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Theorem omv2 6368
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 6352 . . . 4 (𝐴 ∈ On → (𝑦 ∈ V ↦ (𝑦 +o 𝐴)) Fn V)
2 0elon 4321 . . . . 5 ∅ ∈ On
3 rdgival 6286 . . . . 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 1304 . . . 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 281 . . 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 6358 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝐵))
7 onelon 4313 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
8 omexg 6354 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ V)
9 omcl 6364 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
10 simpl 108 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
11 oacl 6363 . . . . . . . . . 10 (((𝐴 ·o 𝑥) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝑥) +o 𝐴) ∈ On)
129, 10, 11syl2anc 409 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝐴 ·o 𝑥) +o 𝐴) ∈ On)
13 oveq1 5788 . . . . . . . . . 10 (𝑦 = (𝐴 ·o 𝑥) → (𝑦 +o 𝐴) = ((𝐴 ·o 𝑥) +o 𝐴))
14 eqid 2140 . . . . . . . . . 10 (𝑦 ∈ V ↦ (𝑦 +o 𝐴)) = (𝑦 ∈ V ↦ (𝑦 +o 𝐴))
1513, 14fvmptg 5504 . . . . . . . . 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 409 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(𝐴 ·o 𝑥)) = ((𝐴 ·o 𝑥) +o 𝐴))
17 omv 6358 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥))
1817fveq2d 5432 . . . . . . . 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 2175 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝐴 ·o 𝑥) +o 𝐴) = ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥)))
207, 19sylan2 284 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥𝐵)) → ((𝐴 ·o 𝑥) +o 𝐴) = ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥)))
2120anassrs 398 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥𝐵) → ((𝐴 ·o 𝑥) +o 𝐴) = ((𝑦 ∈ V ↦ (𝑦 +o 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +o 𝐴)), ∅)‘𝑥)))
2221iuneq2dv 3841 . . . 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 3234 . . 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 2183 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (∅ ∪ 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴)))
25 uncom 3224 . . 3 (∅ ∪ 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴)) = ( 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ ∅)
26 un0 3400 . . 3 ( 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ ∅) = 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴)
2725, 26eqtri 2161 . 2 (∅ ∪ 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴)) = 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴)
2824, 27eqtrdi 2189 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 103   = wceq 1332  wcel 1481  Vcvv 2689  cun 3073  c0 3367   ciun 3820  cmpt 3996  Oncon0 4292   Fn wfn 5125  cfv 5130  (class class class)co 5781  reccrdg 6273   +o coa 6317   ·o comu 6318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-suc 4300  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-oadd 6324  df-omul 6325
This theorem is referenced by:  omsuc  6375
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