Theorem vdwapun 16945

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 12477	. . . . 5 ⊢ (𝐾 ∈ ℕ0 → (𝐾 + 1) ∈ ℕ0)
2		vdwapval 16944	. . . . 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 12500	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐾 ∈ ℂ)
6		ax-1cn 11096	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
7		pncan 11399	. . . . . . . . . . . 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 7383	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (0...((𝐾 + 1) − 1)) = (0...𝐾))
10	9	eleq2d 2822	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑛 ∈ (0...((𝐾 + 1) − 1)) ↔ 𝑛 ∈ (0...𝐾)))
11		nn0uz 12826	. . . . . . . . . . 11 ⊢ ℕ0 = (ℤ≥‘0)
12	4, 11	eleqtrdi 2846	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐾 ∈ (ℤ≥‘0))
13		elfzp12 13557	. . . . . . . . . 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 632	. . . . . . 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 1011	. . . . . . 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 1923	. . . . 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 3062	. . . . 5 ⊢ (∃𝑛 ∈ (0...((𝐾 + 1) − 1))𝑥 = (𝐴 + (𝑛 · 𝐷)) ↔ ∃𝑛(𝑛 ∈ (0...((𝐾 + 1) − 1)) ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))))
21		19.43 1884	. . . . . 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 224	. . . . 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 12182	. . . . . . . . . . 11 ⊢ (𝐷 ∈ ℕ → 𝐷 ∈ ℂ)
26	25	3ad2ant3 1136	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐷 ∈ ℂ)
27	26	mul02d 11344	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (0 · 𝐷) = 0)
28	27	oveq2d 7383	. . . . . . . 8 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴 + (0 · 𝐷)) = (𝐴 + 0))
29		nncn 12182	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ)
30	29	3ad2ant2 1135	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → 𝐴 ∈ ℂ)
31	30	addridd 11346	. . . . . . . 8 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴 + 0) = 𝐴)
32	28, 31	eqtrd 2771	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴 + (0 · 𝐷)) = 𝐴)
33	32	eqeq2d 2747	. . . . . 6 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝑥 = (𝐴 + (0 · 𝐷)) ↔ 𝑥 = 𝐴))
34		c0ex 11138	. . . . . . 7 ⊢ 0 ∈ V
35		oveq1 7374	. . . . . . . . 9 ⊢ (𝑛 = 0 → (𝑛 · 𝐷) = (0 · 𝐷))
36	35	oveq2d 7383	. . . . . . . 8 ⊢ (𝑛 = 0 → (𝐴 + (𝑛 · 𝐷)) = (𝐴 + (0 · 𝐷)))
37	36	eqeq2d 2747	. . . . . . 7 ⊢ (𝑛 = 0 → (𝑥 = (𝐴 + (𝑛 · 𝐷)) ↔ 𝑥 = (𝐴 + (0 · 𝐷))))
38	34, 37	ceqsexv 3478	. . . . . 6 ⊢ (∃𝑛(𝑛 = 0 ∧ 𝑥 = (𝐴 + (𝑛 · 𝐷))) ↔ 𝑥 = (𝐴 + (0 · 𝐷)))
39		velsn 4583	. . . . . 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 484	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝑛 ∈ ((0 + 1)...𝐾))
42		0p1e1 12298	. . . . . . . . . . . . . . 15 ⊢ (0 + 1) = 1
43	42	oveq1i 7377	. . . . . . . . . . . . . 14 ⊢ ((0 + 1)...𝐾) = (1...𝐾)
44	41, 43	eleqtrdi 2846	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝑛 ∈ (1...𝐾))
45		1zzd 12558	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 1 ∈ ℤ)
46	4	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝐾 ∈ ℕ0)
47	46	nn0zd 12549	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝐾 ∈ ℤ)
48		elfzelz 13478	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ((0 + 1)...𝐾) → 𝑛 ∈ ℤ)
49	48	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝑛 ∈ ℤ)
50		fzsubel 13514	. . . . . . . . . . . . . 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 839	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑛 ∈ (1...𝐾) ↔ (𝑛 − 1) ∈ ((1 − 1)...(𝐾 − 1))))
52	44, 51	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑛 − 1) ∈ ((1 − 1)...(𝐾 − 1)))
53		1m1e0 12253	. . . . . . . . . . . . 13 ⊢ (1 − 1) = 0
54	53	oveq1i 7377	. . . . . . . . . . . 12 ⊢ ((1 − 1)...(𝐾 − 1)) = (0...(𝐾 − 1))
55	52, 54	eleqtrdi 2846	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑛 − 1) ∈ (0...(𝐾 − 1)))
56	49	zcnd 12634	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝑛 ∈ ℂ)
57		1cnd 11139	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 1 ∈ ℂ)
58	26	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝐷 ∈ ℂ)
59	56, 57, 58	subdird 11607	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → ((𝑛 − 1) · 𝐷) = ((𝑛 · 𝐷) − (1 · 𝐷)))
60	58	mullidd 11163	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (1 · 𝐷) = 𝐷)
61	60	oveq2d 7383	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → ((𝑛 · 𝐷) − (1 · 𝐷)) = ((𝑛 · 𝐷) − 𝐷))
62	59, 61	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → ((𝑛 − 1) · 𝐷) = ((𝑛 · 𝐷) − 𝐷))
63	62	oveq2d 7383	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝐷 + ((𝑛 − 1) · 𝐷)) = (𝐷 + ((𝑛 · 𝐷) − 𝐷)))
64	56, 58	mulcld 11165	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑛 · 𝐷) ∈ ℂ)
65	58, 64	pncan3d 11508	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝐷 + ((𝑛 · 𝐷) − 𝐷)) = (𝑛 · 𝐷))
66	63, 65	eqtr2d 2772	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝑛 · 𝐷) = (𝐷 + ((𝑛 − 1) · 𝐷)))
67	66	oveq2d 7383	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝐴 + (𝑛 · 𝐷)) = (𝐴 + (𝐷 + ((𝑛 − 1) · 𝐷))))
68	30	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → 𝐴 ∈ ℂ)
69		subcl 11392	. . . . . . . . . . . . . . 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 11165	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → ((𝑛 − 1) · 𝐷) ∈ ℂ)
72	68, 58, 71	addassd 11167	. . . . . . . . . . . 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 2774	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑛 ∈ ((0 + 1)...𝐾)) → (𝐴 + (𝑛 · 𝐷)) = ((𝐴 + 𝐷) + ((𝑛 − 1) · 𝐷)))
74		oveq1 7374	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑛 − 1) → (𝑚 · 𝐷) = ((𝑛 − 1) · 𝐷))
75	74	oveq2d 7383	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 − 1) → ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = ((𝐴 + 𝐷) + ((𝑛 − 1) · 𝐷)))
76	75	rspceeqv 3587	. . . . . . . . . . 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 2740	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 + (𝑛 · 𝐷)) → (𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷)) ↔ (𝐴 + (𝑛 · 𝐷)) = ((𝐴 + 𝐷) + (𝑚 · 𝐷))))
79	78	rexbidv 3161	. . . . . . . . . 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 247	. . . . . . . . 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 453	. . . . . . . 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 1935	. . . . . . 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 484	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ (0...(𝐾 − 1)))
84		0zd 12536	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 0 ∈ ℤ)
85	4	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐾 ∈ ℕ0)
86	85	nn0zd 12549	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐾 ∈ ℤ)
87		peano2zm 12570	. . . . . . . . . . . . . 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 13478	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (0...(𝐾 − 1)) → 𝑚 ∈ ℤ)
90	89	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℤ)
91		1zzd 12558	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 1 ∈ ℤ)
92		fzaddel 13512	. . . . . . . . . . . . 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 839	. . . . . . . . . . . 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 232	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 + 1) ∈ ((0 + 1)...((𝐾 − 1) + 1)))
95	85	nn0cnd 12500	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐾 ∈ ℂ)
96		npcan 11402	. . . . . . . . . . . . 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 7383	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((0 + 1)...((𝐾 − 1) + 1)) = ((0 + 1)...𝐾))
99	94, 98	eleqtrd 2838	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 + 1) ∈ ((0 + 1)...𝐾))
100	30	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐴 ∈ ℂ)
101	26	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝐷 ∈ ℂ)
102	90	zcnd 12634	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 𝑚 ∈ ℂ)
103	102, 101	mulcld 11165	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝑚 · 𝐷) ∈ ℂ)
104	100, 101, 103	addassd 11167	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + (𝐷 + (𝑚 · 𝐷))))
105		1cnd 11139	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → 1 ∈ ℂ)
106	102, 105, 101	adddird 11170	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑚 + 1) · 𝐷) = ((𝑚 · 𝐷) + (1 · 𝐷)))
107	101, 103	addcomd 11348	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐷 + (𝑚 · 𝐷)) = ((𝑚 · 𝐷) + 𝐷))
108	101	mullidd 11163	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (1 · 𝐷) = 𝐷)
109	108	oveq2d 7383	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑚 · 𝐷) + (1 · 𝐷)) = ((𝑚 · 𝐷) + 𝐷))
110	107, 109	eqtr4d 2774	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐷 + (𝑚 · 𝐷)) = ((𝑚 · 𝐷) + (1 · 𝐷)))
111	106, 110	eqtr4d 2774	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝑚 + 1) · 𝐷) = (𝐷 + (𝑚 · 𝐷)))
112	111	oveq2d 7383	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → (𝐴 + ((𝑚 + 1) · 𝐷)) = (𝐴 + (𝐷 + (𝑚 · 𝐷))))
113	104, 112	eqtr4d 2774	. . . . . . . . . 10 ⊢ (((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐾 − 1))) → ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + ((𝑚 + 1) · 𝐷)))
114		ovex 7400	. . . . . . . . . . 11 ⊢ (𝑚 + 1) ∈ V
115		eleq1 2824	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 ∈ ((0 + 1)...𝐾) ↔ (𝑚 + 1) ∈ ((0 + 1)...𝐾)))
116		oveq1 7374	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑚 + 1) → (𝑛 · 𝐷) = ((𝑚 + 1) · 𝐷))
117	116	oveq2d 7383	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑚 + 1) → (𝐴 + (𝑛 · 𝐷)) = (𝐴 + ((𝑚 + 1) · 𝐷)))
118	117	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑚 + 1) → (((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷)) ↔ ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + ((𝑚 + 1) · 𝐷))))
119	115, 118	anbi12d 633	. . . . . . . . . . 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 3548	. . . . . . . . . 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 2740	. . . . . . . . . . 11 ⊢ (𝑥 = ((𝐴 + 𝐷) + (𝑚 · 𝐷)) → (𝑥 = (𝐴 + (𝑛 · 𝐷)) ↔ ((𝐴 + 𝐷) + (𝑚 · 𝐷)) = (𝐴 + (𝑛 · 𝐷))))
123	122	anbi2d 631	. . . . . . . . . 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 1923	. . . . . . . . 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 247	. . . . . . . 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 3138	. . . . . . 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 212	. . . . . 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 12197	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴 + 𝐷) ∈ ℕ)
129	128	3adant1 1131	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴 + 𝐷) ∈ ℕ)
130		vdwapval 16944	. . . . . . 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 1414	. . . . . 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 919	. . . 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 4093	. . . 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 2734	1 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐴 ∈ ℕ ∧ 𝐷 ∈ ℕ) → (𝐴(AP‘(𝐾 + 1))𝐷) = ({𝐴} ∪ ((𝐴 + 𝐷)(AP‘𝐾)𝐷)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3061   ∪ cun 3887  {csn 4567  ‘cfv 6498  (class class class)co 7367  ℂcc 11036  0cc0 11038  1c1 11039   + caddc 11041   · cmul 11043   − cmin 11377  ℕcn 12174  ℕ₀cn0 12437  ℤcz 12524  ℤ_≥cuz 12788  ...cfz 13461  APcvdwa 16936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-n0 12438  df-z 12525  df-uz 12789  df-fz 13462  df-vdwap 16939

This theorem is referenced by:  vdwapid1  16946  vdwap1  16948  vdwlem1  16952  vdwlem5  16956  vdwlem8  16959  vdwlem12  16963