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Theorem infxp 8055
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 7098 . . 3  |-  ( B 
~<  A  ->  B  ~<_  A )
2 infxpabs 8052 . . . . . 6  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  -> 
( A  X.  B
)  ~~  A )
3 infunabs 8047 . . . . . . . . 9  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  ( A  u.  B )  ~~  A )
433expa 1153 . . . . . . . 8  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  B  ~<_  A )  -> 
( A  u.  B
)  ~~  A )
54adantrl 697 . . . . . . 7  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  -> 
( A  u.  B
)  ~~  A )
65ensymd 7121 . . . . . 6  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  ->  A  ~~  ( A  u.  B ) )
7 entr 7122 . . . . . 6  |-  ( ( ( A  X.  B
)  ~~  A  /\  A  ~~  ( A  u.  B ) )  -> 
( A  X.  B
)  ~~  ( A  u.  B ) )
82, 6, 7syl2anc 643 . . . . 5  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  -> 
( A  X.  B
)  ~~  ( A  u.  B ) )
98expr 599 . . . 4  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  B  =/=  (/) )  ->  ( B  ~<_  A  ->  ( A  X.  B )  ~~  ( A  u.  B
) ) )
109adantrl 697 . . 3  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( B  ~<_  A  ->  ( A  X.  B )  ~~  ( A  u.  B )
) )
111, 10syl5 30 . 2  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( B  ~<  A  ->  ( A  X.  B )  ~~  ( A  u.  B )
) )
12 domtri2 7836 . . . 4  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
1312ad2ant2r 728 . . 3  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( A  ~<_  B  <->  -.  B  ~<  A ) )
14 xpcomeng 7163 . . . . . . 7  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card )  ->  ( A  X.  B )  ~~  ( B  X.  A ) )
1514ad2ant2r 728 . . . . . 6  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( A  X.  B )  ~~  ( B  X.  A ) )
1615adantr 452 . . . . 5  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( A  X.  B )  ~~  ( B  X.  A
) )
17 simplrl 737 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  B  e.  dom  card )
18 simplr 732 . . . . . . . 8  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  om  ~<_  A )
19 domtr 7123 . . . . . . . 8  |-  ( ( om  ~<_  A  /\  A  ~<_  B )  ->  om  ~<_  B )
2018, 19sylan 458 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  om  ~<_  B )
21 infn0 7332 . . . . . . . 8  |-  ( om  ~<_  A  ->  A  =/=  (/) )
2221ad3antlr 712 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  A  =/=  (/) )
23 simpr 448 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  A  ~<_  B )
24 infxpabs 8052 . . . . . . 7  |-  ( ( ( B  e.  dom  card  /\  om  ~<_  B )  /\  ( A  =/=  (/)  /\  A  ~<_  B ) )  -> 
( B  X.  A
)  ~~  B )
2517, 20, 22, 23, 24syl22anc 1185 . . . . . 6  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( B  X.  A )  ~~  B )
26 uncom 3455 . . . . . . . 8  |-  ( A  u.  B )  =  ( B  u.  A
)
27 infunabs 8047 . . . . . . . . 9  |-  ( ( B  e.  dom  card  /\ 
om  ~<_  B  /\  A  ~<_  B )  ->  ( B  u.  A )  ~~  B )
2817, 20, 23, 27syl3anc 1184 . . . . . . . 8  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( B  u.  A )  ~~  B )
2926, 28syl5eqbr 4209 . . . . . . 7  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( A  u.  B )  ~~  B )
3029ensymd 7121 . . . . . 6  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  B  ~~  ( A  u.  B
) )
31 entr 7122 . . . . . 6  |-  ( ( ( B  X.  A
)  ~~  B  /\  B  ~~  ( A  u.  B ) )  -> 
( B  X.  A
)  ~~  ( A  u.  B ) )
3225, 30, 31syl2anc 643 . . . . 5  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( B  X.  A )  ~~  ( A  u.  B
) )
33 entr 7122 . . . . 5  |-  ( ( ( A  X.  B
)  ~~  ( B  X.  A )  /\  ( B  X.  A )  ~~  ( A  u.  B
) )  ->  ( A  X.  B )  ~~  ( A  u.  B
) )
3416, 32, 33syl2anc 643 . . . 4  |-  ( ( ( ( A  e. 
dom  card  /\  om  ~<_  A )  /\  ( B  e. 
dom  card  /\  B  =/=  (/) ) )  /\  A  ~<_  B )  ->  ( A  X.  B )  ~~  ( A  u.  B
) )
3534ex 424 . . 3  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( A  ~<_  B  ->  ( A  X.  B )  ~~  ( A  u.  B )
) )
3613, 35sylbird 227 . 2  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  e.  dom  card  /\  B  =/=  (/) ) )  ->  ( -.  B  ~<  A  ->  ( A  X.  B )  ~~  ( A  u.  B )
) )
3711, 36pm2.61d 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 3    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721    =/= wne 2571    u. cun 3282   (/)c0 3592   class class class wbr 4176   omcom 4808    X. cxp 4839   dom cdm 4841    ~~ cen 7069    ~<_ cdom 7070    ~< csdm 7071   cardccrd 7782
This theorem is referenced by:  alephmul  8413
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-2o 6688  df-oadd 6691  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-oi 7439  df-card 7786  df-cda 8008
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