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Theorem nnmsucr 8240
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 7153 . . . . 5 (𝑥 = 𝐵 → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o 𝐵))
2 oveq2 7153 . . . . . 6 (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
3 id 22 . . . . . 6 (𝑥 = 𝐵𝑥 = 𝐵)
42, 3oveq12d 7163 . . . . 5 (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o 𝐵) +o 𝐵))
51, 4eqeq12d 2834 . . . 4 (𝑥 = 𝐵 → ((suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥) ↔ (suc 𝐴 ·o 𝐵) = ((𝐴 ·o 𝐵) +o 𝐵)))
65imbi2d 342 . . 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 7153 . . . . 5 (𝑥 = ∅ → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o ∅))
8 oveq2 7153 . . . . . 6 (𝑥 = ∅ → (𝐴 ·o 𝑥) = (𝐴 ·o ∅))
9 id 22 . . . . . 6 (𝑥 = ∅ → 𝑥 = ∅)
108, 9oveq12d 7163 . . . . 5 (𝑥 = ∅ → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o ∅) +o ∅))
117, 10eqeq12d 2834 . . . 4 (𝑥 = ∅ → ((suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥) ↔ (suc 𝐴 ·o ∅) = ((𝐴 ·o ∅) +o ∅)))
12 oveq2 7153 . . . . 5 (𝑥 = 𝑦 → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o 𝑦))
13 oveq2 7153 . . . . . 6 (𝑥 = 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝑦))
14 id 22 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
1513, 14oveq12d 7163 . . . . 5 (𝑥 = 𝑦 → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o 𝑦) +o 𝑦))
1612, 15eqeq12d 2834 . . . 4 (𝑥 = 𝑦 → ((suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥) ↔ (suc 𝐴 ·o 𝑦) = ((𝐴 ·o 𝑦) +o 𝑦)))
17 oveq2 7153 . . . . 5 (𝑥 = suc 𝑦 → (suc 𝐴 ·o 𝑥) = (suc 𝐴 ·o suc 𝑦))
18 oveq2 7153 . . . . . 6 (𝑥 = suc 𝑦 → (𝐴 ·o 𝑥) = (𝐴 ·o suc 𝑦))
19 id 22 . . . . . 6 (𝑥 = suc 𝑦𝑥 = suc 𝑦)
2018, 19oveq12d 7163 . . . . 5 (𝑥 = suc 𝑦 → ((𝐴 ·o 𝑥) +o 𝑥) = ((𝐴 ·o suc 𝑦) +o suc 𝑦))
2117, 20eqeq12d 2834 . . . 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 7591 . . . . . . 7 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
23 nnm0 8220 . . . . . . 7 (suc 𝐴 ∈ ω → (suc 𝐴 ·o ∅) = ∅)
2422, 23syl 17 . . . . . 6 (𝐴 ∈ ω → (suc 𝐴 ·o ∅) = ∅)
25 nnm0 8220 . . . . . 6 (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
2624, 25eqtr4d 2856 . . . . 5 (𝐴 ∈ ω → (suc 𝐴 ·o ∅) = (𝐴 ·o ∅))
27 peano1 7590 . . . . . . 7 ∅ ∈ ω
28 nnmcl 8227 . . . . . . 7 ((𝐴 ∈ ω ∧ ∅ ∈ ω) → (𝐴 ·o ∅) ∈ ω)
2927, 28mpan2 687 . . . . . 6 (𝐴 ∈ ω → (𝐴 ·o ∅) ∈ ω)
30 nna0 8219 . . . . . 6 ((𝐴 ·o ∅) ∈ ω → ((𝐴 ·o ∅) +o ∅) = (𝐴 ·o ∅))
3129, 30syl 17 . . . . 5 (𝐴 ∈ ω → ((𝐴 ·o ∅) +o ∅) = (𝐴 ·o ∅))
3226, 31eqtr4d 2856 . . . 4 (𝐴 ∈ ω → (suc 𝐴 ·o ∅) = ((𝐴 ·o ∅) +o ∅))
33 oveq1 7152 . . . . . 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 7585 . . . . . . . 8 (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)
35 nnmsuc 8222 . . . . . . . 8 ((suc 𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 ·o suc 𝑦) = ((suc 𝐴 ·o 𝑦) +o suc 𝐴))
3634, 35sylanb 581 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (suc 𝐴 ·o suc 𝑦) = ((suc 𝐴 ·o 𝑦) +o suc 𝐴))
37 nnmcl 8227 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o 𝑦) ∈ ω)
38 peano2b 7585 . . . . . . . . . . . 12 (𝑦 ∈ ω ↔ suc 𝑦 ∈ ω)
39 nnaass 8237 . . . . . . . . . . . 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 1397 . . . . . . . . . . 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 1155 . . . . . . . . . 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 1112 . . . . . . . . 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 567 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·o 𝑦) +o 𝐴) +o suc 𝑦) = ((𝐴 ·o 𝑦) +o (𝐴 +o suc 𝑦)))
44 nnmsuc 8222 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ·o suc 𝑦) = ((𝐴 ·o 𝑦) +o 𝐴))
4544oveq1d 7160 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o suc 𝑦) +o suc 𝑦) = (((𝐴 ·o 𝑦) +o 𝐴) +o suc 𝑦))
46 nnaass 8237 . . . . . . . . . . . . . 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 1397 . . . . . . . . . . . . 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 1155 . . . . . . . . . . . 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 1112 . . . . . . . . . . 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 657 . . . . . . . . . 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 567 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·o 𝑦) +o 𝑦) +o suc 𝐴) = ((𝐴 ·o 𝑦) +o (𝑦 +o suc 𝐴)))
52 nnacom 8232 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o 𝑦) = (𝑦 +o 𝐴))
53 suceq 6249 . . . . . . . . . . . 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 8221 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = suc (𝐴 +o 𝑦))
56 nnasuc 8221 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ 𝐴 ∈ ω) → (𝑦 +o suc 𝐴) = suc (𝑦 +o 𝐴))
5756ancoms 459 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝑦 +o suc 𝐴) = suc (𝑦 +o 𝐴))
5854, 55, 573eqtr4d 2863 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 +o suc 𝑦) = (𝑦 +o suc 𝐴))
5958oveq2d 7161 . . . . . . . . 9 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o 𝑦) +o (𝐴 +o suc 𝑦)) = ((𝐴 ·o 𝑦) +o (𝑦 +o suc 𝐴)))
6051, 59eqtr4d 2856 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → (((𝐴 ·o 𝑦) +o 𝑦) +o suc 𝐴) = ((𝐴 ·o 𝑦) +o (𝐴 +o suc 𝑦)))
6143, 45, 603eqtr4d 2863 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ·o suc 𝑦) +o suc 𝑦) = (((𝐴 ·o 𝑦) +o 𝑦) +o suc 𝐴))
6236, 61eqeq12d 2834 . . . . . 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 247 . . . . 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 414 . . . 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 7599 . . 3 (𝑥 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·o 𝑥) = ((𝐴 ·o 𝑥) +o 𝑥)))
666, 65vtoclga 3571 . 2 (𝐵 ∈ ω → (𝐴 ∈ ω → (suc 𝐴 ·o 𝐵) = ((𝐴 ·o 𝐵) +o 𝐵)))
6766impcom 408 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 396   = wceq 1528  wcel 2105  c0 4288  suc csuc 6186  (class class class)co 7145  ωcom 7569   +o coa 8088   ·o comu 8089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-oadd 8095  df-omul 8096
This theorem is referenced by:  nnmcom  8241
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