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Theorem onmsuc 7849
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 7320 . . . . 5 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
2 nnon 7305 . . . . 5 (suc 𝐵 ∈ ω → suc 𝐵 ∈ On)
31, 2syl 17 . . . 4 (𝐵 ∈ ω → suc 𝐵 ∈ On)
4 omv 7832 . . . 4 ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵))
53, 4sylan2 587 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵))
61adantl 474 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → suc 𝐵 ∈ ω)
7 fvres 6430 . . . 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 2836 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 suc 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘suc 𝐵))
10 ovex 6910 . . . . 5 (𝐴 ·𝑜 𝐵) ∈ V
11 oveq1 6885 . . . . . 6 (𝑥 = (𝐴 ·𝑜 𝐵) → (𝑥 +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
12 eqid 2799 . . . . . 6 (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))
13 ovex 6910 . . . . . 6 ((𝐴 ·𝑜 𝐵) +𝑜 𝐴) ∈ V
1411, 12, 13fvmpt 6507 . . . . 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 7305 . . . . . . 7 (𝐵 ∈ ω → 𝐵 ∈ On)
17 omv 7832 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
1816, 17sylan2 587 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
19 fvres 6430 . . . . . . 7 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
2019adantl 474 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
2118, 20eqtr4d 2836 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵))
2221fveq2d 6415 . . . 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 2847 . . 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 7771 . . . 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 474 . . 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 2836 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘suc 𝐵))
279, 26eqtr4d 2836 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 385   = wceq 1653  wcel 2157  Vcvv 3385  c0 4115  cmpt 4922  cres 5314  Oncon0 5941  suc csuc 5943  cfv 6101  (class class class)co 6878  ωcom 7299  reccrdg 7744   +𝑜 coa 7796   ·𝑜 comu 7797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-omul 7804
This theorem is referenced by:  om1  7862  nnmsuc  7927
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