Theorem nnmsucr 8234

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 7143	. . . . 5 ⊢ (𝑥 = 𝐵 → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o 𝐵))
2		oveq2 7143	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
3		id 22	. . . . . 6 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
4	2, 3	oveq12d 7153	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o 𝐵) +o 𝐵))
5	1, 4	eqeq12d 2814	. . . 4 ⊢ (𝑥 = 𝐵 → ((suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥) ↔ (suc 𝐴 ·o 𝐵) = ((𝐴 ·o 𝐵) +o 𝐵)))
6	5	imbi2d 344	. . 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 7143	. . . . 5 ⊢ (𝑥 = ∅ → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o ∅))
8		oveq2 7143	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
9		id 22	. . . . . 6 ⊢ (𝑥 = ∅ → 𝑥 = ∅)
10	8, 9	oveq12d 7153	. . . . 5 ⊢ (𝑥 = ∅ → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o ∅) +o ∅))
11	7, 10	eqeq12d 2814	. . . 4 ⊢ (𝑥 = ∅ → ((suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥) ↔ (suc 𝐴 ·o ∅) = ((𝐴 ·o ∅) +o ∅)))
12		oveq2 7143	. . . . 5 ⊢ (𝑥 = 𝑦 → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o 𝑦))
13		oveq2 7143	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
14		id 22	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
15	13, 14	oveq12d 7153	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o 𝑦) +o 𝑦))
16	12, 15	eqeq12d 2814	. . . 4 ⊢ (𝑥 = 𝑦 → ((suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥) ↔ (suc 𝐴 ·o 𝑦) = ((𝐴 ·o 𝑦) +o 𝑦)))
17		oveq2 7143	. . . . 5 ⊢ (𝑥 = suc 𝑦 → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o suc 𝑦))
18		oveq2 7143	. . . . . 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 7153	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o suc 𝑦) +o suc 𝑦))
21	17, 20	eqeq12d 2814	. . . 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 7582	. . . . . . 7 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
23		nnm0 8214	. . . . . . 7 ⊢ (suc 𝐴 ∈ ω → (suc 𝐴 ·o ∅) = ∅)
24	22, 23	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ω → (suc 𝐴 ·o ∅) = ∅)
25		nnm0 8214	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
26	24, 25	eqtr4d 2836	. . . . 5 ⊢ (𝐴 ∈ ω → (suc 𝐴 ·o ∅) = (𝐴 ·o ∅))
27		peano1 7581	. . . . . . 7 ⊢ ∅ ∈ ω
28		nnmcl 8221	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ ∅ ∈ ω) → (𝐴 ·o ∅) ∈ ω)
29	27, 28	mpan2 690	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) ∈ ω)
30		nna0 8213	. . . . . 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 2836	. . . 4 ⊢ (𝐴 ∈ ω → (suc 𝐴 ·o ∅) = ((𝐴 ·o ∅) +o ∅))
33		oveq1 7142	. . . . . 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 7576	. . . . . . . 8 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
35		nnmsuc 8216	. . . . . . . 8 ⊢ ((suc 𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 ·o suc 𝑦) = ((suc 𝐴 ·o 𝑦) +o suc 𝐴))
36	34, 35	sylanb 584	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 ·o suc 𝑦) = ((suc 𝐴 ·o 𝑦) +o suc 𝐴))
37		nnmcl 8221	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o 𝑦) ∈ ω)
38		peano2b 7576	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω ↔ suc 𝑦 ∈ ω)
39		nnaass 8231	. . . . . . . . . . . 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 1402	. . . . . . . . . . 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 1160	. . . . . . . . . 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 1117	. . . . . . . . 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 570	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·o 𝑦) +o 𝐴) +o suc 𝑦) = ((𝐴 ·o 𝑦) +o (𝐴 +o suc 𝑦)))
44		nnmsuc 8216	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
45	44	oveq1d 7150	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o suc 𝑦) +o suc 𝑦) = (((𝐴 ·o 𝑦) +o 𝐴) +o suc 𝑦))
46		nnaass 8231	. . . . . . . . . . . . . 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 1402	. . . . . . . . . . . . 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 1160	. . . . . . . . . . . 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 1117	. . . . . . . . . . 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 660	. . . . . . . . . 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 570	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·o 𝑦) +o 𝑦) +o suc 𝐴) = ((𝐴 ·o 𝑦) +o (𝑦 +o suc 𝐴)))
52		nnacom 8226	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) = (𝑦 +o 𝐴))
53		suceq 6224	. . . . . . . . . . . 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 8215	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
56		nnasuc 8215	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +o suc 𝐴) = suc (𝑦 +o 𝐴))
57	56	ancoms 462	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +o suc 𝐴) = suc (𝑦 +o 𝐴))
58	54, 55, 57	3eqtr4d 2843	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = (𝑦 +o suc 𝐴))
59	58	oveq2d 7151	. . . . . . . . 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 2836	. . . . . . . 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 2843	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o suc 𝑦) +o suc 𝑦) = (((𝐴 ·o 𝑦) +o 𝑦) +o suc 𝐴))
62	36, 61	eqeq12d 2814	. . . . . 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 249	. . . . 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 417	. . . 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 7591	. . 3 ⊢ (𝑥 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥)))
66	6, 65	vtoclga 3522	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·o 𝐵) = ((𝐴 ·o 𝐵) +o 𝐵)))
67	66	impcom 411	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ·o 𝐵) = ((𝐴 ·o 𝐵) +o 𝐵))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∅c0 4243  suc csuc 6161  (class class class)co 7135  ωcom 7560   +_o coa 8082   ·_o comu 8083

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089  df-omul 8090

This theorem is referenced by:  nnmcom  8235