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Theorem nnmsucr 6181
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 5599 . . . . 5 (𝑥 = 𝐵 → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 𝐵))
2 oveq2 5599 . . . . . 6 (𝑥 = 𝐵 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐵))
3 id 19 . . . . . 6 (𝑥 = 𝐵𝑥 = 𝐵)
42, 3oveq12d 5609 . . . . 5 (𝑥 = 𝐵 → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵))
51, 4eqeq12d 2097 . . . 4 (𝑥 = 𝐵 → ((suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) ↔ (suc 𝐴 ·𝑜 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵)))
65imbi2d 228 . . 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 5599 . . . . 5 (𝑥 = ∅ → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 ∅))
8 oveq2 5599 . . . . . 6 (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
9 id 19 . . . . . 6 (𝑥 = ∅ → 𝑥 = ∅)
108, 9oveq12d 5609 . . . . 5 (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 ∅) +𝑜 ∅))
117, 10eqeq12d 2097 . . . 4 (𝑥 = ∅ → ((suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) ↔ (suc 𝐴 ·𝑜 ∅) = ((𝐴 ·𝑜 ∅) +𝑜 ∅)))
12 oveq2 5599 . . . . 5 (𝑥 = 𝑦 → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 𝑦))
13 oveq2 5599 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
14 id 19 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
1513, 14oveq12d 5609 . . . . 5 (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 𝑦) +𝑜 𝑦))
1612, 15eqeq12d 2097 . . . 4 (𝑥 = 𝑦 → ((suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) ↔ (suc 𝐴 ·𝑜 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝑦)))
17 oveq2 5599 . . . . 5 (𝑥 = suc 𝑦 → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 suc 𝑦))
18 oveq2 5599 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
19 id 19 . . . . . 6 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
2018, 19oveq12d 5609 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦))
2117, 20eqeq12d 2097 . . . 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 4373 . . . . . . 7 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
23 nnm0 6168 . . . . . . 7 (suc 𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = ∅)
2422, 23syl 14 . . . . . 6 (𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = ∅)
25 nnm0 6168 . . . . . 6 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
2624, 25eqtr4d 2118 . . . . 5 (𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = (𝐴 ·𝑜 ∅))
27 peano1 4372 . . . . . . 7 ∅ ∈ ω
28 nnmcl 6174 . . . . . . 7 ((𝐴 ∈ ω ∧ ∅ ∈ ω) → (𝐴 ·𝑜 ∅) ∈ ω)
2927, 28mpan2 416 . . . . . 6 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) ∈ ω)
30 nna0 6167 . . . . . 6 ((𝐴 ·𝑜 ∅) ∈ ω → ((𝐴 ·𝑜 ∅) +𝑜 ∅) = (𝐴 ·𝑜 ∅))
3129, 30syl 14 . . . . 5 (𝐴 ∈ ω → ((𝐴 ·𝑜 ∅) +𝑜 ∅) = (𝐴 ·𝑜 ∅))
3226, 31eqtr4d 2118 . . . 4 (𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = ((𝐴 ·𝑜 ∅) +𝑜 ∅))
33 oveq1 5598 . . . . . 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 4392 . . . . . . . 8 (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
35 nnmsuc 6170 . . . . . . . 8 ((suc 𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 ·𝑜 suc 𝑦) = ((suc 𝐴 ·𝑜 𝑦) +𝑜 suc 𝐴))
3634, 35sylanb 278 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 ·𝑜 suc 𝑦) = ((suc 𝐴 ·𝑜 𝑦) +𝑜 suc 𝐴))
37 nnmcl 6174 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 𝑦) ∈ ω)
38 peano2b 4392 . . . . . . . . . . . 12 (𝑦 ∈ ω ↔ suc 𝑦 ∈ ω)
39 nnaass 6178 . . . . . . . . . . . 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 1208 . . . . . . . . . . 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 1203 . . . . . . . . . 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 1140 . . . . . . . . 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 389 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝐴) +𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)))
44 nnmsuc 6170 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
4544oveq1d 5606 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦) = (((𝐴 ·𝑜 𝑦) +𝑜 𝐴) +𝑜 suc 𝑦))
46 nnaass 6178 . . . . . . . . . . . . . 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 1208 . . . . . . . . . . . . 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 1203 . . . . . . . . . . . 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 1140 . . . . . . . . . . 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 554 . . . . . . . . . 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 389 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝑦 +𝑜 suc 𝐴)))
52 nnacom 6177 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 𝑦) = (𝑦 +𝑜 𝐴))
53 suceq 4193 . . . . . . . . . . . 12 ((𝐴 +𝑜 𝑦) = (𝑦 +𝑜 𝐴) → suc (𝐴 +𝑜 𝑦) = suc (𝑦 +𝑜 𝐴))
5452, 53syl 14 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → suc (𝐴 +𝑜 𝑦) = suc (𝑦 +𝑜 𝐴))
55 nnasuc 6169 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
56 nnasuc 6169 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +𝑜 suc 𝐴) = suc (𝑦 +𝑜 𝐴))
5756ancoms 264 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +𝑜 suc 𝐴) = suc (𝑦 +𝑜 𝐴))
5854, 55, 573eqtr4d 2125 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = (𝑦 +𝑜 suc 𝐴))
5958oveq2d 5607 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝑦 +𝑜 suc 𝐴)))
6051, 59eqtr4d 2118 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)))
6143, 45, 603eqtr4d 2125 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦) = (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴))
6236, 61eqeq12d 2097 . . . . . 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 154 . . . . 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 114 . . . 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 4379 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥)))
666, 65vtoclga 2675 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·𝑜 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵)))
6766impcom 123 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ·𝑜 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  c0 3269  suc csuc 4156  ωcom 4368  (class class class)co 5591   +𝑜 coa 6110   ·𝑜 comu 6111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-oadd 6117  df-omul 6118
This theorem is referenced by:  nnmcom  6182
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