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Theorem omv 6099
Description: Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.)
Assertion
Ref Expression
omv ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem omv
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0elon 4155 . . 3 ∅ ∈ On
2 omfnex 6093 . . . 4 (𝐴 ∈ On → (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) Fn V)
3 rdgexggg 6026 . . . 4 (((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) Fn V ∧ ∅ ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵) ∈ V)
42, 3syl3an1 1203 . . 3 ((𝐴 ∈ On ∧ ∅ ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵) ∈ V)
51, 4mp3an2 1257 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵) ∈ V)
6 oveq2 5551 . . . . . 6 (𝑦 = 𝐴 → (𝑥 +𝑜 𝑦) = (𝑥 +𝑜 𝐴))
76mpteq2dv 3877 . . . . 5 (𝑦 = 𝐴 → (𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)))
8 rdgeq1 6020 . . . . 5 ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) → rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)), ∅) = rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅))
97, 8syl 14 . . . 4 (𝑦 = 𝐴 → rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)), ∅) = rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅))
109fveq1d 5211 . . 3 (𝑦 = 𝐴 → (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝑧))
11 fveq2 5209 . . 3 (𝑧 = 𝐵 → (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝑧) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
12 df-omul 6070 . . 3 ·𝑜 = (𝑦 ∈ On, 𝑧 ∈ On ↦ (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝑦)), ∅)‘𝑧))
1310, 11, 12ovmpt2g 5666 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵) ∈ V) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
145, 13mpd3an3 1270 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  Vcvv 2602  c0 3258  cmpt 3847  Oncon0 4126   Fn wfn 4927  cfv 4932  (class class class)co 5543  reccrdg 6018   +𝑜 coa 6062   ·𝑜 comu 6063
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-iord 4129  df-on 4131  df-suc 4134  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-oadd 6069  df-omul 6070
This theorem is referenced by:  om0  6102  omcl  6105  omv2  6109
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