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Theorem divap1d 8793
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 8773 . . . 4  |-  ( ph  ->  ( 1  /  B
)  e.  CC )
63, 4recap0d 8774 . . . 4  |-  ( ph  ->  ( 1  /  B
) #  0 )
7 apmul1 8780 . . . 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 1253 . . 3  |-  ( ph  ->  ( A #  B  <->  ( A  x.  ( 1  /  B
) ) #  ( B  x.  ( 1  /  B ) ) ) )
91, 8mpbid 147 . 2  |-  ( ph  ->  ( A  x.  (
1  /  B ) ) #  ( B  x.  ( 1  /  B
) ) )
102, 3, 4divrecapd 8785 . . 3  |-  ( ph  ->  ( A  /  B
)  =  ( A  x.  ( 1  /  B ) ) )
1110eqcomd 2195 . 2  |-  ( ph  ->  ( A  x.  (
1  /  B ) )  =  ( A  /  B ) )
123, 4recidapd 8775 . 2  |-  ( ph  ->  ( B  x.  (
1  /  B ) )  =  1 )
139, 11, 123brtr3d 4052 1  |-  ( ph  ->  ( A  /  B
) #  1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    e. wcel 2160   class class class wbr 4021  (class class class)co 5900   CCcc 7844   0cc0 7846   1c1 7847    x. cmul 7851   # cap 8573    / cdiv 8664
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 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-mulrcl 7945  ax-addcom 7946  ax-mulcom 7947  ax-addass 7948  ax-mulass 7949  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-1rid 7953  ax-0id 7954  ax-rnegex 7955  ax-precex 7956  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-apti 7961  ax-pre-ltadd 7962  ax-pre-mulgt0 7963  ax-pre-mulext 7964
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-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-id 4314  df-po 4317  df-iso 4318  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-iota 5199  df-fun 5240  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-sub 8165  df-neg 8166  df-reap 8567  df-ap 8574  df-div 8665
This theorem is referenced by: (None)
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