Theorem nnmsucr 6573

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 5951	. . . . 5 ⊢ (𝑥 = 𝐵 → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o 𝐵))
2		oveq2 5951	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
3		id 19	. . . . . 6 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
4	2, 3	oveq12d 5961	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o 𝐵) +o 𝐵))
5	1, 4	eqeq12d 2219	. . . 4 ⊢ (𝑥 = 𝐵 → ((suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥) ↔ (suc 𝐴 ·o 𝐵) = ((𝐴 ·o 𝐵) +o 𝐵)))
6	5	imbi2d 230	. . 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 5951	. . . . 5 ⊢ (𝑥 = ∅ → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o ∅))
8		oveq2 5951	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
9		id 19	. . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
10	8, 9	oveq12d 5961	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o ∅) +o ∅))
11	7, 10	eqeq12d 2219	. . . 4 ⊢ (𝑥 = ∅ → ((suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥) ↔ (suc 𝐴 ·o ∅) = ((𝐴 ·o ∅) +o ∅)))
12		oveq2 5951	. . . . 5 ⊢ (𝑥 = 𝑦 → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o 𝑦))
13		oveq2 5951	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
14		id 19	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
15	13, 14	oveq12d 5961	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o 𝑦) +o 𝑦))
16	12, 15	eqeq12d 2219	. . . 4 ⊢ (𝑥 = 𝑦 → ((suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥) ↔ (suc 𝐴 ·o 𝑦) = ((𝐴 ·o 𝑦) +o 𝑦)))
17		oveq2 5951	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o suc 𝑦))
18		oveq2 5951	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
19		id 19	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → 𝑥 = suc 𝑦)
20	18, 19	oveq12d 5961	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o suc 𝑦) +o suc 𝑦))
21	17, 20	eqeq12d 2219	. . . 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 4642	. . . . . . 7 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
23		nnm0 6560	. . . . . . 7 ⊢ (suc 𝐴 ∈ ω → (suc 𝐴 ·o ∅) = ∅)
24	22, 23	syl 14	. . . . . 6 ⊢ (𝐴 ∈ ω → (suc 𝐴 ·o ∅) = ∅)
25		nnm0 6560	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
26	24, 25	eqtr4d 2240	. . . . 5 ⊢ (𝐴 ∈ ω → (suc 𝐴 ·o ∅) = (𝐴 ·o ∅))
27		peano1 4641	. . . . . . 7 ⊢ ∅ ∈ ω
28		nnmcl 6566	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ ∅ ∈ ω) → (𝐴 ·o ∅) ∈ ω)
29	27, 28	mpan2 425	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) ∈ ω)
30		nna0 6559	. . . . . 6 ⊢ ((𝐴 ·o ∅) ∈ ω → ((𝐴 ·o ∅) +o ∅) = (𝐴 ·o ∅))
31	29, 30	syl 14	. . . . 5 ⊢ (𝐴 ∈ ω → ((𝐴 ·o ∅) +o ∅) = (𝐴 ·o ∅))
32	26, 31	eqtr4d 2240	. . . 4 ⊢ (𝐴 ∈ ω → (suc 𝐴 ·o ∅) = ((𝐴 ·o ∅) +o ∅))
33		oveq1 5950	. . . . . 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 4662	. . . . . . . 8 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
35		nnmsuc 6562	. . . . . . . 8 ⊢ ((suc 𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 ·o suc 𝑦) = ((suc 𝐴 ·o 𝑦) +o suc 𝐴))
36	34, 35	sylanb 284	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 ·o suc 𝑦) = ((suc 𝐴 ·o 𝑦) +o suc 𝐴))
37		nnmcl 6566	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o 𝑦) ∈ ω)
38		peano2b 4662	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω ↔ suc 𝑦 ∈ ω)
39		nnaass 6570	. . . . . . . . . . . 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 1287	. . . . . . . . . . 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 1282	. . . . . . . . . 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 1206	. . . . . . . . 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 397	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·o 𝑦) +o 𝐴) +o suc 𝑦) = ((𝐴 ·o 𝑦) +o (𝐴 +o suc 𝑦)))
44		nnmsuc 6562	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
45	44	oveq1d 5958	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o suc 𝑦) +o suc 𝑦) = (((𝐴 ·o 𝑦) +o 𝐴) +o suc 𝑦))
46		nnaass 6570	. . . . . . . . . . . . . 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 1287	. . . . . . . . . . . . 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 1282	. . . . . . . . . . . 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 1206	. . . . . . . . . . 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 589	. . . . . . . . . 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 397	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·o 𝑦) +o 𝑦) +o suc 𝐴) = ((𝐴 ·o 𝑦) +o (𝑦 +o suc 𝐴)))
52		nnacom 6569	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) = (𝑦 +o 𝐴))
53		suceq 4448	. . . . . . . . . . . 12 ⊢ ((𝐴 +o 𝑦) = (𝑦 +o 𝐴) → suc (𝐴 +o 𝑦) = suc (𝑦 +o 𝐴))
54	52, 53	syl 14	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝐴 +o 𝑦) = suc (𝑦 +o 𝐴))
55		nnasuc 6561	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
56		nnasuc 6561	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +o suc 𝐴) = suc (𝑦 +o 𝐴))
57	56	ancoms 268	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +o suc 𝐴) = suc (𝑦 +o 𝐴))
58	54, 55, 57	3eqtr4d 2247	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = (𝑦 +o suc 𝐴))
59	58	oveq2d 5959	. . . . . . . . 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 2240	. . . . . . . 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 2247	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o suc 𝑦) +o suc 𝑦) = (((𝐴 ·o 𝑦) +o 𝑦) +o suc 𝐴))
62	36, 61	eqeq12d 2219	. . . . . 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 156	. . . . 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 116	. . . 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 4648	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥)))
66	6, 65	vtoclga 2838	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·o 𝐵) = ((𝐴 ·o 𝐵) +o 𝐵)))
67	66	impcom 125	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ·o 𝐵) = ((𝐴 ·o 𝐵) +o 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1372   ∈ wcel 2175  ∅c0 3459  suc csuc 4411  ωcom 4637  (class class class)co 5943   +_o coa 6498   ·_o comu 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-oadd 6505  df-omul 6506

This theorem is referenced by:  nnmcom  6574