Theorem nnmsucr 7705

Description: Multiplication with successor. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
nnmsucr	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ·𝑜 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵))

Proof of Theorem nnmsucr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . 5 ⊢ (𝑥 = 𝐵 → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 𝐵))
2		oveq2 6658	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐵))
3		id 22	. . . . . 6 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
4	2, 3	oveq12d 6668	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵))
5	1, 4	eqeq12d 2637	. . . 4 ⊢ (𝑥 = 𝐵 → ((suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) ↔ (suc 𝐴 ·𝑜 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵)))
6	5	imbi2d 330	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ ω → (suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥)) ↔ (𝐴 ∈ ω → (suc 𝐴 ·𝑜 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵))))
7		oveq2 6658	. . . . 5 ⊢ (𝑥 = ∅ → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 ∅))
8		oveq2 6658	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
9		id 22	. . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
10	8, 9	oveq12d 6668	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 ∅) +𝑜 ∅))
11	7, 10	eqeq12d 2637	. . . 4 ⊢ (𝑥 = ∅ → ((suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) ↔ (suc 𝐴 ·𝑜 ∅) = ((𝐴 ·𝑜 ∅) +𝑜 ∅)))
12		oveq2 6658	. . . . 5 ⊢ (𝑥 = 𝑦 → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 𝑦))
13		oveq2 6658	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
14		id 22	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
15	13, 14	oveq12d 6668	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 𝑦) +𝑜 𝑦))
16	12, 15	eqeq12d 2637	. . . 4 ⊢ (𝑥 = 𝑦 → ((suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) ↔ (suc 𝐴 ·𝑜 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝑦)))
17		oveq2 6658	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 suc 𝑦))
18		oveq2 6658	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
19		id 22	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
20	18, 19	oveq12d 6668	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦))
21	17, 20	eqeq12d 2637	. . . 4 ⊢ (𝑥 = suc 𝑦 → ((suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) ↔ (suc 𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦)))
22		peano2 7086	. . . . . . 7 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
23		nnm0 7685	. . . . . . 7 ⊢ (suc 𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = ∅)
24	22, 23	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = ∅)
25		nnm0 7685	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
26	24, 25	eqtr4d 2659	. . . . 5 ⊢ (𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = (𝐴 ·𝑜 ∅))
27		peano1 7085	. . . . . . 7 ⊢ ∅ ∈ ω
28		nnmcl 7692	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ ∅ ∈ ω) → (𝐴 ·𝑜 ∅) ∈ ω)
29	27, 28	mpan2 707	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) ∈ ω)
30		nna0 7684	. . . . . 6 ⊢ ((𝐴 ·𝑜 ∅) ∈ ω → ((𝐴 ·𝑜 ∅) +𝑜 ∅) = (𝐴 ·𝑜 ∅))
31	29, 30	syl 17	. . . . 5 ⊢ (𝐴 ∈ ω → ((𝐴 ·𝑜 ∅) +𝑜 ∅) = (𝐴 ·𝑜 ∅))
32	26, 31	eqtr4d 2659	. . . 4 ⊢ (𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = ((𝐴 ·𝑜 ∅) +𝑜 ∅))
33		oveq1 6657	. . . . . 6 ⊢ ((suc 𝐴 ·𝑜 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝑦) → ((suc 𝐴 ·𝑜 𝑦) +𝑜 suc 𝐴) = (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴))
34		peano2b 7081	. . . . . . . 8 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
35		nnmsuc 7687	. . . . . . . 8 ⊢ ((suc 𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 ·𝑜 suc 𝑦) = ((suc 𝐴 ·𝑜 𝑦) +𝑜 suc 𝐴))
36	34, 35	sylanb 489	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 ·𝑜 suc 𝑦) = ((suc 𝐴 ·𝑜 𝑦) +𝑜 suc 𝐴))
37		nnmcl 7692	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 𝑦) ∈ ω)
38		peano2b 7081	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω ↔ suc 𝑦 ∈ ω)
39		nnaass 7702	. . . . . . . . . . . 12 ⊢ (((𝐴 ·𝑜 𝑦) ∈ ω ∧ 𝐴 ∈ ω ∧ suc 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝐴) +𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)))
40	38, 39	syl3an3b 1364	. . . . . . . . . . 11 ⊢ (((𝐴 ·𝑜 𝑦) ∈ ω ∧ 𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝐴) +𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)))
41	37, 40	syl3an1 1359	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 𝑦 ∈ ω) ∧ 𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝐴) +𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)))
42	41	3expb 1266	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝑦 ∈ ω) ∧ (𝐴 ∈ ω ∧ 𝑦 ∈ ω)) → (((𝐴 ·𝑜 𝑦) +𝑜 𝐴) +𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)))
43	42	anidms 677	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝐴) +𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)))
44		nnmsuc 7687	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
45	44	oveq1d 6665	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦) = (((𝐴 ·𝑜 𝑦) +𝑜 𝐴) +𝑜 suc 𝑦))
46		nnaass 7702	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ·𝑜 𝑦) ∈ ω ∧ 𝑦 ∈ ω ∧ suc 𝐴 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝑦 +𝑜 suc 𝐴)))
47	34, 46	syl3an3b 1364	. . . . . . . . . . . . 13 ⊢ (((𝐴 ·𝑜 𝑦) ∈ ω ∧ 𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝑦 +𝑜 suc 𝐴)))
48	37, 47	syl3an1 1359	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ω ∧ 𝑦 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝑦 +𝑜 suc 𝐴)))
49	48	3expb 1266	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ω ∧ 𝑦 ∈ ω) ∧ (𝑦 ∈ ω ∧ 𝐴 ∈ ω)) → (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝑦 +𝑜 suc 𝐴)))
50	49	an42s 870	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 𝑦 ∈ ω) ∧ (𝐴 ∈ ω ∧ 𝑦 ∈ ω)) → (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝑦 +𝑜 suc 𝐴)))
51	50	anidms 677	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝑦 +𝑜 suc 𝐴)))
52		nnacom 7697	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 𝑦) = (𝑦 +𝑜 𝐴))
53		suceq 5790	. . . . . . . . . . . 12 ⊢ ((𝐴 +𝑜 𝑦) = (𝑦 +𝑜 𝐴) → suc (𝐴 +𝑜 𝑦) = suc (𝑦 +𝑜 𝐴))
54	52, 53	syl 17	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝐴 +𝑜 𝑦) = suc (𝑦 +𝑜 𝐴))
55		nnasuc 7686	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
56		nnasuc 7686	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +𝑜 suc 𝐴) = suc (𝑦 +𝑜 𝐴))
57	56	ancoms 469	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +𝑜 suc 𝐴) = suc (𝑦 +𝑜 𝐴))
58	54, 55, 57	3eqtr4d 2666	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = (𝑦 +𝑜 suc 𝐴))
59	58	oveq2d 6666	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝑦 +𝑜 suc 𝐴)))
60	51, 59	eqtr4d 2659	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)))
61	43, 45, 60	3eqtr4d 2666	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦) = (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴))
62	36, 61	eqeq12d 2637	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((suc 𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦) ↔ ((suc 𝐴 ·𝑜 𝑦) +𝑜 suc 𝐴) = (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴)))
63	33, 62	syl5ibr 236	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((suc 𝐴 ·𝑜 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝑦) → (suc 𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦)))
64	63	expcom 451	. . . 4 ⊢ (𝑦 ∈ ω → (𝐴 ∈ ω → ((suc 𝐴 ·𝑜 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝑦) → (suc 𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦))))
65	11, 16, 21, 32, 64	finds2 7094	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥)))
66	6, 65	vtoclga 3272	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·𝑜 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵)))
67	66	impcom 446	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ·𝑜 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∅c0 3915  suc csuc 5725  (class class class)co 6650  ωcom 7065   +_𝑜 coa 7557   ·_𝑜 comu 7558

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-omul 7565

This theorem is referenced by:  nnmcom  7706