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Theorem onmsuc 7496
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 6978 . . . . 5 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
2 nnon 6963 . . . . 5 (suc 𝐵 ∈ ω → suc 𝐵 ∈ On)
31, 2syl 17 . . . 4 (𝐵 ∈ ω → suc 𝐵 ∈ On)
4 omv 7479 . . . 4 ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵))
53, 4sylan2 490 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵))
61adantl 481 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → suc 𝐵 ∈ ω)
7 fvres 6117 . . . 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 2647 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 suc 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘suc 𝐵))
10 ovex 6577 . . . . 5 (𝐴 ·𝑜 𝐵) ∈ V
11 oveq1 6556 . . . . . 6 (𝑥 = (𝐴 ·𝑜 𝐵) → (𝑥 +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
12 eqid 2610 . . . . . 6 (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))
13 ovex 6577 . . . . . 6 ((𝐴 ·𝑜 𝐵) +𝑜 𝐴) ∈ V
1411, 12, 13fvmpt 6191 . . . . 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 6963 . . . . . . 7 (𝐵 ∈ ω → 𝐵 ∈ On)
17 omv 7479 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
1816, 17sylan2 490 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
19 fvres 6117 . . . . . . 7 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
2019adantl 481 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
2118, 20eqtr4d 2647 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵))
2221fveq2d 6107 . . . 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 2658 . . 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 7419 . . . 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 481 . . 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 2647 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘suc 𝐵))
279, 26eqtr4d 2647 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 383   = wceq 1475  wcel 1977  Vcvv 3173  c0 3874  cmpt 4643  cres 5040  Oncon0 5640  suc csuc 5642  cfv 5804  (class class class)co 6549  ωcom 6957  reccrdg 7392   +𝑜 coa 7444   ·𝑜 comu 7445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-omul 7452
This theorem is referenced by:  om1  7509  nnmsuc  7574
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