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Theorem omv2 6109
Description: Value of ordinal multiplication. (Contributed by Jim Kingdon, 23-Aug-2019.)
Assertion
Ref Expression
omv2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem omv2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 omfnex 6093 . . . 4 (𝐴 ∈ On → (𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)) Fn V)
2 0elon 4155 . . . . 5 ∅ ∈ On
3 rdgival 6031 . . . . 5 (((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)) Fn V ∧ ∅ ∈ On ∧ 𝐵 ∈ On) → (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝐵) = (∅ ∪ 𝑥𝐵 ((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))))
42, 3mp3an2 1257 . . . 4 (((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)) Fn V ∧ 𝐵 ∈ On) → (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝐵) = (∅ ∪ 𝑥𝐵 ((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))))
51, 4sylan 277 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝐵) = (∅ ∪ 𝑥𝐵 ((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))))
6 omv 6099 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝐵))
7 onelon 4147 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
8 omexg 6095 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ V)
9 omcl 6105 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ On)
10 simpl 107 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
11 oacl 6104 . . . . . . . . . 10 (((𝐴 ·𝑜 𝑥) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) ∈ On)
129, 10, 11syl2anc 403 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) ∈ On)
13 oveq1 5550 . . . . . . . . . 10 (𝑦 = (𝐴 ·𝑜 𝑥) → (𝑦 +𝑜 𝐴) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
14 eqid 2082 . . . . . . . . . 10 (𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)) = (𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))
1513, 14fvmptg 5280 . . . . . . . . 9 (((𝐴 ·𝑜 𝑥) ∈ V ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) ∈ On) → ((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))‘(𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
168, 12, 15syl2anc 403 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))‘(𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
17 omv 6099 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))
1817fveq2d 5213 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))‘(𝐴 ·𝑜 𝑥)) = ((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥)))
1916, 18eqtr3d 2116 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥)))
207, 19sylan2 280 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥𝐵)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥)))
2120anassrs 392 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥𝐵) → ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥)))
2221iuneq2dv 3707 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = 𝑥𝐵 ((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥)))
2322uneq2d 3127 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∪ 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)) = (∅ ∪ 𝑥𝐵 ((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))))
245, 6, 233eqtr4d 2124 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (∅ ∪ 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
25 uncom 3117 . . 3 (∅ ∪ 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)) = ( 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) ∪ ∅)
26 un0 3285 . . 3 ( 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) ∪ ∅) = 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)
2725, 26eqtri 2102 . 2 (∅ ∪ 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)) = 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)
2824, 27syl6eq 2130 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  Vcvv 2602  cun 2972  c0 3258   ciun 3686  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:  omsuc  6116
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