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Theorem mulap0 8642
Description: The product of two numbers apart from zero is apart from zero. Lemma 2.15 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 22-Feb-2020.)
Assertion
Ref Expression
mulap0  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( A  x.  B ) #  0 )

Proof of Theorem mulap0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 recexap 8641 . . 3  |-  ( ( B  e.  CC  /\  B #  0 )  ->  E. x  e.  CC  ( B  x.  x )  =  1 )
21adantl 277 . 2  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  E. x  e.  CC  ( B  x.  x
)  =  1 )
3 simpllr 534 . . . 4  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  A #  0
)
4 simplll 533 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  A  e.  CC )
5 simplrl 535 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  B  e.  CC )
6 simprl 529 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  x  e.  CC )
74, 5, 6mulassd 8012 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  ( ( A  x.  B )  x.  x )  =  ( A  x.  ( B  x.  x ) ) )
8 simprr 531 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  ( B  x.  x )  =  1 )
98oveq2d 5913 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  ( A  x.  ( B  x.  x
) )  =  ( A  x.  1 ) )
104mulridd 8005 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  ( A  x.  1 )  =  A )
117, 9, 103eqtrd 2226 . . . 4  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  ( ( A  x.  B )  x.  x )  =  A )
126mul02d 8380 . . . 4  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  ( 0  x.  x )  =  0 )
133, 11, 123brtr4d 4050 . . 3  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  ( ( A  x.  B )  x.  x ) #  ( 0  x.  x ) )
144, 5mulcld 8009 . . . 4  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  ( A  x.  B )  e.  CC )
15 0cnd 7981 . . . 4  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  0  e.  CC )
16 mulext1 8600 . . . 4  |-  ( ( ( A  x.  B
)  e.  CC  /\  0  e.  CC  /\  x  e.  CC )  ->  (
( ( A  x.  B )  x.  x
) #  ( 0  x.  x )  ->  ( A  x.  B ) #  0 ) )
1714, 15, 6, 16syl3anc 1249 . . 3  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  ( (
( A  x.  B
)  x.  x ) #  ( 0  x.  x
)  ->  ( A  x.  B ) #  0 ) )
1813, 17mpd 13 . 2  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  ( A  x.  B ) #  0 )
192, 18rexlimddv 2612 1  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  ( A  x.  B ) #  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   E.wrex 2469   class class class wbr 4018  (class class class)co 5897   CCcc 7840   0cc0 7842   1c1 7843    x. cmul 7847   # cap 8569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570
This theorem is referenced by:  mulap0b  8643  mulap0i  8644  mulap0d  8646  divmuldivap  8700  divdivdivap  8701  divmuleqap  8705  divadddivap  8715  conjmulap  8717  expcl2lemap  10566  expclzaplem  10578  lgsne0  14917
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