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Theorem infxpidm2 8281
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 8824. (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 8221 . . . . . 6  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
21ensymd 7457 . . . . 5  |-  ( A  e.  dom  card  ->  A 
~~  ( card `  A
) )
3 xpen 7571 . . . . 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 8212 . . . 4  |-  ( card `  A )  e.  On
7 cardom 8254 . . . . 5  |-  ( card `  om )  =  om
8 omelon 7950 . . . . . . . 8  |-  om  e.  On
9 onenon 8217 . . . . . . . 8  |-  ( om  e.  On  ->  om  e.  dom  card )
108, 9ax-mp 5 . . . . . . 7  |-  om  e.  dom  card
11 carddom2 8245 . . . . . . 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 3496 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  om  C_  ( card `  A
) )
15 infxpen 8279 . . . 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 7458 . . 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 7458 . 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 1758    C_ wss 3423   class class class wbr 4387   Oncon0 4814    X. cxp 4933   dom cdm 4935   ` cfv 5513   omcom 6573    ~~ cen 7404    ~<_ cdom 7405   cardccrd 8203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-inf2 7945
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-se 4775  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-isom 5522  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-oi 7822  df-card 8207
This theorem is referenced by:  infpwfien  8330  mappwen  8380  infcdaabs  8473  infxpdom  8478  fin67  8662  infxpidm  8824  ttac  29520
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