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Theorem nnmsucr 7939
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.
StepHypRef Expression
1 oveq2 6879 . . . . 5 (𝑥 = 𝐵 → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 𝐵))
2 oveq2 6879 . . . . . 6 (𝑥 = 𝐵 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐵))
3 id 22 . . . . . 6 (𝑥 = 𝐵𝑥 = 𝐵)
42, 3oveq12d 6889 . . . . 5 (𝑥 = 𝐵 → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵))
51, 4eqeq12d 2820 . . . 4 (𝑥 = 𝐵 → ((suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) ↔ (suc 𝐴 ·𝑜 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵)))
65imbi2d 331 . . 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 6879 . . . . 5 (𝑥 = ∅ → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 ∅))
8 oveq2 6879 . . . . . 6 (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
9 id 22 . . . . . 6 (𝑥 = ∅ → 𝑥 = ∅)
108, 9oveq12d 6889 . . . . 5 (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 ∅) +𝑜 ∅))
117, 10eqeq12d 2820 . . . 4 (𝑥 = ∅ → ((suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) ↔ (suc 𝐴 ·𝑜 ∅) = ((𝐴 ·𝑜 ∅) +𝑜 ∅)))
12 oveq2 6879 . . . . 5 (𝑥 = 𝑦 → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 𝑦))
13 oveq2 6879 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
14 id 22 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
1513, 14oveq12d 6889 . . . . 5 (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 𝑦) +𝑜 𝑦))
1612, 15eqeq12d 2820 . . . 4 (𝑥 = 𝑦 → ((suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) ↔ (suc 𝐴 ·𝑜 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝑦)))
17 oveq2 6879 . . . . 5 (𝑥 = suc 𝑦 → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 suc 𝑦))
18 oveq2 6879 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
19 id 22 . . . . . 6 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
2018, 19oveq12d 6889 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦))
2117, 20eqeq12d 2820 . . . 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 7313 . . . . . . 7 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
23 nnm0 7919 . . . . . . 7 (suc 𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = ∅)
2422, 23syl 17 . . . . . 6 (𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = ∅)
25 nnm0 7919 . . . . . 6 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
2624, 25eqtr4d 2842 . . . . 5 (𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = (𝐴 ·𝑜 ∅))
27 peano1 7312 . . . . . . 7 ∅ ∈ ω
28 nnmcl 7926 . . . . . . 7 ((𝐴 ∈ ω ∧ ∅ ∈ ω) → (𝐴 ·𝑜 ∅) ∈ ω)
2927, 28mpan2 674 . . . . . 6 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) ∈ ω)
30 nna0 7918 . . . . . 6 ((𝐴 ·𝑜 ∅) ∈ ω → ((𝐴 ·𝑜 ∅) +𝑜 ∅) = (𝐴 ·𝑜 ∅))
3129, 30syl 17 . . . . 5 (𝐴 ∈ ω → ((𝐴 ·𝑜 ∅) +𝑜 ∅) = (𝐴 ·𝑜 ∅))
3226, 31eqtr4d 2842 . . . 4 (𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = ((𝐴 ·𝑜 ∅) +𝑜 ∅))
33 oveq1 6878 . . . . . 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 7308 . . . . . . . 8 (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
35 nnmsuc 7921 . . . . . . . 8 ((suc 𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 ·𝑜 suc 𝑦) = ((suc 𝐴 ·𝑜 𝑦) +𝑜 suc 𝐴))
3634, 35sylanb 572 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 ·𝑜 suc 𝑦) = ((suc 𝐴 ·𝑜 𝑦) +𝑜 suc 𝐴))
37 nnmcl 7926 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 𝑦) ∈ ω)
38 peano2b 7308 . . . . . . . . . . . 12 (𝑦 ∈ ω ↔ suc 𝑦 ∈ ω)
39 nnaass 7936 . . . . . . . . . . . 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)))
4038, 39syl3an3b 1517 . . . . . . . . . . 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)))
4137, 40syl3an1 1195 . . . . . . . . . 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)))
42413expb 1142 . . . . . . . . 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)))
4342anidms 558 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝐴) +𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)))
44 nnmsuc 7921 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
4544oveq1d 6886 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦) = (((𝐴 ·𝑜 𝑦) +𝑜 𝐴) +𝑜 suc 𝑦))
46 nnaass 7936 . . . . . . . . . . . . . 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)))
4734, 46syl3an3b 1517 . . . . . . . . . . . . 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)))
4837, 47syl3an1 1195 . . . . . . . . . . . 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)))
49483expb 1142 . . . . . . . . . . 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)))
5049an42s 643 . . . . . . . . . 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)))
5150anidms 558 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝑦 +𝑜 suc 𝐴)))
52 nnacom 7931 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 𝑦) = (𝑦 +𝑜 𝐴))
53 suceq 6000 . . . . . . . . . . . 12 ((𝐴 +𝑜 𝑦) = (𝑦 +𝑜 𝐴) → suc (𝐴 +𝑜 𝑦) = suc (𝑦 +𝑜 𝐴))
5452, 53syl 17 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝐴 +𝑜 𝑦) = suc (𝑦 +𝑜 𝐴))
55 nnasuc 7920 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
56 nnasuc 7920 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +𝑜 suc 𝐴) = suc (𝑦 +𝑜 𝐴))
5756ancoms 448 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +𝑜 suc 𝐴) = suc (𝑦 +𝑜 𝐴))
5854, 55, 573eqtr4d 2849 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = (𝑦 +𝑜 suc 𝐴))
5958oveq2d 6887 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝑦 +𝑜 suc 𝐴)))
6051, 59eqtr4d 2842 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)))
6143, 45, 603eqtr4d 2849 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦) = (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴))
6236, 61eqeq12d 2820 . . . . . 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)))
6333, 62syl5ibr 237 . . . . 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)))
6463expcom 400 . . . 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))))
6511, 16, 21, 32, 64finds2 7321 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥)))
666, 65vtoclga 3464 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·𝑜 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵)))
6766impcom 396 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ·𝑜 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2158  c0 4113  suc csuc 5935  (class class class)co 6871  ωcom 7292   +𝑜 coa 7790   ·𝑜 comu 7791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-ral 3100  df-rex 3101  df-reu 3102  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-pss 3782  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4627  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-tr 4943  df-id 5216  df-eprel 5221  df-po 5229  df-so 5230  df-fr 5267  df-we 5269  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-pred 5890  df-ord 5936  df-on 5937  df-lim 5938  df-suc 5939  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-om 7293  df-wrecs 7639  df-recs 7701  df-rdg 7739  df-oadd 7797  df-omul 7798
This theorem is referenced by:  nnmcom  7940
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