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Theorem infxp 7843
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 6891 . . 3  |-  ( B 
~<  A  ->  B  ~<_  A )
2 infxpabs 7840 . . . . . 6  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  -> 
( A  X.  B
)  ~~  A )
3 infunabs 7835 . . . . . . . . 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 6912 . . . . . . 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 6915 . . . . . 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 7624 . . . 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 6956 . . . . . . 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 6916 . . . . . . . 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 7121 . . . . . . . . 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 7840 . . . . . . 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 3321 . . . . . . . 8  |-  ( A  u.  B )  =  ( B  u.  A
)
29 infunabs 7835 . . . . . . . . 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 4058 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( A  u.  B )  ~~  B )
32 ensym 6912 . . . . . . 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 6915 . . . . . 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 6915 . . . . 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 1686    =/= wne 2448    u. cun 3152   (/)c0 3457   class class class wbr 4025   omcom 4658    X. cxp 4689   dom cdm 4691    ~~ cen 6862    ~<_ cdom 6863    ~< csdm 6864   cardccrd 7570
This theorem is referenced by:  alephmul  8202  infxpg  25106
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-oi 7227  df-card 7574  df-cda 7796
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