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Theorem infxp 7725
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 6775 . . 3  |-  ( B 
~<  A  ->  B  ~<_  A )
2 infxpabs 7722 . . . . . 6  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  -> 
( A  X.  B
)  ~~  A )
3 infunabs 7717 . . . . . . . . 9  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  ( A  u.  B )  ~~  A )
433expa 1156 . . . . . . . 8  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  B  ~<_  A )  -> 
( A  u.  B
)  ~~  A )
54adantrl 699 . . . . . . 7  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  -> 
( A  u.  B
)  ~~  A )
6 ensym 6796 . . . . . . 7  |-  ( ( A  u.  B ) 
~~  A  ->  A  ~~  ( A  u.  B
) )
75, 6syl 17 . . . . . 6  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  ->  A  ~~  ( A  u.  B ) )
8 entr 6798 . . . . . 6  |-  ( ( ( A  X.  B
)  ~~  A  /\  A  ~~  ( A  u.  B ) )  -> 
( A  X.  B
)  ~~  ( A  u.  B ) )
92, 7, 8syl2anc 645 . . . . 5  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  -> 
( A  X.  B
)  ~~  ( A  u.  B ) )
109expr 601 . . . 4  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  B  =/=  (/) )  ->  ( B  ~<_  A  ->  ( A  X.  B )  ~~  ( A  u.  B
) ) )
1110adantrl 699 . . 3  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( B  ~<_  A  ->  ( A  X.  B )  ~~  ( A  u.  B )
) )
121, 11syl5 30 . 2  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( B  ~<  A  ->  ( A  X.  B )  ~~  ( A  u.  B )
) )
13 domtri2 7506 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
1413ad2ant2r 730 . . 3  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
15 xpcomeng 6839 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  X.  B )  ~~  ( B  X.  A ) )
1615ad2ant2r 730 . . . . . 6  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( A  X.  B )  ~~  ( B  X.  A ) )
1716adantr 453 . . . . 5  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( A  X.  B )  ~~  ( B  X.  A
) )
18 simplrl 739 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  B  e.  dom  card )
19 simplr 734 . . . . . . . 8  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  om  ~<_  A )
20 domtr 6799 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  ~<_  B )  ->  om  ~<_  B )
2119, 20sylan 459 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  om  ~<_  B )
22 infn0 7004 . . . . . . . . 9  |-  ( om  ~<_  A  ->  A  =/=  (/) )
2322ad2antlr 710 . . . . . . . 8  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  A  =/=  (/) )
2423adantr 453 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  A  =/=  (/) )
25 simpr 449 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  A  ~<_  B )
26 infxpabs 7722 . . . . . . 7  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( A  =/=  (/)  /\  A  ~<_  B ) )  -> 
( B  X.  A
)  ~~  B )
2718, 21, 24, 25, 26syl22anc 1188 . . . . . 6  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( B  X.  A )  ~~  B )
28 uncom 3229 . . . . . . . 8  |-  ( A  u.  B )  =  ( B  u.  A
)
29 infunabs 7717 . . . . . . . . 9  |-  ( ( B  e.  dom  card  /\ 
om  ~<_  B  /\  A  ~<_  B )  ->  ( B  u.  A )  ~~  B )
3018, 21, 25, 29syl3anc 1187 . . . . . . . 8  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( B  u.  A )  ~~  B )
3128, 30syl5eqbr 3953 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( A  u.  B )  ~~  B )
32 ensym 6796 . . . . . . 7  |-  ( ( A  u.  B ) 
~~  B  ->  B  ~~  ( A  u.  B
) )
3331, 32syl 17 . . . . . 6  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  B  ~~  ( A  u.  B
) )
34 entr 6798 . . . . . 6  |-  ( ( ( B  X.  A
)  ~~  B  /\  B  ~~  ( A  u.  B ) )  -> 
( B  X.  A
)  ~~  ( A  u.  B ) )
3527, 33, 34syl2anc 645 . . . . 5  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( B  X.  A )  ~~  ( A  u.  B
) )
36 entr 6798 . . . . 5  |-  ( ( ( A  X.  B
)  ~~  ( B  X.  A )  /\  ( B  X.  A )  ~~  ( A  u.  B
) )  ->  ( A  X.  B )  ~~  ( A  u.  B
) )
3717, 35, 36syl2anc 645 . . . 4  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( A  X.  B )  ~~  ( A  u.  B
) )
3837ex 425 . . 3  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( A  ~<_  B  ->  ( A  X.  B )  ~~  ( A  u.  B )
) )
3914, 38sylbird 228 . 2  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( -.  B  ~<  A  ->  ( A  X.  B )  ~~  ( A  u.  B )
) )
4012, 39pm2.61d 152 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 5    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1621    =/= wne 2412    u. cun 3076   (/)c0 3362   class class class wbr 3920   omcom 4547    X. cxp 4578   dom cdm 4580    ~~ cen 6746    ~<_ cdom 6747    ~< csdm 6748   cardccrd 7452
This theorem is referenced by:  alephmul  8080  infxpg  24260
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-2o 6366  df-oadd 6369  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-oi 7109  df-card 7456  df-cda 7678
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