Theorem nnmsucr 8663

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 7439	. . . . 5 ⊢ (𝑥 = 𝐵 → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o 𝐵))
2		oveq2 7439	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
3		id 22	. . . . . 6 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
4	2, 3	oveq12d 7449	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o 𝐵) +o 𝐵))
5	1, 4	eqeq12d 2753	. . . 4 ⊢ (𝑥 = 𝐵 → ((suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥) ↔ (suc 𝐴 ·o 𝐵) = ((𝐴 ·o 𝐵) +o 𝐵)))
6	5	imbi2d 340	. . 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 7439	. . . . 5 ⊢ (𝑥 = ∅ → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o ∅))
8		oveq2 7439	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
9		id 22	. . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
10	8, 9	oveq12d 7449	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o ∅) +o ∅))
11	7, 10	eqeq12d 2753	. . . 4 ⊢ (𝑥 = ∅ → ((suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥) ↔ (suc 𝐴 ·o ∅) = ((𝐴 ·o ∅) +o ∅)))
12		oveq2 7439	. . . . 5 ⊢ (𝑥 = 𝑦 → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o 𝑦))
13		oveq2 7439	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
14		id 22	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
15	13, 14	oveq12d 7449	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o 𝑦) +o 𝑦))
16	12, 15	eqeq12d 2753	. . . 4 ⊢ (𝑥 = 𝑦 → ((suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥) ↔ (suc 𝐴 ·o 𝑦) = ((𝐴 ·o 𝑦) +o 𝑦)))
17		oveq2 7439	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o suc 𝑦))
18		oveq2 7439	. . . . . 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 7449	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o suc 𝑦) +o suc 𝑦))
21	17, 20	eqeq12d 2753	. . . 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 7912	. . . . . . 7 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
23		nnm0 8643	. . . . . . 7 ⊢ (suc 𝐴 ∈ ω → (suc 𝐴 ·o ∅) = ∅)
24	22, 23	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ω → (suc 𝐴 ·o ∅) = ∅)
25		nnm0 8643	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
26	24, 25	eqtr4d 2780	. . . . 5 ⊢ (𝐴 ∈ ω → (suc 𝐴 ·o ∅) = (𝐴 ·o ∅))
27		peano1 7910	. . . . . . 7 ⊢ ∅ ∈ ω
28		nnmcl 8650	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ ∅ ∈ ω) → (𝐴 ·o ∅) ∈ ω)
29	27, 28	mpan2 691	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) ∈ ω)
30		nna0 8642	. . . . . 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 2780	. . . 4 ⊢ (𝐴 ∈ ω → (suc 𝐴 ·o ∅) = ((𝐴 ·o ∅) +o ∅))
33		oveq1 7438	. . . . . 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 7904	. . . . . . . 8 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
35		nnmsuc 8645	. . . . . . . 8 ⊢ ((suc 𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 ·o suc 𝑦) = ((suc 𝐴 ·o 𝑦) +o suc 𝐴))
36	34, 35	sylanb 581	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 ·o suc 𝑦) = ((suc 𝐴 ·o 𝑦) +o suc 𝐴))
37		nnmcl 8650	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o 𝑦) ∈ ω)
38		peano2b 7904	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω ↔ suc 𝑦 ∈ ω)
39		nnaass 8660	. . . . . . . . . . . 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 1407	. . . . . . . . . . 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 1164	. . . . . . . . . 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 1121	. . . . . . . . 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 566	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·o 𝑦) +o 𝐴) +o suc 𝑦) = ((𝐴 ·o 𝑦) +o (𝐴 +o suc 𝑦)))
44		nnmsuc 8645	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
45	44	oveq1d 7446	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o suc 𝑦) +o suc 𝑦) = (((𝐴 ·o 𝑦) +o 𝐴) +o suc 𝑦))
46		nnaass 8660	. . . . . . . . . . . . . 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 1407	. . . . . . . . . . . . 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 1164	. . . . . . . . . . . 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 1121	. . . . . . . . . . 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 661	. . . . . . . . . 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 566	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·o 𝑦) +o 𝑦) +o suc 𝐴) = ((𝐴 ·o 𝑦) +o (𝑦 +o suc 𝐴)))
52		nnacom 8655	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) = (𝑦 +o 𝐴))
53		suceq 6450	. . . . . . . . . . . 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 8644	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
56		nnasuc 8644	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +o suc 𝐴) = suc (𝑦 +o 𝐴))
57	56	ancoms 458	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +o suc 𝐴) = suc (𝑦 +o 𝐴))
58	54, 55, 57	3eqtr4d 2787	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = (𝑦 +o suc 𝐴))
59	58	oveq2d 7447	. . . . . . . . 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 2780	. . . . . . . 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 2787	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o suc 𝑦) +o suc 𝑦) = (((𝐴 ·o 𝑦) +o 𝑦) +o suc 𝐴))
62	36, 61	eqeq12d 2753	. . . . . 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	imbitrrid 246	. . . . 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 413	. . . 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 7920	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥)))
66	6, 65	vtoclga 3577	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·o 𝐵) = ((𝐴 ·o 𝐵) +o 𝐵)))
67	66	impcom 407	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ·o 𝐵) = ((𝐴 ·o 𝐵) +o 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∅c0 4333  suc csuc 6386  (class class class)co 7431  ωcom 7887   +_o coa 8503   ·_o comu 8504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510  df-omul 8511

This theorem is referenced by:  nnmcom  8664