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Theorem omcl 6487
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 6481 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) )
2 0elon 4410 . . . 4  |-  (/)  e.  On
32a1i 9 . . 3  |-  ( A  e.  On  ->  (/)  e.  On )
4 vex 2755 . . . . . . 7  |-  y  e. 
_V
5 oacl 6486 . . . . . . 7  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( y  +o  A
)  e.  On )
6 oveq1 5904 . . . . . . . 8  |-  ( x  =  y  ->  (
x  +o  A )  =  ( y  +o  A ) )
7 eqid 2189 . . . . . . . 8  |-  ( x  e.  _V  |->  ( x  +o  A ) )  =  ( x  e. 
_V  |->  ( x  +o  A ) )
86, 7fvmptg 5613 . . . . . . 7  |-  ( ( y  e.  _V  /\  ( y  +o  A
)  e.  On )  ->  ( ( x  e.  _V  |->  ( x  +o  A ) ) `
 y )  =  ( y  +o  A
) )
94, 5, 8sylancr 414 . . . . . 6  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( ( x  e. 
_V  |->  ( x  +o  A ) ) `  y )  =  ( y  +o  A ) )
109, 5eqeltrd 2266 . . . . 5  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( ( x  e. 
_V  |->  ( x  +o  A ) ) `  y )  e.  On )
1110ancoms 268 . . . 4  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( x  e. 
_V  |->  ( x  +o  A ) ) `  y )  e.  On )
1211ralrimiva 2563 . . 3  |-  ( A  e.  On  ->  A. y  e.  On  ( ( x  e.  _V  |->  ( x  +o  A ) ) `
 y )  e.  On )
133, 12rdgon 6412 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) ) `  B )  e.  On )
141, 13eqeltrd 2266 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   _Vcvv 2752   (/)c0 3437    |-> cmpt 4079   Oncon0 4381   ` cfv 5235  (class class class)co 5897   reccrdg 6395    +o coa 6439    .o comu 6440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-oadd 6446  df-omul 6447
This theorem is referenced by:  oeicl  6488  omv2  6491  omsuc  6498
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