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Theorem mulap0 8427
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 8426 . . 3  |-  ( ( B  e.  CC  /\  B #  0 )  ->  E. x  e.  CC  ( B  x.  x )  =  1 )
21adantl 275 . 2  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  E. x  e.  CC  ( B  x.  x
)  =  1 )
3 simpllr 523 . . . 4  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  A #  0
)
4 simplll 522 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  A  e.  CC )
5 simplrl 524 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  B  e.  CC )
6 simprl 520 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  x  e.  CC )
74, 5, 6mulassd 7801 . . . . 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 521 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  ( B  x.  x )  =  1 )
98oveq2d 5790 . . . . 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 ) )
104mulid1d 7795 . . . . 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 2176 . . . 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 8166 . . . 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 3960 . . 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 7798 . . . 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 7771 . . . 4  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  0  e.  CC )
16 mulext1 8386 . . . 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 1216 . . 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 2554 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 103    = wceq 1331    e. wcel 1480   E.wrex 2417   class class class wbr 3929  (class class class)co 5774   CCcc 7630   0cc0 7632   1c1 7633    x. cmul 7637   # cap 8355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356
This theorem is referenced by:  mulap0b  8428  mulap0i  8429  mulap0d  8431  divmuldivap  8484  divdivdivap  8485  divmuleqap  8489  divadddivap  8499  conjmulap  8501  expcl2lemap  10317  expclzaplem  10329
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