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Theorem omcl 6364
Description: Closure law for ordinal multiplication. Proposition 8.16 of [TakeutiZaring] p. 57. (Contributed by NM, 3-Aug-2004.) (Constructive proof by Jim Kingdon, 26-Jul-2019.)
Assertion
Ref Expression
omcl ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)

Proof of Theorem omcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omv 6358 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵))
2 0elon 4321 . . . 4 ∅ ∈ On
32a1i 9 . . 3 (𝐴 ∈ On → ∅ ∈ On)
4 vex 2692 . . . . . . 7 𝑦 ∈ V
5 oacl 6363 . . . . . . 7 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 +o 𝐴) ∈ On)
6 oveq1 5788 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 +o 𝐴) = (𝑦 +o 𝐴))
7 eqid 2140 . . . . . . . 8 (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +o 𝐴))
86, 7fvmptg 5504 . . . . . . 7 ((𝑦 ∈ V ∧ (𝑦 +o 𝐴) ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘𝑦) = (𝑦 +o 𝐴))
94, 5, 8sylancr 411 . . . . . 6 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘𝑦) = (𝑦 +o 𝐴))
109, 5eqeltrd 2217 . . . . 5 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘𝑦) ∈ On)
1110ancoms 266 . . . 4 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘𝑦) ∈ On)
1211ralrimiva 2508 . . 3 (𝐴 ∈ On → ∀𝑦 ∈ On ((𝑥 ∈ V ↦ (𝑥 +o 𝐴))‘𝑦) ∈ On)
133, 12rdgon 6290 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 +o 𝐴)), ∅)‘𝐵) ∈ On)
141, 13eqeltrd 2217 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wcel 1481  Vcvv 2689  c0 3367  cmpt 3996  Oncon0 4292  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:  oeicl  6365  omv2  6368  omsuc  6375
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