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Theorem omxpen 7173
Description: The cardinal and ordinal products are always equinumerous. Exercise 10 of [TakeutiZaring] p. 89. (Contributed by Mario Carneiro, 3-Mar-2013.)
Assertion
Ref Expression
omxpen  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  ~~  ( A  X.  B ) )

Proof of Theorem omxpen
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpcomeng 7163 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  X.  B
)  ~~  ( B  X.  A ) )
2 xpexg 4952 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( B  X.  A
)  e.  _V )
32ancoms 440 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  X.  A
)  e.  _V )
4 omcl 6743 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
5 eqid 2408 . . . . 5  |-  ( x  e.  B ,  y  e.  A  |->  ( ( A  .o  x )  +o  y ) )  =  ( x  e.  B ,  y  e.  A  |->  ( ( A  .o  x )  +o  y ) )
65omxpenlem 7172 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( x  e.  B ,  y  e.  A  |->  ( ( A  .o  x )  +o  y
) ) : ( B  X.  A ) -1-1-onto-> ( A  .o  B ) )
7 f1oen2g 7087 . . . 4  |-  ( ( ( B  X.  A
)  e.  _V  /\  ( A  .o  B
)  e.  On  /\  ( x  e.  B ,  y  e.  A  |->  ( ( A  .o  x )  +o  y
) ) : ( B  X.  A ) -1-1-onto-> ( A  .o  B ) )  ->  ( B  X.  A )  ~~  ( A  .o  B ) )
83, 4, 6, 7syl3anc 1184 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  X.  A
)  ~~  ( A  .o  B ) )
9 entr 7122 . . 3  |-  ( ( ( A  X.  B
)  ~~  ( B  X.  A )  /\  ( B  X.  A )  ~~  ( A  .o  B
) )  ->  ( A  X.  B )  ~~  ( A  .o  B
) )
101, 8, 9syl2anc 643 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  X.  B
)  ~~  ( A  .o  B ) )
1110ensymd 7121 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  ~~  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721   _Vcvv 2920   class class class wbr 4176   Oncon0 4545    X. cxp 4839   -1-1-onto->wf1o 5416  (class class class)co 6044    e. cmpt2 6046    +o coa 6684    .o comu 6685    ~~ cen 7069
This theorem is referenced by:  xpnum  7798  infxpenc2  7863
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-omul 6692  df-er 6868  df-en 7073
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