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Theorem omv2 6456
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 6440 . . . 4 (𝐴 ∈ On → (𝑦 ∈ V ↦ (𝑦 +o 𝐴)) Fn V)
2 0elon 4386 . . . . 5 ∅ ∈ On
3 rdgival 6373 . . . . 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 1325 . . . 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 283 . . 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 6446 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝐵))
7 onelon 4378 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
8 omexg 6442 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ V)
9 omcl 6452 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
10 simpl 109 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
11 oacl 6451 . . . . . . . . . 10 (((𝐴 ·o 𝑥) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝑥) +o 𝐴) ∈ On)
129, 10, 11syl2anc 411 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝐴 ·o 𝑥) +o 𝐴) ∈ On)
13 oveq1 5872 . . . . . . . . . 10 (𝑦 = (𝐴 ·o 𝑥) → (𝑦 +o 𝐴) = ((𝐴 ·o 𝑥) +o 𝐴))
14 eqid 2175 . . . . . . . . . 10 (𝑦 ∈ V ↦ (𝑦 +o 𝐴)) = (𝑦 ∈ V ↦ (𝑦 +o 𝐴))
1513, 14fvmptg 5584 . . . . . . . . 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 411 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(𝐴 ·o 𝑥)) = ((𝐴 ·o 𝑥) +o 𝐴))
17 omv 6446 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥))
1817fveq2d 5511 . . . . . . . 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 2210 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝐴 ·o 𝑥) +o 𝐴) = ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥)))
207, 19sylan2 286 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥𝐵)) → ((𝐴 ·o 𝑥) +o 𝐴) = ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥)))
2120anassrs 400 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥𝐵) → ((𝐴 ·o 𝑥) +o 𝐴) = ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥)))
2221iuneq2dv 3903 . . . 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 3287 . . 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 2218 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (∅ ∪ 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴)))
25 uncom 3277 . . 3 (∅ ∪ 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴)) = ( 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ ∅)
26 un0 3454 . . 3 ( 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ ∅) = 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴)
2725, 26eqtri 2196 . 2 (∅ ∪ 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴)) = 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴)
2824, 27eqtrdi 2224 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 104   = wceq 1353  wcel 2146  Vcvv 2735  cun 3125  c0 3420   ciun 3882  cmpt 4059  Oncon0 4357   Fn wfn 5203  cfv 5208  (class class class)co 5865  reccrdg 6360   +o coa 6404   ·o comu 6405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-oadd 6411  df-omul 6412
This theorem is referenced by:  omsuc  6463
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