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Theorem onmsuc 7594
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 7071 . . . . 5 (𝐵 ∈ ω → suc 𝐵 ∈ ω)
2 nnon 7056 . . . . 5 (suc 𝐵 ∈ ω → suc 𝐵 ∈ On)
31, 2syl 17 . . . 4 (𝐵 ∈ ω → suc 𝐵 ∈ On)
4 omv 7577 . . . 4 ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 ·𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵))
53, 4sylan2 491 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘suc 𝐵))
61adantl 482 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → suc 𝐵 ∈ ω)
7 fvres 6194 . . . 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 2657 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 suc 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘suc 𝐵))
10 ovex 6663 . . . . 5 (𝐴 ·𝑜 𝐵) ∈ V
11 oveq1 6642 . . . . . 6 (𝑥 = (𝐴 ·𝑜 𝐵) → (𝑥 +𝑜 𝐴) = ((𝐴 ·𝑜 𝐵) +𝑜 𝐴))
12 eqid 2620 . . . . . 6 (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))
13 ovex 6663 . . . . . 6 ((𝐴 ·𝑜 𝐵) +𝑜 𝐴) ∈ V
1411, 12, 13fvmpt 6269 . . . . 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 7056 . . . . . . 7 (𝐵 ∈ ω → 𝐵 ∈ On)
17 omv 7577 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
1816, 17sylan2 491 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
19 fvres 6194 . . . . . . 7 (𝐵 ∈ ω → ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
2019adantl 482 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
2118, 20eqtr4d 2657 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘𝐵))
2221fveq2d 6182 . . . 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 2668 . . 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 7517 . . . 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 482 . . 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 2657 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) +𝑜 𝐴) = ((rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅) ↾ ω)‘suc 𝐵))
279, 26eqtr4d 2657 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 384   = wceq 1481  wcel 1988  Vcvv 3195  c0 3907  cmpt 4720  cres 5106  Oncon0 5711  suc csuc 5713  cfv 5876  (class class class)co 6635  ωcom 7050  reccrdg 7490   +𝑜 coa 7542   ·𝑜 comu 7543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-omul 7550
This theorem is referenced by:  om1  7607  nnmsuc  7672
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