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Theorem nnmsucr 6231
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 5642 . . . . 5 (𝑥 = 𝐵 → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 𝐵))
2 oveq2 5642 . . . . . 6 (𝑥 = 𝐵 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐵))
3 id 19 . . . . . 6 (𝑥 = 𝐵𝑥 = 𝐵)
42, 3oveq12d 5652 . . . . 5 (𝑥 = 𝐵 → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵))
51, 4eqeq12d 2102 . . . 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 5642 . . . . 5 (𝑥 = ∅ → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 ∅))
8 oveq2 5642 . . . . . 6 (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
9 id 19 . . . . . 6 (𝑥 = ∅ → 𝑥 = ∅)
108, 9oveq12d 5652 . . . . 5 (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 ∅) +𝑜 ∅))
117, 10eqeq12d 2102 . . . 4 (𝑥 = ∅ → ((suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) ↔ (suc 𝐴 ·𝑜 ∅) = ((𝐴 ·𝑜 ∅) +𝑜 ∅)))
12 oveq2 5642 . . . . 5 (𝑥 = 𝑦 → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 𝑦))
13 oveq2 5642 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
14 id 19 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
1513, 14oveq12d 5652 . . . . 5 (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 𝑦) +𝑜 𝑦))
1612, 15eqeq12d 2102 . . . 4 (𝑥 = 𝑦 → ((suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) ↔ (suc 𝐴 ·𝑜 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝑦)))
17 oveq2 5642 . . . . 5 (𝑥 = suc 𝑦 → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 suc 𝑦))
18 oveq2 5642 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
19 id 19 . . . . . 6 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
2018, 19oveq12d 5652 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦))
2117, 20eqeq12d 2102 . . . 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 4400 . . . . . . 7 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
23 nnm0 6218 . . . . . . 7 (suc 𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = ∅)
2422, 23syl 14 . . . . . 6 (𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = ∅)
25 nnm0 6218 . . . . . 6 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
2624, 25eqtr4d 2123 . . . . 5 (𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = (𝐴 ·𝑜 ∅))
27 peano1 4399 . . . . . . 7 ∅ ∈ ω
28 nnmcl 6224 . . . . . . 7 ((𝐴 ∈ ω ∧ ∅ ∈ ω) → (𝐴 ·𝑜 ∅) ∈ ω)
2927, 28mpan2 416 . . . . . 6 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) ∈ ω)
30 nna0 6217 . . . . . 6 ((𝐴 ·𝑜 ∅) ∈ ω → ((𝐴 ·𝑜 ∅) +𝑜 ∅) = (𝐴 ·𝑜 ∅))
3129, 30syl 14 . . . . 5 (𝐴 ∈ ω → ((𝐴 ·𝑜 ∅) +𝑜 ∅) = (𝐴 ·𝑜 ∅))
3226, 31eqtr4d 2123 . . . 4 (𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = ((𝐴 ·𝑜 ∅) +𝑜 ∅))
33 oveq1 5641 . . . . . 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 4419 . . . . . . . 8 (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
35 nnmsuc 6220 . . . . . . . 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 6224 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 𝑦) ∈ ω)
38 peano2b 4419 . . . . . . . . . . . 12 (𝑦 ∈ ω ↔ suc 𝑦 ∈ ω)
39 nnaass 6228 . . . . . . . . . . . 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 1212 . . . . . . . . . . 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 1207 . . . . . . . . . 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 1144 . . . . . . . . 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 6220 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
4544oveq1d 5649 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦) = (((𝐴 ·𝑜 𝑦) +𝑜 𝐴) +𝑜 suc 𝑦))
46 nnaass 6228 . . . . . . . . . . . . . 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 1212 . . . . . . . . . . . . 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 1207 . . . . . . . . . . . 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 1144 . . . . . . . . . . 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 556 . . . . . . . . . 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 6227 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 𝑦) = (𝑦 +𝑜 𝐴))
53 suceq 4220 . . . . . . . . . . . 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 6219 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
56 nnasuc 6219 . . . . . . . . . . . 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 2130 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = (𝑦 +𝑜 suc 𝐴))
5958oveq2d 5650 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝑦 +𝑜 suc 𝐴)))
6051, 59eqtr4d 2123 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)))
6143, 45, 603eqtr4d 2130 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦) = (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴))
6236, 61eqeq12d 2102 . . . . . 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 4406 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥)))
666, 65vtoclga 2685 . 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 1289  wcel 1438  c0 3284  suc csuc 4183  ωcom 4395  (class class class)co 5634   +𝑜 coa 6160   ·𝑜 comu 6161
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-oadd 6167  df-omul 6168
This theorem is referenced by:  nnmcom  6232
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