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Theorem omv2 6460
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 6444 . . . 4 (𝐴 ∈ On → (𝑦 ∈ V ↦ (𝑦 +o 𝐴)) Fn V)
2 0elon 4389 . . . . 5 ∅ ∈ On
3 rdgival 6377 . . . . 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 6450 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝐵))
7 onelon 4381 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
8 omexg 6446 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ V)
9 omcl 6456 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
10 simpl 109 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
11 oacl 6455 . . . . . . . . . 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 5876 . . . . . . . . . 10 (𝑦 = (𝐴 ·o 𝑥) → (𝑦 +o 𝐴) = ((𝐴 ·o 𝑥) +o 𝐴))
14 eqid 2177 . . . . . . . . . 10 (𝑦 ∈ V ↦ (𝑦 +o 𝐴)) = (𝑦 ∈ V ↦ (𝑦 +o 𝐴))
1513, 14fvmptg 5588 . . . . . . . . 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 6450 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥))
1817fveq2d 5515 . . . . . . . 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 2212 . . . . . . 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 3905 . . . 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 3289 . . 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 2220 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (∅ ∪ 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴)))
25 uncom 3279 . . 3 (∅ ∪ 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴)) = ( 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ ∅)
26 un0 3456 . . 3 ( 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ ∅) = 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴)
2725, 26eqtri 2198 . 2 (∅ ∪ 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴)) = 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴)
2824, 27eqtrdi 2226 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 2148  Vcvv 2737  cun 3127  c0 3422   ciun 3884  cmpt 4061  Oncon0 4360   Fn wfn 5207  cfv 5212  (class class class)co 5869  reccrdg 6364   +o coa 6408   ·o comu 6409
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-oadd 6415  df-omul 6416
This theorem is referenced by:  omsuc  6467
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