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Theorem omsuc 7470
Description: Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
omsuc ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))

Proof of Theorem omsuc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 rdgsuc 7384 . . 3 (𝐵 ∈ On → (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵)))
21adantl 480 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵)))
3 suceloni 6882 . . 3 (𝐵 ∈ On → suc 𝐵 ∈ On)
4 omv 7456 . . 3 ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵))
53, 4sylan2 489 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵))
6 ovex 6555 . . . 4 (𝐴 ·𝑜 𝐵) ∈ V
7 oveq1 6534 . . . . 5 (𝑥 = (𝐴 ·𝑜 𝐵) → (𝑥 +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
8 eqid 2609 . . . . 5 (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))
9 ovex 6555 . . . . 5 ((𝐴 ·𝑜 𝐵) +𝑜 𝐴) ∈ V
107, 8, 9fvmpt 6176 . . . 4 ((𝐴 ·𝑜 𝐵) ∈ V → ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘(𝐴 ·𝑜 𝐵)) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
116, 10ax-mp 5 . . 3 ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘(𝐴 ·𝑜 𝐵)) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴)
12 omv 7456 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
1312fveq2d 6092 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘(𝐴 ·𝑜 𝐵)) = ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵)))
1411, 13syl5eqr 2657 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐴) = ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵)))
152, 5, 143eqtr4d 2653 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  c0 3873  cmpt 4637  Oncon0 5626  suc csuc 5628  cfv 5790  (class class class)co 6527  reccrdg 7369   +𝑜 coa 7421   ·𝑜 comu 7422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-omul 7429
This theorem is referenced by:  omcl  7480  om0r  7483  om1r  7487  omordi  7510  omwordri  7516  omlimcl  7522  odi  7523  omass  7524  oneo  7525  omeulem1  7526  omeulem2  7527  oeoelem  7542  oaabs2  7589  omxpenlem  7923  cantnflt  8429  cantnflem1d  8445  infxpenc  8701
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