	. . . . 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 783	. . . . . 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 6885	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝑑‘𝑖) = (𝑑‘𝑘))
36	35	oveq2d 7435	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑎 + (𝑑‘𝑖)) = (𝑎 + (𝑑‘𝑘)))
37	36, 35	oveq12d 7437	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ((𝑎 + (𝑑‘𝑖))(AP‘𝐾)(𝑑‘𝑖)) = ((𝑎 + (𝑑‘𝑘))(AP‘𝐾)(𝑑‘𝑘)))
38	36	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝐺‘(𝑎 + (𝑑‘𝑖))) = (𝐺‘(𝑎 + (𝑑‘𝑘))))
39	38	sneqd 4603	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → {(𝐺‘(𝑎 + (𝑑‘𝑖)))} = {(𝐺‘(𝑎 + (𝑑‘𝑘)))})
40	39	imaeq2d 6064	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑖)))}) = (^◡𝐺 “ {(𝐺‘(𝑎 + (𝑑‘𝑘)))}))
41	37, 40	sseq12d 3971	. . . . . . 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	cbvralvw 3245	. . . . . 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 221	. . . . 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 5217	. . . . 5 ⊢ (𝑖 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑖)))) = (𝑘 ∈ (1...𝑀) ↦ (𝐺‘(𝑎 + (𝑑‘𝑘))))
45		simprr 785	. . . . 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 2765	. . . . 5 ⊢ (𝑎 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1))) = (𝑎 + (𝑊 · ((𝐴 + (𝑉 − 𝐷)) − 1)))
47		eqid 2765	. . . . 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 17070	. . . 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 418	. . 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 3224	. 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 243	1 ⊢ (𝜑 → (⟨𝑀, 𝐾⟩ PolyAP 𝐺 → (⟨(𝑀 + 1), 𝐾⟩ PolyAP 𝐻 ∨ (𝐾 + 1) MonoAP 𝐺)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2146 ∀wral 3081 ∃wrex 3091 ⊆ wss 3906 ifcif 4489 {csn 4591 ⟨cop 4597 class class class wbr 5111 ↦ cmpt 5194 ^◡ccnv 5662 ran crn 5664 “ cima 5666 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ↑_m cmap 8830 Fincfn 8949 0cc0 11117 1c1 11118 + caddc 11120 · cmul 11122 − cmin 11458 ℕcn 12250 2c2 12312 ℕ₀cn0 12521 ℤ_≥cuz 12880 ...cfz 13553 ♯chash 14386 APcvdwa 17049 MonoAP cvdwm 17050 PolyAP cvdwp 17051

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-oadd 8463 df-er 8700 df-map 8832 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-dju 9903 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-nn 12251 df-2 12320 df-n0 12522 df-z 12609 df-uz 12881 df-rp 13035 df-fz 13554 df-hash 14387 df-vdwap 17052 df-vdwmc 17053 df-vdwpc 17054

This theorem is used by: vdwlem9 17073

W3C validator