MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  infxpidm2 Unicode version

Theorem infxpidm2 7858
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 8397. (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 7800 . . . . . 6  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
21ensymd 7121 . . . . 5  |-  ( A  e.  dom  card  ->  A 
~~  ( card `  A
) )
3 xpen 7233 . . . . 5  |-  ( ( A  ~~  ( card `  A )  /\  A  ~~  ( card `  A
) )  ->  ( A  X.  A )  ~~  ( ( card `  A
)  X.  ( card `  A ) ) )
42, 2, 3syl2anc 643 . . . 4  |-  ( A  e.  dom  card  ->  ( A  X.  A ) 
~~  ( ( card `  A )  X.  ( card `  A ) ) )
54adantr 452 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  X.  A
)  ~~  ( ( card `  A )  X.  ( card `  A
) ) )
6 cardon 7791 . . . 4  |-  ( card `  A )  e.  On
7 cardom 7833 . . . . 5  |-  ( card `  om )  =  om
8 omelon 7561 . . . . . . . 8  |-  om  e.  On
9 onenon 7796 . . . . . . . 8  |-  ( om  e.  On  ->  om  e.  dom  card )
108, 9ax-mp 8 . . . . . . 7  |-  om  e.  dom  card
11 carddom2 7824 . . . . . . 7  |-  ( ( om  e.  dom  card  /\  A  e.  dom  card )  ->  ( ( card `  om )  C_  ( card `  A )  <->  om  ~<_  A ) )
1210, 11mpan 652 . . . . . 6  |-  ( A  e.  dom  card  ->  ( ( card `  om )  C_  ( card `  A
)  <->  om  ~<_  A ) )
1312biimpar 472 . . . . 5  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( card `  om )  C_  ( card `  A )
)
147, 13syl5eqssr 3357 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  ->  om  C_  ( card `  A
) )
15 infxpen 7856 . . . 4  |-  ( ( ( card `  A
)  e.  On  /\  om  C_  ( card `  A
) )  ->  (
( card `  A )  X.  ( card `  A
) )  ~~  ( card `  A ) )
166, 14, 15sylancr 645 . . 3  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( ( card `  A
)  X.  ( card `  A ) )  ~~  ( card `  A )
)
17 entr 7122 . . 3  |-  ( ( ( A  X.  A
)  ~~  ( ( card `  A )  X.  ( card `  A
) )  /\  (
( card `  A )  X.  ( card `  A
) )  ~~  ( card `  A ) )  ->  ( A  X.  A )  ~~  ( card `  A ) )
185, 16, 17syl2anc 643 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  X.  A
)  ~~  ( card `  A ) )
191adantr 452 . 2  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( card `  A )  ~~  A )
20 entr 7122 . 2  |-  ( ( ( A  X.  A
)  ~~  ( card `  A )  /\  ( card `  A )  ~~  A )  ->  ( A  X.  A )  ~~  A )
2118, 19, 20syl2anc 643 1  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A )  -> 
( A  X.  A
)  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721    C_ wss 3284   class class class wbr 4176   Oncon0 4545   omcom 4808    X. cxp 4839   dom cdm 4841   ` cfv 5417    ~~ cen 7069    ~<_ cdom 7070   cardccrd 7782
This theorem is referenced by:  infpwfien  7903  mappwen  7953  infcdaabs  8046  infxpdom  8051  fin67  8235  infxpidm  8397  ttac  27001
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  ax-inf2 7556
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-se 4506  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-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-oi 7439  df-card 7786
  Copyright terms: Public domain W3C validator