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Theorem mulap0 8220
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 8219 . . 3  |-  ( ( B  e.  CC  /\  B #  0 )  ->  E. x  e.  CC  ( B  x.  x )  =  1 )
21adantl 272 . 2  |-  ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  ->  E. x  e.  CC  ( B  x.  x
)  =  1 )
3 simpllr 502 . . . 4  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  A #  0
)
4 simplll 501 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  A  e.  CC )
5 simplrl 503 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  B  e.  CC )
6 simprl 499 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  x  e.  CC )
74, 5, 6mulassd 7608 . . . . 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 500 . . . . . 6  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  ( B  x.  x )  =  1 )
98oveq2d 5706 . . . . 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 7602 . . . . 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 2131 . . . 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 7967 . . . 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 3897 . . 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 7605 . . . 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 7578 . . . 4  |-  ( ( ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) )  /\  ( x  e.  CC  /\  ( B  x.  x
)  =  1 ) )  ->  0  e.  CC )
16 mulext1 8186 . . . 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 1181 . . 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 2507 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 1296    e. wcel 1445   E.wrex 2371   class class class wbr 3867  (class class class)co 5690   CCcc 7445   0cc0 7447   1c1 7448    x. cmul 7452   # cap 8155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-po 4147  df-iso 4148  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156
This theorem is referenced by:  mulap0b  8221  mulap0i  8222  mulap0d  8224  divmuldivap  8276  divdivdivap  8277  divmuleqap  8281  divadddivap  8291  conjmulap  8293  expcl2lemap  10082  expclzaplem  10094
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