Theorem vdwapun 15678

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 11333	. . . . 5 ⊢ (𝐾 ∈ ℕ0 → (𝐾 + 1) ∈ ℕ0)
2		vdwapval 15677	. . . . 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 1359	. . . 4 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 ∈ (𝐴(AP‘(𝐾 + 1))𝐷) ↔ ∃𝑛 ∈ (0...((𝐾 + 1) − 1))𝑥 = (𝐴 + (𝑛 · 𝐷))))
4		simp1 1061	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐾 ∈ ℕ0)
5	4	nn0cnd 11353	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐾 ∈ ℂ)
6		ax-1cn 9994	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
7		pncan 10287	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 + 1) − 1) = 𝐾)
8	5, 6, 7	sylancl 694	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → ((𝐾 + 1) − 1) = 𝐾)
9	8	oveq2d 6666	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (0...((𝐾 + 1) − 1)) = (0...𝐾))
10	9	eleq2d 2687	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑛 ∈ (0...((𝐾 + 1) − 1)) ↔ 𝑛 ∈ (0...𝐾)))
11		nn0uz 11722	. . . . . . . . . . 11 ⊢ ℕ0 = (ℤ≥‘0)
12	4, 11	syl6eleq 2711	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐾 ∈ (ℤ≥‘0))
13		elfzp12 12419	. . . . . . . . . 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 268	. . . . . . . 8 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑛 ∈ (0...((𝐾 + 1) − 1)) ↔ (𝑛 = 0 ∨ 𝑛 ∈ ((0 + 1)...𝐾))))
16	15	anbi1d 741	. . . . . . 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 912	. . . . . . 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	syl6bb 276	. . . . . 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 1850	. . . . 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 2918	. . . . 5 ⊢ (∃𝑛 ∈ (0...((𝐾 + 1) − 1))𝑥 = (𝐴 + (𝑛 · 𝐷)) ↔ ∃𝑛(𝑛 ∈ (0...((𝐾 + 1) − 1)) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))))
21		19.43 1810	. . . . . 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 214	. . . . 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 303	. . . 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 268	. . 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 11028	. . . . . . . . . . 11 ⊢ (𝐷 ∈ ℕ → 𝐷 ∈ ℂ)
26	25	3ad2ant3 1084	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐷 ∈ ℂ)
27	26	mul02d 10234	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (0 · 𝐷) = 0)
28	27	oveq2d 6666	. . . . . . . 8 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴 + (0 · 𝐷)) = (𝐴 + 0))
29		nncn 11028	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
30	29	3ad2ant2 1083	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐴 ∈ ℂ)
31	30	addid1d 10236	. . . . . . . 8 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴 + 0) = 𝐴)
32	28, 31	eqtrd 2656	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴 + (0 · 𝐷)) = 𝐴)
33	32	eqeq2d 2632	. . . . . 6 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 = (𝐴 + (0 · 𝐷)) ↔ 𝑥 = 𝐴))
34		c0ex 10034	. . . . . . 7 ⊢ 0 ∈ V
35		oveq1 6657	. . . . . . . . 9 ⊢ (𝑛 = 0 → (𝑛 · 𝐷) = (0 · 𝐷))
36	35	oveq2d 6666	. . . . . . . 8 ⊢ (𝑛 = 0 → (𝐴 + (𝑛 · 𝐷)) = (𝐴 + (0 · 𝐷)))
37	36	eqeq2d 2632	. . . . . . 7 ⊢ (𝑛 = 0 → (𝑥 = (𝐴 + (𝑛 · 𝐷)) ↔ 𝑥 = (𝐴 + (0 · 𝐷))))
38	34, 37	ceqsexv 3242	. . . . . 6 ⊢ (∃𝑛(𝑛 = 0 ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ↔ 𝑥 = (𝐴 + (0 · 𝐷)))
39		velsn 4193	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
40	33, 38, 39	3bitr4g 303	. . . . 5 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (∃𝑛(𝑛 = 0 ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ↔ 𝑥 ∈ {𝐴}))
41		simpr 477	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝑛 ∈ ((0 + 1)...𝐾))
42		0p1e1 11132	. . . . . . . . . . . . . . 15 ⊢ (0 + 1) = 1
43	42	oveq1i 6660	. . . . . . . . . . . . . 14 ⊢ ((0 + 1)...𝐾) = (1...𝐾)
44	41, 43	syl6eleq 2711	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝑛 ∈ (1...𝐾))
45		1zzd 11408	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 1 ∈ ℤ)
46	4	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝐾 ∈ ℕ0)
47	46	nn0zd 11480	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝐾 ∈ ℤ)
48		elfzelz 12342	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ((0 + 1)...𝐾) → 𝑛 ∈ ℤ)
49	48	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝑛 ∈ ℤ)
50		fzsubel 12377	. . . . . . . . . . . . . 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 1327	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑛 ∈ (1...𝐾) ↔ (𝑛 − 1) ∈ ((1 − 1)...(𝐾 − 1))))
52	44, 51	mpbid 222	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑛 − 1) ∈ ((1 − 1)...(𝐾 − 1)))
53		1m1e0 11089	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
54	53	oveq1i 6660	. . . . . . . . . . . 12 ⊢ ((1 − 1)...(𝐾 − 1)) = (0...(𝐾 − 1))
55	52, 54	syl6eleq 2711	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑛 − 1) ∈ (0...(𝐾 − 1)))
56	49	zcnd 11483	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝑛 ∈ ℂ)
57		1cnd 10056	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 1 ∈ ℂ)
58	26	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝐷 ∈ ℂ)
59	56, 57, 58	subdird 10487	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → ((𝑛 − 1) · 𝐷) = ((𝑛 · 𝐷) − (1 · 𝐷)))
60	58	mulid2d 10058	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (1 · 𝐷) = 𝐷)
61	60	oveq2d 6666	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → ((𝑛 · 𝐷) − (1 · 𝐷)) = ((𝑛 · 𝐷) − 𝐷))
62	59, 61	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → ((𝑛 − 1) · 𝐷) = ((𝑛 · 𝐷) − 𝐷))
63	62	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝐷 + ((𝑛 − 1) · 𝐷)) = (𝐷 + ((𝑛 · 𝐷) − 𝐷)))
64	56, 58	mulcld 10060	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑛 · 𝐷) ∈ ℂ)
65	58, 64	pncan3d 10395	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝐷 + ((𝑛 · 𝐷) − 𝐷)) = (𝑛 · 𝐷))
66	63, 65	eqtr2d 2657	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑛 · 𝐷) = (𝐷 + ((𝑛 − 1) · 𝐷)))
67	66	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝐴 + (𝑛 · 𝐷)) = (𝐴 + (𝐷 + ((𝑛 − 1) · 𝐷))))
68	30	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝐴 ∈ ℂ)
69		subcl 10280	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑛 − 1) ∈ ℂ)
70	56, 6, 69	sylancl 694	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑛 − 1) ∈ ℂ)
71	70, 58	mulcld 10060	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → ((𝑛 − 1) · 𝐷) ∈ ℂ)
72	68, 58, 71	addassd 10062	. . . . . . . . . . . 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 2659	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝐴 + (𝑛 · 𝐷)) = ((𝐴 + 𝐷) + ((𝑛 − 1) · 𝐷)))
74		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑛 − 1) → (𝑚 · 𝐷) = ((𝑛 − 1) · 𝐷))
75	74	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 − 1) → ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = ((𝐴 + 𝐷) + ((𝑛 − 1) · 𝐷)))
76	75	eqeq2d 2632	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 − 1) → ((𝐴 + (𝑛 · 𝐷)) = ((𝐴 + 𝐷) + (𝑚 · 𝐷)) ↔ (𝐴 + (𝑛 · 𝐷)) = ((𝐴 + 𝐷) + ((𝑛 − 1) · 𝐷))))
77	76	rspcev 3309	. . . . . . . . . . 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)))
78	55, 73, 77	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → ∃𝑚 ∈ (0...(𝐾 − 1))(𝐴 + (𝑛 · 𝐷)) = ((𝐴 + 𝐷) + (𝑚 · 𝐷)))
79		eqeq1 2626	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 + (𝑛 · 𝐷)) → (𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷)) ↔ (𝐴 + (𝑛 · 𝐷)) = ((𝐴 + 𝐷) + (𝑚 · 𝐷))))
80	79	rexbidv 3052	. . . . . . . . . 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))))
81	78, 80	syl5ibrcom 237	. . . . . . . . 9 ⊢ (((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	expimpd 629	. . . . . . . 8 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → ((𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) → ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷))))
83	82	exlimdv 1861	. . . . . . 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))))
84		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ (0...(𝐾 − 1)))
85		0zd 11389	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 0 ∈ ℤ)
86	4	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐾 ∈ ℕ0)
87	86	nn0zd 11480	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐾 ∈ ℤ)
88		peano2zm 11420	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ ℤ → (𝐾 − 1) ∈ ℤ)
89	87, 88	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐾 − 1) ∈ ℤ)
90		elfzelz 12342	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (0...(𝐾 − 1)) → 𝑚 ∈ ℤ)
91	90	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℤ)
92		1zzd 11408	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 1 ∈ ℤ)
93		fzaddel 12375	. . . . . . . . . . . . 13 ⊢ (((0 ∈ ℤ ∧ (𝐾 − 1) ∈ ℤ) ∧ (𝑚 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑚 ∈ (0...(𝐾 − 1)) ↔ (𝑚 + 1) ∈ ((0 + 1)...((𝐾 − 1) + 1))))
94	85, 89, 91, 92, 93	syl22anc 1327	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 ∈ (0...(𝐾 − 1)) ↔ (𝑚 + 1) ∈ ((0 + 1)...((𝐾 − 1) + 1))))
95	84, 94	mpbid 222	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 + 1) ∈ ((0 + 1)...((𝐾 − 1) + 1)))
96	86	nn0cnd 11353	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐾 ∈ ℂ)
97		npcan 10290	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐾 − 1) + 1) = 𝐾)
98	96, 6, 97	sylancl 694	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐾 − 1) + 1) = 𝐾)
99	98	oveq2d 6666	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((0 + 1)...((𝐾 − 1) + 1)) = ((0 + 1)...𝐾))
100	95, 99	eleqtrd 2703	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 + 1) ∈ ((0 + 1)...𝐾))
101	30	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐴 ∈ ℂ)
102	26	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐷 ∈ ℂ)
103	91	zcnd 11483	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
104	103, 102	mulcld 10060	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝐷) ∈ ℂ)
105	101, 102, 104	addassd 10062	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + (𝐷 + (𝑚 · 𝐷))))
106		1cnd 10056	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 1 ∈ ℂ)
107	103, 106, 102	adddird 10065	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑚 + 1) · 𝐷) = ((𝑚 · 𝐷) + (1 · 𝐷)))
108	102, 104	addcomd 10238	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐷 + (𝑚 · 𝐷)) = ((𝑚 · 𝐷) + 𝐷))
109	102	mulid2d 10058	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (1 · 𝐷) = 𝐷)
110	109	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑚 · 𝐷) + (1 · 𝐷)) = ((𝑚 · 𝐷) + 𝐷))
111	108, 110	eqtr4d 2659	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐷 + (𝑚 · 𝐷)) = ((𝑚 · 𝐷) + (1 · 𝐷)))
112	107, 111	eqtr4d 2659	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑚 + 1) · 𝐷) = (𝐷 + (𝑚 · 𝐷)))
113	112	oveq2d 6666	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + ((𝑚 + 1) · 𝐷)) = (𝐴 + (𝐷 + (𝑚 · 𝐷))))
114	105, 113	eqtr4d 2659	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + ((𝑚 + 1) · 𝐷)))
115		ovex 6678	. . . . . . . . . . 11 ⊢ (𝑚 + 1) ∈ V
116		eleq1 2689	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 ∈ ((0 + 1)...𝐾) ↔ (𝑚 + 1) ∈ ((0 + 1)...𝐾)))
117		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 · 𝐷) = ((𝑚 + 1) · 𝐷))
118	117	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑚 + 1) → (𝐴 + (𝑛 · 𝐷)) = (𝐴 + ((𝑚 + 1) · 𝐷)))
119	118	eqeq2d 2632	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑚 + 1) → (((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷)) ↔ ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + ((𝑚 + 1) · 𝐷))))
120	116, 119	anbi12d 747	. . . . . . . . . . 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)))))
121	115, 120	spcev 3300	. . . . . . . . . 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))))
122	100, 114, 121	syl2anc 693	. . . . . . . . 9 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ∃𝑛(𝑛 ∈ ((0 + 1)...𝐾) ∧ ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))))
123		eqeq1 2626	. . . . . . . . . . 11 ⊢ (𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷)) → (𝑥 = (𝐴 + (𝑛 · 𝐷)) ↔ ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))))
124	123	anbi2d 740	. . . . . . . . . 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)))))
125	124	exbidv 1850	. . . . . . . . 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)))))
126	122, 125	syl5ibrcom 237	. . . . . . . 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)))))
127	126	rexlimdva 3031	. . . . . . 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)))))
128	83, 127	impbid 202	. . . . . 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))))
129		nnaddcl 11042	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴 + 𝐷) ∈ ℕ)
130	129	3adant1 1079	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴 + 𝐷) ∈ ℕ)
131		vdwapval 15677	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ0 ∧ (𝐴 + 𝐷) ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 ∈ ((𝐴 + 𝐷)(AP‘𝐾)𝐷) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷))))
132	130, 131	syld3an2 1373	. . . . . 6 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 ∈ ((𝐴 + 𝐷)(AP‘𝐾)𝐷) ↔ ∃𝑚 ∈ (0...(𝐾 − 1))𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷))))
133	128, 132	bitr4d 271	. . . . 5 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (∃𝑛(𝑛 ∈ ((0 + 1)...𝐾) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ↔ 𝑥 ∈ ((𝐴 + 𝐷)(AP‘𝐾)𝐷)))
134	40, 133	orbi12d 746	. . . 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))))
135		elun 3753	. . . 4 ⊢ (𝑥 ∈ ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘𝐾)𝐷)) ↔ (𝑥 ∈ {𝐴} ∨ 𝑥 ∈ ((𝐴 + 𝐷)(AP‘𝐾)𝐷)))
136	134, 135	syl6bbr 278	. . 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))))
137	24, 136	bitrd 268	. 2 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 ∈ (𝐴(AP‘(𝐾 + 1))𝐷) ↔ 𝑥 ∈ ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘𝐾)𝐷))))
138	137	eqrdv 2620	1 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘(𝐾 + 1))𝐷) = ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘𝐾)𝐷)))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃wrex 2913   ∪ cun 3572  {csn 4177  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   − cmin 10266  ℕcn 11020  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  APcvdwa 15669

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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-vdwap 15672

This theorem is referenced by:  vdwapid1  15679  vdwap1  15681  vdwlem1  15685  vdwlem5  15689  vdwlem8  15692  vdwlem12  15696