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Theorem infxpabs 8048
Description: Absorption law for multiplication with an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
infxpabs  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  -> 
( A  X.  B
)  ~~  A )

Proof of Theorem infxpabs
StepHypRef Expression
1 infxpdom 8047 . . . 4  |-  ( ( A  e.  dom  card  /\ 
om  ~<_  A  /\  B  ~<_  A )  ->  ( A  X.  B )  ~<_  A )
213expa 1153 . . 3  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  B  ~<_  A )  -> 
( A  X.  B
)  ~<_  A )
32adantrl 697 . 2  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  -> 
( A  X.  B
)  ~<_  A )
4 simpll 731 . . 3  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  ->  A  e.  dom  card )
5 numdom 7875 . . . 4  |-  ( ( A  e.  dom  card  /\  B  ~<_  A )  ->  B  e.  dom  card )
65ad2ant2rl 730 . . 3  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  ->  B  e.  dom  card )
7 simprl 733 . . 3  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  ->  B  =/=  (/) )
8 xpdom3 7165 . . 3  |-  ( ( A  e.  dom  card  /\  B  e.  dom  card  /\  B  =/=  (/) )  ->  A  ~<_  ( A  X.  B ) )
94, 6, 7, 8syl3anc 1184 . 2  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  ->  A  ~<_  ( A  X.  B ) )
10 sbth 7186 . 2  |-  ( ( ( A  X.  B
)  ~<_  A  /\  A  ~<_  ( A  X.  B
) )  ->  ( A  X.  B )  ~~  A )
113, 9, 10syl2anc 643 1  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  -> 
( A  X.  B
)  ~~  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721    =/= wne 2567   (/)c0 3588   class class class wbr 4172   omcom 4804    X. cxp 4835   dom cdm 4837    ~~ cen 7065    ~<_ cdom 7066   cardccrd 7778
This theorem is referenced by:  infxp  8051  infmap2  8054
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-card 7782
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