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Theorem infxpidm2 8390
Description: The Cartesian product of an infinite set with itself is idempotent. This theorem provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. See also infxpidm 8933. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
infxpidm2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  X.  A
)  ~~  A )

Proof of Theorem infxpidm2
StepHypRef Expression
1 cardid2 8330 . . . . . 6  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
21ensymd 7563 . . . . 5  |-  ( A  e.  dom  card  ->  A 
~~  ( card `  A
) )
3 xpen 7677 . . . . 5  |-  ( ( A  ~~  ( card `  A )  /\  A  ~~  ( card `  A
) )  ->  ( A  X.  A )  ~~  ( ( card `  A
)  X.  ( card `  A ) ) )
42, 2, 3syl2anc 661 . . . 4  |-  ( A  e.  dom  card  ->  ( A  X.  A ) 
~~  ( ( card `  A )  X.  ( card `  A ) ) )
54adantr 465 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  X.  A
)  ~~  ( ( card `  A )  X.  ( card `  A
) ) )
6 cardon 8321 . . . 4  |-  ( card `  A )  e.  On
7 cardom 8363 . . . . 5  |-  ( card `  om )  =  om
8 omelon 8059 . . . . . . . 8  |-  om  e.  On
9 onenon 8326 . . . . . . . 8  |-  ( om  e.  On  ->  om  e.  dom  card )
108, 9ax-mp 5 . . . . . . 7  |-  om  e.  dom  card
11 carddom2 8354 . . . . . . 7  |-  ( ( om  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  om )  C_  ( card `  A )  <->  om  ~<_  A ) )
1210, 11mpan 670 . . . . . 6  |-  ( A  e.  dom  card  ->  ( ( card `  om )  C_  ( card `  A
)  <->  om  ~<_  A ) )
1312biimpar 485 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( card `  om )  C_  ( card `  A )
)
147, 13syl5eqssr 3549 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  om  C_  ( card `  A
) )
15 infxpen 8388 . . . 4  |-  ( ( ( card `  A
)  e.  On  /\  om  C_  ( card `  A
) )  ->  (
( card `  A )  X.  ( card `  A
) )  ~~  ( card `  A ) )
166, 14, 15sylancr 663 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( card `  A
)  X.  ( card `  A ) )  ~~  ( card `  A )
)
17 entr 7564 . . 3  |-  ( ( ( A  X.  A
)  ~~  ( ( card `  A )  X.  ( card `  A
) )  /\  (
( card `  A )  X.  ( card `  A
) )  ~~  ( card `  A ) )  ->  ( A  X.  A )  ~~  ( card `  A ) )
185, 16, 17syl2anc 661 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  X.  A
)  ~~  ( card `  A ) )
191adantr 465 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( card `  A )  ~~  A )
20 entr 7564 . 2  |-  ( ( ( A  X.  A
)  ~~  ( card `  A )  /\  ( card `  A )  ~~  A )  ->  ( A  X.  A )  ~~  A )
2118, 19, 20syl2anc 661 1  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  X.  A
)  ~~  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767    C_ wss 3476   class class class wbr 4447   Oncon0 4878    X. cxp 4997   dom cdm 4999   ` cfv 5586   omcom 6678    ~~ cen 7510    ~<_ cdom 7511   cardccrd 8312
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  ax-inf2 8054
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-se 4839  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-isom 5595  df-riota 6243  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-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-oi 7931  df-card 8316
This theorem is referenced by:  infpwfien  8439  mappwen  8489  infcdaabs  8582  infxpdom  8587  fin67  8771  infxpidm  8933  ttac  30582
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