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Theorem onmsuc 7474
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 6956 . . . . 5 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
2 nnon 6941 . . . . 5 (suc 𝐵 ∈ ω → suc 𝐵 ∈ On)
31, 2syl 17 . . . 4 (𝐵 ∈ ω → suc 𝐵 ∈ On)
4 omv 7457 . . . 4 ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵))
53, 4sylan2 489 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵))
61adantl 480 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → suc 𝐵 ∈ ω)
7 fvres 6102 . . . 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 2646 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 suc 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘suc 𝐵))
10 ovex 6555 . . . . 5 (𝐴 ·𝑜 𝐵) ∈ V
11 oveq1 6534 . . . . . 6 (𝑥 = (𝐴 ·𝑜 𝐵) → (𝑥 +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
12 eqid 2609 . . . . . 6 (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))
13 ovex 6555 . . . . . 6 ((𝐴 ·𝑜 𝐵) +𝑜 𝐴) ∈ V
1411, 12, 13fvmpt 6176 . . . . 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 6941 . . . . . . 7 (𝐵 ∈ ω → 𝐵 ∈ On)
17 omv 7457 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
1816, 17sylan2 489 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
19 fvres 6102 . . . . . . 7 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
2019adantl 480 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
2118, 20eqtr4d 2646 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵))
2221fveq2d 6092 . . . 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 2657 . . 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 7397 . . . 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 480 . . 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 2646 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘suc 𝐵))
279, 26eqtr4d 2646 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 1474  wcel 1976  Vcvv 3172  c0 3873  cmpt 4637  cres 5030  Oncon0 5626  suc csuc 5628  cfv 5790  (class class class)co 6527  ωcom 6935  reccrdg 7370   +𝑜 coa 7422   ·𝑜 comu 7423
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-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
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-om 6936  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-omul 7430
This theorem is referenced by:  om1  7487  nnmsuc  7552
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