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Theorem omcl 6071
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 6065 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵))
2 omfnex 6059 . . 3 (𝐴 ∈ On → (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) Fn V)
3 0elon 4156 . . . 4 ∅ ∈ On
43a1i 9 . . 3 (𝐴 ∈ On → ∅ ∈ On)
5 vex 2577 . . . . . . 7 𝑦 ∈ V
6 oacl 6070 . . . . . . 7 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 +𝑜 𝐴) ∈ On)
7 oveq1 5546 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 +𝑜 𝐴) = (𝑦 +𝑜 𝐴))
8 eqid 2056 . . . . . . . 8 (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))
97, 8fvmptg 5275 . . . . . . 7 ((𝑦 ∈ V ∧ (𝑦 +𝑜 𝐴) ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘𝑦) = (𝑦 +𝑜 𝐴))
105, 6, 9sylancr 399 . . . . . 6 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘𝑦) = (𝑦 +𝑜 𝐴))
1110, 6eqeltrd 2130 . . . . 5 ((𝑦 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘𝑦) ∈ On)
1211ancoms 259 . . . 4 ((𝐴 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘𝑦) ∈ On)
1312ralrimiva 2409 . . 3 (𝐴 ∈ On → ∀𝑦 ∈ On ((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴))‘𝑦) ∈ On)
142, 4, 13rdgon 6003 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (rec((𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)), ∅)‘𝐵) ∈ On)
151, 14eqeltrd 2130 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  Vcvv 2574  c0 3251  cmpt 3845  Oncon0 4127  cfv 4929  (class class class)co 5539  reccrdg 5986   +𝑜 coa 6028   ·𝑜 comu 6029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-oadd 6035  df-omul 6036
This theorem is referenced by:  oeicl  6072  omv2  6075  omsuc  6081
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