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Theorem omv2 6045
 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 6029 . . . 4 (𝐴 ∈ On → (𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)) Fn V)
2 0elon 4129 . . . . 5 ∅ ∈ On
3 rdgival 5969 . . . . 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 1220 . . . 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 267 . . 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 6035 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝐵))
7 onelon 4121 . . . . . . 7 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
8 omexg 6031 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ V)
9 omcl 6041 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ On)
10 simpl 102 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
11 oacl 6040 . . . . . . . . . 10 (((𝐴 ·𝑜 𝑥) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) ∈ On)
129, 10, 11syl2anc 391 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) ∈ On)
13 oveq1 5519 . . . . . . . . . 10 (𝑦 = (𝐴 ·𝑜 𝑥) → (𝑦 +𝑜 𝐴) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
14 eqid 2040 . . . . . . . . . 10 (𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)) = (𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))
1513, 14fvmptg 5248 . . . . . . . . 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 391 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))‘(𝐴 ·𝑜 𝑥)) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
17 omv 6035 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) = (rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥))
1817fveq2d 5182 . . . . . . . 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 2074 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥)))
207, 19sylan2 270 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵 ∈ On ∧ 𝑥𝐵)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥)))
2120anassrs 380 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑥𝐵) → ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) = ((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴))‘(rec((𝑦 ∈ V ↦ (𝑦 +𝑜 𝐴)), ∅)‘𝑥)))
2221iuneq2dv 3678 . . . 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 3097 . . 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 2082 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) = (∅ ∪ 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
25 uncom 3087 . . 3 (∅ ∪ 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)) = ( 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) ∪ ∅)
26 un0 3251 . . 3 ( 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴) ∪ ∅) = 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)
2725, 26eqtri 2060 . 2 (∅ ∪ 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)) = 𝑥𝐵 ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)
2824, 27syl6eq 2088 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 97   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ∪ cun 2915  ∅c0 3224  ∪ ciun 3657   ↦ cmpt 3818  Oncon0 4100   Fn wfn 4897  ‘cfv 4902  (class class class)co 5512  reccrdg 5956   +𝑜 coa 5998   ·𝑜 comu 5999 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006 This theorem is referenced by:  omsuc  6051
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