Theorem nnmsucr 6541

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 5926	. . . . 5 ⊢ (𝑥 = 𝐵 → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o 𝐵))
2		oveq2 5926	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
3		id 19	. . . . . 6 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
4	2, 3	oveq12d 5936	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o 𝐵) +o 𝐵))
5	1, 4	eqeq12d 2208	. . . 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 5926	. . . . 5 ⊢ (𝑥 = ∅ → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o ∅))
8		oveq2 5926	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
9		id 19	. . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
10	8, 9	oveq12d 5936	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o ∅) +o ∅))
11	7, 10	eqeq12d 2208	. . . 4 ⊢ (𝑥 = ∅ → ((suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥) ↔ (suc 𝐴 ·o ∅) = ((𝐴 ·o ∅) +o ∅)))
12		oveq2 5926	. . . . 5 ⊢ (𝑥 = 𝑦 → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o 𝑦))
13		oveq2 5926	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
14		id 19	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
15	13, 14	oveq12d 5936	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o 𝑦) +o 𝑦))
16	12, 15	eqeq12d 2208	. . . 4 ⊢ (𝑥 = 𝑦 → ((suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥) ↔ (suc 𝐴 ·o 𝑦) = ((𝐴 ·o 𝑦) +o 𝑦)))
17		oveq2 5926	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o suc 𝑦))
18		oveq2 5926	. . . . . 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 5936	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o suc 𝑦) +o suc 𝑦))
21	17, 20	eqeq12d 2208	. . . 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 4627	. . . . . . 7 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
23		nnm0 6528	. . . . . . 7 ⊢ (suc 𝐴 ∈ ω → (suc 𝐴 ·o ∅) = ∅)
24	22, 23	syl 14	. . . . . 6 ⊢ (𝐴 ∈ ω → (suc 𝐴 ·o ∅) = ∅)
25		nnm0 6528	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
26	24, 25	eqtr4d 2229	. . . . 5 ⊢ (𝐴 ∈ ω → (suc 𝐴 ·o ∅) = (𝐴 ·o ∅))
27		peano1 4626	. . . . . . 7 ⊢ ∅ ∈ ω
28		nnmcl 6534	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ ∅ ∈ ω) → (𝐴 ·o ∅) ∈ ω)
29	27, 28	mpan2 425	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) ∈ ω)
30		nna0 6527	. . . . . 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 2229	. . . 4 ⊢ (𝐴 ∈ ω → (suc 𝐴 ·o ∅) = ((𝐴 ·o ∅) +o ∅))
33		oveq1 5925	. . . . . 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 4647	. . . . . . . 8 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
35		nnmsuc 6530	. . . . . . . 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 6534	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o 𝑦) ∈ ω)
38		peano2b 4647	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω ↔ suc 𝑦 ∈ ω)
39		nnaass 6538	. . . . . . . . . . . 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 6530	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
45	44	oveq1d 5933	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o suc 𝑦) +o suc 𝑦) = (((𝐴 ·o 𝑦) +o 𝐴) +o suc 𝑦))
46		nnaass 6538	. . . . . . . . . . . . . 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 6537	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) = (𝑦 +o 𝐴))
53		suceq 4433	. . . . . . . . . . . 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 6529	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
56		nnasuc 6529	. . . . . . . . . . . 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 2236	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = (𝑦 +o suc 𝐴))
59	58	oveq2d 5934	. . . . . . . . 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 2229	. . . . . . . 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 2236	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o suc 𝑦) +o suc 𝑦) = (((𝐴 ·o 𝑦) +o 𝑦) +o suc 𝐴))
62	36, 61	eqeq12d 2208	. . . . . 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 4633	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥)))
66	6, 65	vtoclga 2826	. 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 1364   ∈ wcel 2164  ∅c0 3446  suc csuc 4396  ωcom 4622  (class class class)co 5918   +_o coa 6466   ·_o comu 6467

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-oadd 6473  df-omul 6474

This theorem is referenced by:  nnmcom  6542