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Theorem infxpidm2 7577
Description: The cross 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 8117. (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 7519 . . . . . 6  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
2 ensym 6843 . . . . . 6  |-  ( (
card `  A )  ~~  A  ->  A  ~~  ( card `  A )
)
31, 2syl 17 . . . . 5  |-  ( A  e.  dom  card  ->  A 
~~  ( card `  A
) )
4 xpen 6957 . . . . 5  |-  ( ( A  ~~  ( card `  A )  /\  A  ~~  ( card `  A
) )  ->  ( A  X.  A )  ~~  ( ( card `  A
)  X.  ( card `  A ) ) )
53, 3, 4syl2anc 645 . . . 4  |-  ( A  e.  dom  card  ->  ( A  X.  A ) 
~~  ( ( card `  A )  X.  ( card `  A ) ) )
65adantr 453 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  X.  A
)  ~~  ( ( card `  A )  X.  ( card `  A
) ) )
7 cardon 7510 . . . 4  |-  ( card `  A )  e.  On
8 cardom 7552 . . . . 5  |-  ( card `  om )  =  om
9 omelon 7280 . . . . . . . 8  |-  om  e.  On
10 onenon 7515 . . . . . . . 8  |-  ( om  e.  On  ->  om  e.  dom  card )
119, 10ax-mp 10 . . . . . . 7  |-  om  e.  dom  card
12 carddom2 7543 . . . . . . 7  |-  ( ( om  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  om )  C_  ( card `  A )  <->  om  ~<_  A ) )
1311, 12mpan 654 . . . . . 6  |-  ( A  e.  dom  card  ->  ( ( card `  om )  C_  ( card `  A
)  <->  om  ~<_  A ) )
1413biimpar 473 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( card `  om )  C_  ( card `  A )
)
158, 14syl5eqssr 3165 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  om  C_  ( card `  A
) )
16 infxpen 7575 . . . 4  |-  ( ( ( card `  A
)  e.  On  /\  om  C_  ( card `  A
) )  ->  (
( card `  A )  X.  ( card `  A
) )  ~~  ( card `  A ) )
177, 15, 16sylancr 647 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( card `  A
)  X.  ( card `  A ) )  ~~  ( card `  A )
)
18 entr 6846 . . 3  |-  ( ( ( A  X.  A
)  ~~  ( ( card `  A )  X.  ( card `  A
) )  /\  (
( card `  A )  X.  ( card `  A
) )  ~~  ( card `  A ) )  ->  ( A  X.  A )  ~~  ( card `  A ) )
196, 17, 18syl2anc 645 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  X.  A
)  ~~  ( card `  A ) )
201adantr 453 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( card `  A )  ~~  A )
21 entr 6846 . 2  |-  ( ( ( A  X.  A
)  ~~  ( card `  A )  /\  ( card `  A )  ~~  A )  ->  ( A  X.  A )  ~~  A )
2219, 20, 21syl2anc 645 1  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  X.  A
)  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1621    C_ wss 3094   class class class wbr 3963   Oncon0 4329   omcom 4593    X. cxp 4624   dom cdm 4626   ` cfv 4638    ~~ cen 6793    ~<_ cdom 6794   cardccrd 7501
This theorem is referenced by:  infpwfien  7622  mappwen  7672  infcdaabs  7765  infxpdom  7770  fin67  7954  infxpidm  8117  ttac  26461
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-oadd 6416  df-er 6593  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-oi 7158  df-card 7505
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