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Theorem divap1d 8365
Description: If two complex numbers are apart, their quotient is apart from one. (Contributed by Jim Kingdon, 20-Mar-2020.)
Hypotheses
Ref Expression
divcld.1  |-  ( ph  ->  A  e.  CC )
divcld.2  |-  ( ph  ->  B  e.  CC )
divclapd.3  |-  ( ph  ->  B #  0 )
divap1d.4  |-  ( ph  ->  A #  B )
Assertion
Ref Expression
divap1d  |-  ( ph  ->  ( A  /  B
) #  1 )

Proof of Theorem divap1d
StepHypRef Expression
1 divap1d.4 . . 3  |-  ( ph  ->  A #  B )
2 divcld.1 . . . 4  |-  ( ph  ->  A  e.  CC )
3 divcld.2 . . . 4  |-  ( ph  ->  B  e.  CC )
4 divclapd.3 . . . . 5  |-  ( ph  ->  B #  0 )
53, 4recclapd 8345 . . . 4  |-  ( ph  ->  ( 1  /  B
)  e.  CC )
63, 4recap0d 8346 . . . 4  |-  ( ph  ->  ( 1  /  B
) #  0 )
7 apmul1 8352 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  (
( 1  /  B
)  e.  CC  /\  ( 1  /  B
) #  0 ) )  ->  ( A #  B  <->  ( A  x.  ( 1  /  B ) ) #  ( B  x.  (
1  /  B ) ) ) )
82, 3, 5, 6, 7syl112anc 1185 . . 3  |-  ( ph  ->  ( A #  B  <->  ( A  x.  ( 1  /  B
) ) #  ( B  x.  ( 1  /  B ) ) ) )
91, 8mpbid 146 . 2  |-  ( ph  ->  ( A  x.  (
1  /  B ) ) #  ( B  x.  ( 1  /  B
) ) )
102, 3, 4divrecapd 8357 . . 3  |-  ( ph  ->  ( A  /  B
)  =  ( A  x.  ( 1  /  B ) ) )
1110eqcomd 2100 . 2  |-  ( ph  ->  ( A  x.  (
1  /  B ) )  =  ( A  /  B ) )
123, 4recidapd 8347 . 2  |-  ( ph  ->  ( B  x.  (
1  /  B ) )  =  1 )
139, 11, 123brtr3d 3896 1  |-  ( ph  ->  ( A  /  B
) #  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    e. wcel 1445   class class class wbr 3867  (class class class)co 5690   CCcc 7445   0cc0 7447   1c1 7448    x. cmul 7452   # cap 8155    / cdiv 8236
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-rmo 2378  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  df-div 8237
This theorem is referenced by: (None)
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