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Theorem infxp 7837
Description: Absorption law for multiplication with an infinite cardinal. Equivalent to Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
infxp  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( A  X.  B )  ~~  ( A  u.  B )
)

Proof of Theorem infxp
StepHypRef Expression
1 sdomdom 6885 . . 3  |-  ( B 
~<  A  ->  B  ~<_  A )
2 infxpabs 7834 . . . . . 6  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  -> 
( A  X.  B
)  ~~  A )
3 infunabs 7829 . . . . . . . . 9  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  ( A  u.  B )  ~~  A )
433expa 1151 . . . . . . . 8  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  B  ~<_  A )  -> 
( A  u.  B
)  ~~  A )
54adantrl 696 . . . . . . 7  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  -> 
( A  u.  B
)  ~~  A )
6 ensym 6906 . . . . . . 7  |-  ( ( A  u.  B ) 
~~  A  ->  A  ~~  ( A  u.  B
) )
75, 6syl 15 . . . . . 6  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  ->  A  ~~  ( A  u.  B ) )
8 entr 6909 . . . . . 6  |-  ( ( ( A  X.  B
)  ~~  A  /\  A  ~~  ( A  u.  B ) )  -> 
( A  X.  B
)  ~~  ( A  u.  B ) )
92, 7, 8syl2anc 642 . . . . 5  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  -> 
( A  X.  B
)  ~~  ( A  u.  B ) )
109expr 598 . . . 4  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  B  =/=  (/) )  ->  ( B  ~<_  A  ->  ( A  X.  B )  ~~  ( A  u.  B
) ) )
1110adantrl 696 . . 3  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( B  ~<_  A  ->  ( A  X.  B )  ~~  ( A  u.  B )
) )
121, 11syl5 28 . 2  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( B  ~<  A  ->  ( A  X.  B )  ~~  ( A  u.  B )
) )
13 domtri2 7618 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
1413ad2ant2r 727 . . 3  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
15 xpcomeng 6950 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  X.  B )  ~~  ( B  X.  A ) )
1615ad2ant2r 727 . . . . . 6  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( A  X.  B )  ~~  ( B  X.  A ) )
1716adantr 451 . . . . 5  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( A  X.  B )  ~~  ( B  X.  A
) )
18 simplrl 736 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  B  e.  dom  card )
19 simplr 731 . . . . . . . 8  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  om  ~<_  A )
20 domtr 6910 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  ~<_  B )  ->  om  ~<_  B )
2119, 20sylan 457 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  om  ~<_  B )
22 infn0 7115 . . . . . . . . 9  |-  ( om  ~<_  A  ->  A  =/=  (/) )
2322ad2antlr 707 . . . . . . . 8  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  A  =/=  (/) )
2423adantr 451 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  A  =/=  (/) )
25 simpr 447 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  A  ~<_  B )
26 infxpabs 7834 . . . . . . 7  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( A  =/=  (/)  /\  A  ~<_  B ) )  -> 
( B  X.  A
)  ~~  B )
2718, 21, 24, 25, 26syl22anc 1183 . . . . . 6  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( B  X.  A )  ~~  B )
28 uncom 3320 . . . . . . . 8  |-  ( A  u.  B )  =  ( B  u.  A
)
29 infunabs 7829 . . . . . . . . 9  |-  ( ( B  e.  dom  card  /\ 
om  ~<_  B  /\  A  ~<_  B )  ->  ( B  u.  A )  ~~  B )
3018, 21, 25, 29syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( B  u.  A )  ~~  B )
3128, 30syl5eqbr 4057 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( A  u.  B )  ~~  B )
32 ensym 6906 . . . . . . 7  |-  ( ( A  u.  B ) 
~~  B  ->  B  ~~  ( A  u.  B
) )
3331, 32syl 15 . . . . . 6  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  B  ~~  ( A  u.  B
) )
34 entr 6909 . . . . . 6  |-  ( ( ( B  X.  A
)  ~~  B  /\  B  ~~  ( A  u.  B ) )  -> 
( B  X.  A
)  ~~  ( A  u.  B ) )
3527, 33, 34syl2anc 642 . . . . 5  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( B  X.  A )  ~~  ( A  u.  B
) )
36 entr 6909 . . . . 5  |-  ( ( ( A  X.  B
)  ~~  ( B  X.  A )  /\  ( B  X.  A )  ~~  ( A  u.  B
) )  ->  ( A  X.  B )  ~~  ( A  u.  B
) )
3717, 35, 36syl2anc 642 . . . 4  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( A  X.  B )  ~~  ( A  u.  B
) )
3837ex 423 . . 3  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( A  ~<_  B  ->  ( A  X.  B )  ~~  ( A  u.  B )
) )
3914, 38sylbird 226 . 2  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( -.  B  ~<  A  ->  ( A  X.  B )  ~~  ( A  u.  B )
) )
4012, 39pm2.61d 150 1  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( A  X.  B )  ~~  ( A  u.  B )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1685    =/= wne 2447    u. cun 3151   (/)c0 3456   class class class wbr 4024   omcom 4655    X. cxp 4686    dom cdm 4688    ~~ cen 6856    ~<_ cdom 6857    ~< csdm 6858   cardccrd 7564
This theorem is referenced by:  alephmul  8196  infxpg  24505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-2o 6476  df-oadd 6479  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-oi 7221  df-card 7568  df-cda 7790
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