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Theorem omv 6758
 Description: Value of ordinal multiplication. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 23-Aug-2014.)
Assertion
Ref Expression
omv
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem omv
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6091 . . . . 5
21mpteq2dv 4298 . . . 4
3 rdgeq1 6671 . . . 4
42, 3syl 16 . . 3
54fveq1d 5732 . 2
6 fveq2 5730 . 2
7 df-omul 6731 . 2
8 fvex 5744 . 2
95, 6, 7, 8ovmpt2 6211 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  cvv 2958  c0 3630   cmpt 4268  con0 4583  cfv 5456  (class class class)co 6083  crdg 6669   coa 6723   comu 6724 This theorem is referenced by:  om0  6763  omsuc  6772  onmsuc  6775  omlim  6779 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-recs 6635  df-rdg 6670  df-omul 6731
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