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Theorem omcl 6072
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 6066 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  +o  A ) ) ,  (/) ) `  B ) )
2 omfnex 6060 . . 3  |-  ( A  e.  On  ->  (
x  e.  _V  |->  ( x  +o  A ) )  Fn  _V )
3 0elon 4157 . . . 4  |-  (/)  e.  On
43a1i 9 . . 3  |-  ( A  e.  On  ->  (/)  e.  On )
5 vex 2577 . . . . . . 7  |-  y  e. 
_V
6 oacl 6071 . . . . . . 7  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( y  +o  A
)  e.  On )
7 oveq1 5547 . . . . . . . 8  |-  ( x  =  y  ->  (
x  +o  A )  =  ( y  +o  A ) )
8 eqid 2056 . . . . . . . 8  |-  ( x  e.  _V  |->  ( x  +o  A ) )  =  ( x  e. 
_V  |->  ( x  +o  A ) )
97, 8fvmptg 5276 . . . . . . 7  |-  ( ( y  e.  _V  /\  ( y  +o  A
)  e.  On )  ->  ( ( x  e.  _V  |->  ( x  +o  A ) ) `
 y )  =  ( y  +o  A
) )
105, 6, 9sylancr 399 . . . . . 6  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( ( x  e. 
_V  |->  ( x  +o  A ) ) `  y )  =  ( y  +o  A ) )
1110, 6eqeltrd 2130 . . . . 5  |-  ( ( y  e.  On  /\  A  e.  On )  ->  ( ( x  e. 
_V  |->  ( x  +o  A ) ) `  y )  e.  On )
1211ancoms 259 . . . 4  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( x  e. 
_V  |->  ( x  +o  A ) ) `  y )  e.  On )
1312ralrimiva 2409 . . 3  |-  ( A  e.  On  ->  A. y  e.  On  ( ( x  e.  _V  |->  ( x  +o  A ) ) `
 y )  e.  On )
142, 4, 13rdgon 6004 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( rec ( ( x  e.  _V  |->  ( x  +o  A ) ) ,  (/) ) `  B )  e.  On )
151, 14eqeltrd 2130 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    e. wcel 1409   _Vcvv 2574   (/)c0 3252    |-> cmpt 3846   Oncon0 4128   ` cfv 4930  (class class class)co 5540   reccrdg 5987    +o coa 6029    .o comu 6030
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 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290
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 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-oadd 6036  df-omul 6037
This theorem is referenced by:  oeicl  6073  omv2  6076  omsuc  6082
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