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Theorem onmsuc 7763
Description: Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
Assertion
Ref Expression
onmsuc ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))

Proof of Theorem onmsuc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 peano2 7233 . . . . 5 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
2 nnon 7218 . . . . 5 (suc 𝐵 ∈ ω → suc 𝐵 ∈ On)
31, 2syl 17 . . . 4 (𝐵 ∈ ω → suc 𝐵 ∈ On)
4 omv 7746 . . . 4 ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵))
53, 4sylan2 580 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵))
61adantl 467 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → suc 𝐵 ∈ ω)
7 fvres 6348 . . . 4 (suc 𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵))
86, 7syl 17 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵))
95, 8eqtr4d 2808 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 suc 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘suc 𝐵))
10 ovex 6823 . . . . 5 (𝐴 ·𝑜 𝐵) ∈ V
11 oveq1 6800 . . . . . 6 (𝑥 = (𝐴 ·𝑜 𝐵) → (𝑥 +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
12 eqid 2771 . . . . . 6 (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))
13 ovex 6823 . . . . . 6 ((𝐴 ·𝑜 𝐵) +𝑜 𝐴) ∈ V
1411, 12, 13fvmpt 6424 . . . . 5 ((𝐴 ·𝑜 𝐵) ∈ V → ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘(𝐴 ·𝑜 𝐵)) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
1510, 14ax-mp 5 . . . 4 ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘(𝐴 ·𝑜 𝐵)) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)
16 nnon 7218 . . . . . . 7 (𝐵 ∈ ω → 𝐵 ∈ On)
17 omv 7746 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
1816, 17sylan2 580 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
19 fvres 6348 . . . . . . 7 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
2019adantl 467 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
2118, 20eqtr4d 2808 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵))
2221fveq2d 6336 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘(𝐴 ·𝑜 𝐵)) = ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵)))
2315, 22syl5eqr 2819 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐴) = ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵)))
24 frsuc 7685 . . . 4 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵)))
2524adantl 467 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵)))
2623, 25eqtr4d 2808 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘suc 𝐵))
279, 26eqtr4d 2808 1 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  c0 4063  cmpt 4863  cres 5251  Oncon0 5866  suc csuc 5868  cfv 6031  (class class class)co 6793  ωcom 7212  reccrdg 7658   +𝑜 coa 7710   ·𝑜 comu 7711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-omul 7718
This theorem is referenced by:  om1  7776  nnmsuc  7841
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