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Theorem vdwapun 16907

Description: Remove the first element of an arithmetic progression. (Contributed by Mario Carneiro, 11-Sep-2014.)

**Assertion**
Ref	Expression
vdwapun	⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘(𝐾 + 1))𝐷) = ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘𝐾)𝐷)))

Proof of Theorem vdwapun Dummy variables 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2nn0 12512	. . . . 5 ⊢ (𝐾 ∈ ℕ₀ → (𝐾 + 1) ∈ ℕ₀)
2		vdwapval 16906	. . . . 5 ⊢ (((𝐾 + 1) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 ∈ (𝐴(AP‘(𝐾 + 1))𝐷) ↔ ∃𝑛 ∈ (0...((𝐾 + 1) − 1))𝑥 = (𝐴 + (𝑛 · 𝐷))))
3	1, 2	syl3an1 1164	. . . 4 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 ∈ (𝐴(AP‘(𝐾 + 1))𝐷) ↔ ∃𝑛 ∈ (0...((𝐾 + 1) − 1))𝑥 = (𝐴 + (𝑛 · 𝐷))))
4		simp1 1137	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐾 ∈ ℕ₀)
5	4	nn0cnd 12534	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐾 ∈ ℂ)
6		ax-1cn 11168	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
7		pncan 11466	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 + 1) − 1) = 𝐾)
8	5, 6, 7	sylancl 587	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → ((𝐾 + 1) − 1) = 𝐾)
9	8	oveq2d 7425	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (0...((𝐾 + 1) − 1)) = (0...𝐾))
10	9	eleq2d 2820	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑛 ∈ (0...((𝐾 + 1) − 1)) ↔ 𝑛 ∈ (0...𝐾)))
11		nn0uz 12864	. . . . . . . . . . 11 ⊢ ℕ₀ = (ℤ_≥‘0)
12	4, 11	eleqtrdi 2844	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐾 ∈ (ℤ_≥‘0))
13		elfzp12 13580	. . . . . . . . . 10 ⊢ (𝐾 ∈ (ℤ_≥‘0) → (𝑛 ∈ (0...𝐾) ↔ (𝑛 = 0 ∨ 𝑛 ∈ ((0 + 1)...𝐾))))
14	12, 13	syl 17	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑛 ∈ (0...𝐾) ↔ (𝑛 = 0 ∨ 𝑛 ∈ ((0 + 1)...𝐾))))
15	10, 14	bitrd 279	. . . . . . . 8 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑛 ∈ (0...((𝐾 + 1) − 1)) ↔ (𝑛 = 0 ∨ 𝑛 ∈ ((0 + 1)...𝐾))))
16	15	anbi1d 631	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → ((𝑛 ∈ (0...((𝐾 + 1) − 1)) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ↔ ((𝑛 = 0 ∨ 𝑛 ∈ ((0 + 1)...𝐾)) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷)))))
17		andir 1008	. . . . . . 7 ⊢ (((𝑛 = 0 ∨ 𝑛 ∈ ((0 + 1)...𝐾)) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ↔ ((𝑛 = 0 ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ∨ (𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷)))))
18	16, 17	bitrdi 287	. . . . . 6 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → ((𝑛 ∈ (0...((𝐾 + 1) − 1)) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ↔ ((𝑛 = 0 ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ∨ (𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))))))
19	18	exbidv 1925	. . . . 5 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (∃𝑛(𝑛 ∈ (0...((𝐾 + 1) − 1)) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ↔ ∃𝑛((𝑛 = 0 ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ∨ (𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))))))
20		df-rex 3072	. . . . 5 ⊢ (∃𝑛 ∈ (0...((𝐾 + 1) − 1))𝑥 = (𝐴 + (𝑛 · 𝐷)) ↔ ∃𝑛(𝑛 ∈ (0...((𝐾 + 1) − 1)) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))))
21		19.43 1886	. . . . . 6 ⊢ (∃𝑛((𝑛 = 0 ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ∨ (𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷)))) ↔ (∃𝑛(𝑛 = 0 ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ∨ ∃𝑛(𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷)))))
22	21	bicomi 223	. . . . 5 ⊢ ((∃𝑛(𝑛 = 0 ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ∨ ∃𝑛(𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷)))) ↔ ∃𝑛((𝑛 = 0 ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ∨ (𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷)))))
23	19, 20, 22	3bitr4g 314	. . . 4 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (∃𝑛 ∈ (0...((𝐾 + 1) − 1))𝑥 = (𝐴 + (𝑛 · 𝐷)) ↔ (∃𝑛(𝑛 = 0 ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ∨ ∃𝑛(𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))))))
24	3, 23	bitrd 279	. . 3 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 ∈ (𝐴(AP‘(𝐾 + 1))𝐷) ↔ (∃𝑛(𝑛 = 0 ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ∨ ∃𝑛(𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))))))
25		nncn 12220	. . . . . . . . . . 11 ⊢ (𝐷 ∈ ℕ → 𝐷 ∈ ℂ)
26	25	3ad2ant3 1136	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐷 ∈ ℂ)
27	26	mul02d 11412	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (0 · 𝐷) = 0)
28	27	oveq2d 7425	. . . . . . . 8 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴 + (0 · 𝐷)) = (𝐴 + 0))
29		nncn 12220	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
30	29	3ad2ant2 1135	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐴 ∈ ℂ)
31	30	addridd 11414	. . . . . . . 8 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴 + 0) = 𝐴)
32	28, 31	eqtrd 2773	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴 + (0 · 𝐷)) = 𝐴)
33	32	eqeq2d 2744	. . . . . 6 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 = (𝐴 + (0 · 𝐷)) ↔ 𝑥 = 𝐴))
34		c0ex 11208	. . . . . . 7 ⊢ 0 ∈ V
35		oveq1 7416	. . . . . . . . 9 ⊢ (𝑛 = 0 → (𝑛 · 𝐷) = (0 · 𝐷))
36	35	oveq2d 7425	. . . . . . . 8 ⊢ (𝑛 = 0 → (𝐴 + (𝑛 · 𝐷)) = (𝐴 + (0 · 𝐷)))
37	36	eqeq2d 2744	. . . . . . 7 ⊢ (𝑛 = 0 → (𝑥 = (𝐴 + (𝑛 · 𝐷)) ↔ 𝑥 = (𝐴 + (0 · 𝐷))))
38	34, 37	ceqsexv 3526	. . . . . 6 ⊢ (∃𝑛(𝑛 = 0 ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ↔ 𝑥 = (𝐴 + (0 · 𝐷)))
39		velsn 4645	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
40	33, 38, 39	3bitr4g 314	. . . . 5 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (∃𝑛(𝑛 = 0 ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ↔ 𝑥 ∈ {𝐴}))
41		simpr 486	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝑛 ∈ ((0 + 1)...𝐾))
42		0p1e1 12334	. . . . . . . . . . . . . . 15 ⊢ (0 + 1) = 1
43	42	oveq1i 7419	. . . . . . . . . . . . . 14 ⊢ ((0 + 1)...𝐾) = (1...𝐾)
44	41, 43	eleqtrdi 2844	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝑛 ∈ (1...𝐾))
45		1zzd 12593	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 1 ∈ ℤ)
46	4	adantr 482	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝐾 ∈ ℕ₀)
47	46	nn0zd 12584	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝐾 ∈ ℤ)
48		elfzelz 13501	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ((0 + 1)...𝐾) → 𝑛 ∈ ℤ)
49	48	adantl 483	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝑛 ∈ ℤ)
50		fzsubel 13537	. . . . . . . . . . . . . 14 ⊢ (((1 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (1...𝐾) ↔ (𝑛 − 1) ∈ ((1 − 1)...(𝐾 − 1))))
51	45, 47, 49, 45, 50	syl22anc 838	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑛 ∈ (1...𝐾) ↔ (𝑛 − 1) ∈ ((1 − 1)...(𝐾 − 1))))
52	44, 51	mpbid 231	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑛 − 1) ∈ ((1 − 1)...(𝐾 − 1)))
53		1m1e0 12284	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
54	53	oveq1i 7419	. . . . . . . . . . . 12 ⊢ ((1 − 1)...(𝐾 − 1)) = (0...(𝐾 − 1))
55	52, 54	eleqtrdi 2844	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑛 − 1) ∈ (0...(𝐾 − 1)))
56	49	zcnd 12667	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝑛 ∈ ℂ)
57		1cnd 11209	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 1 ∈ ℂ)
58	26	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝐷 ∈ ℂ)
59	56, 57, 58	subdird 11671	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → ((𝑛 − 1) · 𝐷) = ((𝑛 · 𝐷) − (1 · 𝐷)))
60	58	mullidd 11232	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (1 · 𝐷) = 𝐷)
61	60	oveq2d 7425	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → ((𝑛 · 𝐷) − (1 · 𝐷)) = ((𝑛 · 𝐷) − 𝐷))
62	59, 61	eqtrd 2773	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → ((𝑛 − 1) · 𝐷) = ((𝑛 · 𝐷) − 𝐷))
63	62	oveq2d 7425	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝐷 + ((𝑛 − 1) · 𝐷)) = (𝐷 + ((𝑛 · 𝐷) − 𝐷)))
64	56, 58	mulcld 11234	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑛 · 𝐷) ∈ ℂ)
65	58, 64	pncan3d 11574	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝐷 + ((𝑛 · 𝐷) − 𝐷)) = (𝑛 · 𝐷))
66	63, 65	eqtr2d 2774	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑛 · 𝐷) = (𝐷 + ((𝑛 − 1) · 𝐷)))
67	66	oveq2d 7425	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝐴 + (𝑛 · 𝐷)) = (𝐴 + (𝐷 + ((𝑛 − 1) · 𝐷))))
68	30	adantr 482	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝐴 ∈ ℂ)
69		subcl 11459	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑛 − 1) ∈ ℂ)
70	56, 6, 69	sylancl 587	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑛 − 1) ∈ ℂ)
71	70, 58	mulcld 11234	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → ((𝑛 − 1) · 𝐷) ∈ ℂ)
72	68, 58, 71	addassd 11236	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → ((𝐴 + 𝐷) + ((𝑛 − 1) · 𝐷)) = (𝐴 + (𝐷 + ((𝑛 − 1) · 𝐷))))
73	67, 72	eqtr4d 2776	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝐴 + (𝑛 · 𝐷)) = ((𝐴 + 𝐷) + ((𝑛 − 1) · 𝐷)))
74		oveq1 7416	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 − 1) → (𝑚 · 𝐷) = ((𝑛 − 1) · 𝐷))
75	74	oveq2d 7425	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 − 1) → ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = ((𝐴 + 𝐷) + ((𝑛 − 1) · 𝐷)))
76	75	rspceeqv 3634	. . . . . . . . . . 11 ⊢ (((𝑛 − 1) ∈ (0...(𝐾 − 1)) ∧ (𝐴 + (𝑛 · 𝐷)) = ((𝐴 + 𝐷) + ((𝑛 − 1) · 𝐷))) → ∃𝑚 ∈ (0...(𝐾 − 1))(𝐴 + (𝑛 · 𝐷)) = ((𝐴 + 𝐷) + (𝑚 · 𝐷)))
77	55, 73, 76	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → ∃𝑚 ∈ (0...(𝐾 − 1))(𝐴 + (𝑛 · 𝐷)) = ((𝐴 + 𝐷) + (𝑚 · 𝐷)))
78		eqeq1 2737	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 + (𝑛 · 𝐷)) → (𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷)) ↔ (𝐴 + (𝑛 · 𝐷)) = ((𝐴 + 𝐷) + (𝑚 · 𝐷))))
79	78	rexbidv 3179	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴 + (𝑛 · 𝐷)) → (∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷)) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))(𝐴 + (𝑛 · 𝐷)) = ((𝐴 + 𝐷) + (𝑚 · 𝐷))))
80	77, 79	syl5ibrcom 246	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑥 = (𝐴 + (𝑛 · 𝐷)) → ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷))))
81	80	expimpd 455	. . . . . . . 8 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → ((𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) → ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷))))
82	81	exlimdv 1937	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (∃𝑛(𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) → ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷))))
83		simpr 486	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ (0...(𝐾 − 1)))
84		0zd 12570	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 0 ∈ ℤ)
85	4	adantr 482	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐾 ∈ ℕ₀)
86	85	nn0zd 12584	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐾 ∈ ℤ)
87		peano2zm 12605	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℤ → (𝐾 − 1) ∈ ℤ)
88	86, 87	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐾 − 1) ∈ ℤ)
89		elfzelz 13501	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (0...(𝐾 − 1)) → 𝑚 ∈ ℤ)
90	89	adantl 483	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℤ)
91		1zzd 12593	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 1 ∈ ℤ)
92		fzaddel 13535	. . . . . . . . . . . . 13 ⊢ (((0 ∈ ℤ ∧ (𝐾 − 1) ∈ ℤ) ∧ (𝑚 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑚 ∈ (0...(𝐾 − 1)) ↔ (𝑚 + 1) ∈ ((0 + 1)...((𝐾 − 1) + 1))))
93	84, 88, 90, 91, 92	syl22anc 838	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 ∈ (0...(𝐾 − 1)) ↔ (𝑚 + 1) ∈ ((0 + 1)...((𝐾 − 1) + 1))))
94	83, 93	mpbid 231	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 + 1) ∈ ((0 + 1)...((𝐾 − 1) + 1)))
95	85	nn0cnd 12534	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐾 ∈ ℂ)
96		npcan 11469	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 − 1) + 1) = 𝐾)
97	95, 6, 96	sylancl 587	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐾 − 1) + 1) = 𝐾)
98	97	oveq2d 7425	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((0 + 1)...((𝐾 − 1) + 1)) = ((0 + 1)...𝐾))
99	94, 98	eleqtrd 2836	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 + 1) ∈ ((0 + 1)...𝐾))
100	30	adantr 482	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐴 ∈ ℂ)
101	26	adantr 482	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐷 ∈ ℂ)
102	90	zcnd 12667	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
103	102, 101	mulcld 11234	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝐷) ∈ ℂ)
104	100, 101, 103	addassd 11236	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + (𝐷 + (𝑚 · 𝐷))))
105		1cnd 11209	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 1 ∈ ℂ)
106	102, 105, 101	adddird 11239	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑚 + 1) · 𝐷) = ((𝑚 · 𝐷) + (1 · 𝐷)))
107	101, 103	addcomd 11416	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐷 + (𝑚 · 𝐷)) = ((𝑚 · 𝐷) + 𝐷))
108	101	mullidd 11232	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (1 · 𝐷) = 𝐷)
109	108	oveq2d 7425	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑚 · 𝐷) + (1 · 𝐷)) = ((𝑚 · 𝐷) + 𝐷))
110	107, 109	eqtr4d 2776	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐷 + (𝑚 · 𝐷)) = ((𝑚 · 𝐷) + (1 · 𝐷)))
111	106, 110	eqtr4d 2776	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑚 + 1) · 𝐷) = (𝐷 + (𝑚 · 𝐷)))
112	111	oveq2d 7425	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + ((𝑚 + 1) · 𝐷)) = (𝐴 + (𝐷 + (𝑚 · 𝐷))))
113	104, 112	eqtr4d 2776	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + ((𝑚 + 1) · 𝐷)))
114		ovex 7442	. . . . . . . . . . 11 ⊢ (𝑚 + 1) ∈ V
115		eleq1 2822	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 ∈ ((0 + 1)...𝐾) ↔ (𝑚 + 1) ∈ ((0 + 1)...𝐾)))
116		oveq1 7416	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 · 𝐷) = ((𝑚 + 1) · 𝐷))
117	116	oveq2d 7425	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑚 + 1) → (𝐴 + (𝑛 · 𝐷)) = (𝐴 + ((𝑚 + 1) · 𝐷)))
118	117	eqeq2d 2744	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑚 + 1) → (((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷)) ↔ ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + ((𝑚 + 1) · 𝐷))))
119	115, 118	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑚 + 1) → ((𝑛 ∈ ((0 + 1)...𝐾) ∧ ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))) ↔ ((𝑚 + 1) ∈ ((0 + 1)...𝐾) ∧ ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + ((𝑚 + 1) · 𝐷)))))
120	114, 119	spcev 3597	. . . . . . . . . 10 ⊢ (((𝑚 + 1) ∈ ((0 + 1)...𝐾) ∧ ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + ((𝑚 + 1) · 𝐷))) → ∃𝑛(𝑛 ∈ ((0 + 1)...𝐾) ∧ ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))))
121	99, 113, 120	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ∃𝑛(𝑛 ∈ ((0 + 1)...𝐾) ∧ ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))))
122		eqeq1 2737	. . . . . . . . . . 11 ⊢ (𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷)) → (𝑥 = (𝐴 + (𝑛 · 𝐷)) ↔ ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))))
123	122	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷)) → ((𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ↔ (𝑛 ∈ ((0 + 1)...𝐾) ∧ ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷)))))
124	123	exbidv 1925	. . . . . . . . 9 ⊢ (𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷)) → (∃𝑛(𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ↔ ∃𝑛(𝑛 ∈ ((0 + 1)...𝐾) ∧ ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷)))))
125	121, 124	syl5ibrcom 246	. . . . . . . 8 ⊢ (((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷)) → ∃𝑛(𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷)))))
126	125	rexlimdva 3156	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷)) → ∃𝑛(𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷)))))
127	82, 126	impbid 211	. . . . . 6 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (∃𝑛(𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷))))
128		nnaddcl 12235	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴 + 𝐷) ∈ ℕ)
129	128	3adant1 1131	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴 + 𝐷) ∈ ℕ)
130		vdwapval 16906	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ₀ ∧ (𝐴 + 𝐷) ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 ∈ ((𝐴 + 𝐷)(AP‘𝐾)𝐷) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷))))
131	129, 130	syld3an2 1412	. . . . . 6 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 ∈ ((𝐴 + 𝐷)(AP‘𝐾)𝐷) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷))))
132	127, 131	bitr4d 282	. . . . 5 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (∃𝑛(𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ↔ 𝑥 ∈ ((𝐴 + 𝐷)(AP‘𝐾)𝐷)))
133	40, 132	orbi12d 918	. . . 4 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → ((∃𝑛(𝑛 = 0 ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ∨ ∃𝑛(𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷)))) ↔ (𝑥 ∈ {𝐴} ∨ 𝑥 ∈ ((𝐴 + 𝐷)(AP‘𝐾)𝐷))))
134		elun 4149	. . . 4 ⊢ (𝑥 ∈ ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘𝐾)𝐷)) ↔ (𝑥 ∈ {𝐴} ∨ 𝑥 ∈ ((𝐴 + 𝐷)(AP‘𝐾)𝐷)))
135	133, 134	bitr4di 289	. . 3 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → ((∃𝑛(𝑛 = 0 ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ∨ ∃𝑛(𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷)))) ↔ 𝑥 ∈ ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘𝐾)𝐷))))
136	24, 135	bitrd 279	. 2 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 ∈ (𝐴(AP‘(𝐾 + 1))𝐷) ↔ 𝑥 ∈ ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘𝐾)𝐷))))
137	136	eqrdv 2731	1 ⊢ ((𝐾 ∈ ℕ₀ ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘(𝐾 + 1))𝐷) = ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘𝐾)𝐷)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∃wrex 3071 ∪ cun 3947 {csn 4629 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 − cmin 11444 ℕcn 12212 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 ...cfz 13484 APcvdwa 16898

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-vdwap 16901

This theorem is referenced by: vdwapid1 16908 vdwap1 16910 vdwlem1 16914 vdwlem5 16918 vdwlem8 16921 vdwlem12 16925

W3C validator