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Theorem omcl 6363
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  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )

Proof of Theorem omcl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omv 6357 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) )
2 0elon 4320 . . . 4  |-  (/)  e.  On
32a1i 9 . . 3  |-  ( A  e.  On  ->  (/)  e.  On )
4 vex 2692 . . . . . . 7  |-  y  e. 
_V
5 oacl 6362 . . . . . . 7  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( y  +o  A
)  e.  On )
6 oveq1 5787 . . . . . . . 8  |-  ( x  =  y  ->  (
x  +o  A )  =  ( y  +o  A ) )
7 eqid 2140 . . . . . . . 8  |-  ( x  e.  _V  |->  ( x  +o  A ) )  =  ( x  e. 
_V  |->  ( x  +o  A ) )
86, 7fvmptg 5503 . . . . . . 7  |-  ( ( y  e.  _V  /\  ( y  +o  A
)  e.  On )  ->  ( ( x  e.  _V  |->  ( x  +o  A ) ) `
 y )  =  ( y  +o  A
) )
94, 5, 8sylancr 411 . . . . . 6  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( ( x  e. 
_V  |->  ( x  +o  A ) ) `  y )  =  ( y  +o  A ) )
109, 5eqeltrd 2217 . . . . 5  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( ( x  e. 
_V  |->  ( x  +o  A ) ) `  y )  e.  On )
1110ancoms 266 . . . 4  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( x  e. 
_V  |->  ( x  +o  A ) ) `  y )  e.  On )
1211ralrimiva 2508 . . 3  |-  ( A  e.  On  ->  A. y  e.  On  ( ( x  e.  _V  |->  ( x  +o  A ) ) `
 y )  e.  On )
133, 12rdgon 6289 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) ) `  B )  e.  On )
141, 13eqeltrd 2217 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481   _Vcvv 2689   (/)c0 3366    |-> cmpt 3995   Oncon0 4291   ` cfv 5129  (class class class)co 5780   reccrdg 6272    +o coa 6316    .o comu 6317
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 4049  ax-sep 4052  ax-nul 4060  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458
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 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-iun 3821  df-br 3936  df-opab 3996  df-mpt 3997  df-tr 4033  df-id 4221  df-iord 4294  df-on 4296  df-suc 4299  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-res 4557  df-ima 4558  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-f1 5134  df-fo 5135  df-f1o 5136  df-fv 5137  df-ov 5783  df-oprab 5784  df-mpo 5785  df-1st 6044  df-2nd 6045  df-recs 6208  df-irdg 6273  df-oadd 6323  df-omul 6324
This theorem is referenced by:  oeicl  6364  omv2  6367  omsuc  6374
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