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Theorem nnmsucr 6097
 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 5547 . . . . 5 (𝑥 = 𝐵 → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 𝐵))
2 oveq2 5547 . . . . . 6 (𝑥 = 𝐵 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝐵))
3 id 19 . . . . . 6 (𝑥 = 𝐵𝑥 = 𝐵)
42, 3oveq12d 5557 . . . . 5 (𝑥 = 𝐵 → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵))
51, 4eqeq12d 2070 . . . 4 (𝑥 = 𝐵 → ((suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) ↔ (suc 𝐴 ·𝑜 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵)))
65imbi2d 223 . . 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 5547 . . . . 5 (𝑥 = ∅ → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 ∅))
8 oveq2 5547 . . . . . 6 (𝑥 = ∅ → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 ∅))
9 id 19 . . . . . 6 (𝑥 = ∅ → 𝑥 = ∅)
108, 9oveq12d 5557 . . . . 5 (𝑥 = ∅ → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 ∅) +𝑜 ∅))
117, 10eqeq12d 2070 . . . 4 (𝑥 = ∅ → ((suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) ↔ (suc 𝐴 ·𝑜 ∅) = ((𝐴 ·𝑜 ∅) +𝑜 ∅)))
12 oveq2 5547 . . . . 5 (𝑥 = 𝑦 → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 𝑦))
13 oveq2 5547 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 𝑦))
14 id 19 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
1513, 14oveq12d 5557 . . . . 5 (𝑥 = 𝑦 → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 𝑦) +𝑜 𝑦))
1612, 15eqeq12d 2070 . . . 4 (𝑥 = 𝑦 → ((suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) ↔ (suc 𝐴 ·𝑜 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝑦)))
17 oveq2 5547 . . . . 5 (𝑥 = suc 𝑦 → (suc 𝐴 ·𝑜 𝑥) = (suc 𝐴 ·𝑜 suc 𝑦))
18 oveq2 5547 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·𝑜 𝑥) = (𝐴 ·𝑜 suc 𝑦))
19 id 19 . . . . . 6 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
2018, 19oveq12d 5557 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 ·𝑜 𝑥) +𝑜 𝑥) = ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦))
2117, 20eqeq12d 2070 . . . 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 4345 . . . . . . 7 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
23 nnm0 6084 . . . . . . 7 (suc 𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = ∅)
2422, 23syl 14 . . . . . 6 (𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = ∅)
25 nnm0 6084 . . . . . 6 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
2624, 25eqtr4d 2091 . . . . 5 (𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = (𝐴 ·𝑜 ∅))
27 peano1 4344 . . . . . . 7 ∅ ∈ ω
28 nnmcl 6090 . . . . . . 7 ((𝐴 ∈ ω ∧ ∅ ∈ ω) → (𝐴 ·𝑜 ∅) ∈ ω)
2927, 28mpan2 409 . . . . . 6 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) ∈ ω)
30 nna0 6083 . . . . . 6 ((𝐴 ·𝑜 ∅) ∈ ω → ((𝐴 ·𝑜 ∅) +𝑜 ∅) = (𝐴 ·𝑜 ∅))
3129, 30syl 14 . . . . 5 (𝐴 ∈ ω → ((𝐴 ·𝑜 ∅) +𝑜 ∅) = (𝐴 ·𝑜 ∅))
3226, 31eqtr4d 2091 . . . 4 (𝐴 ∈ ω → (suc 𝐴 ·𝑜 ∅) = ((𝐴 ·𝑜 ∅) +𝑜 ∅))
33 oveq1 5546 . . . . . 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 4364 . . . . . . . 8 (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
35 nnmsuc 6086 . . . . . . . 8 ((suc 𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 ·𝑜 suc 𝑦) = ((suc 𝐴 ·𝑜 𝑦) +𝑜 suc 𝐴))
3634, 35sylanb 272 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 ·𝑜 suc 𝑦) = ((suc 𝐴 ·𝑜 𝑦) +𝑜 suc 𝐴))
37 nnmcl 6090 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 𝑦) ∈ ω)
38 peano2b 4364 . . . . . . . . . . . 12 (𝑦 ∈ ω ↔ suc 𝑦 ∈ ω)
39 nnaass 6094 . . . . . . . . . . . 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 1184 . . . . . . . . . . 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 1179 . . . . . . . . . 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 1116 . . . . . . . . 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 383 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝐴) +𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)))
44 nnmsuc 6086 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·𝑜 suc 𝑦) = ((𝐴 ·𝑜 𝑦) +𝑜 𝐴))
4544oveq1d 5554 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦) = (((𝐴 ·𝑜 𝑦) +𝑜 𝐴) +𝑜 suc 𝑦))
46 nnaass 6094 . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . 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 1179 . . . . . . . . . . . 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 1116 . . . . . . . . . . 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 531 . . . . . . . . . 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 383 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝑦 +𝑜 suc 𝐴)))
52 nnacom 6093 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 𝑦) = (𝑦 +𝑜 𝐴))
53 suceq 4166 . . . . . . . . . . . 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 6085 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = suc (𝐴 +𝑜 𝑦))
56 nnasuc 6085 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +𝑜 suc 𝐴) = suc (𝑦 +𝑜 𝐴))
5756ancoms 259 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +𝑜 suc 𝐴) = suc (𝑦 +𝑜 𝐴))
5854, 55, 573eqtr4d 2098 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +𝑜 suc 𝑦) = (𝑦 +𝑜 suc 𝐴))
5958oveq2d 5555 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝑦 +𝑜 suc 𝐴)))
6051, 59eqtr4d 2091 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴) = ((𝐴 ·𝑜 𝑦) +𝑜 (𝐴 +𝑜 suc 𝑦)))
6143, 45, 603eqtr4d 2098 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·𝑜 suc 𝑦) +𝑜 suc 𝑦) = (((𝐴 ·𝑜 𝑦) +𝑜 𝑦) +𝑜 suc 𝐴))
6236, 61eqeq12d 2070 . . . . . 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 149 . . . . 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 113 . . . 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 4351 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·𝑜 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝑥)))
666, 65vtoclga 2636 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·𝑜 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐵)))
6766impcom 120 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 101   = wceq 1259   ∈ wcel 1409  ∅c0 3251  suc csuc 4129  ωcom 4340  (class class class)co 5539   +𝑜 coa 6028   ·𝑜 comu 6029 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-oadd 6035  df-omul 6036 This theorem is referenced by:  nnmcom  6098
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