MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  omxpen Structured version   Unicode version

Theorem omxpen 7616
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 7606 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  X.  B
)  ~~  ( B  X.  A ) )
2 xpexg 6709 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( B  X.  A
)  e.  _V )
32ancoms 453 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  X.  A
)  e.  _V )
4 omcl 7183 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
5 eqid 2467 . . . . 5  |-  ( x  e.  B ,  y  e.  A  |->  ( ( A  .o  x )  +o  y ) )  =  ( x  e.  B ,  y  e.  A  |->  ( ( A  .o  x )  +o  y ) )
65omxpenlem 7615 . . . 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 7529 . . . 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 1228 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  X.  A
)  ~~  ( A  .o  B ) )
9 entr 7564 . . 3  |-  ( ( ( A  X.  B
)  ~~  ( B  X.  A )  /\  ( B  X.  A )  ~~  ( A  .o  B
) )  ->  ( A  X.  B )  ~~  ( A  .o  B
) )
101, 8, 9syl2anc 661 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  X.  B
)  ~~  ( A  .o  B ) )
1110ensymd 7563 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  ~~  ( A  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   _Vcvv 3113   class class class wbr 4447   Oncon0 4878    X. cxp 4997   -1-1-onto->wf1o 5585  (class class class)co 6282    |-> cmpt2 6284    +o coa 7124    .o comu 7125    ~~ cen 7510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-omul 7132  df-er 7308  df-en 7514
This theorem is referenced by:  xpnum  8328  infxpenc2  8395  infxpenc2OLD  8399
  Copyright terms: Public domain W3C validator