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Theorem conjmulap 8703
Description: Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. (Contributed by Jim Kingdon, 26-Feb-2020.)
Assertion
Ref Expression
conjmulap  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( ( 1  /  P )  +  ( 1  /  Q ) )  =  1  <->  ( ( P  -  1 )  x.  ( Q  -  1 ) )  =  1 ) )

Proof of Theorem conjmulap
StepHypRef Expression
1 simpll 527 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  P  e.  CC )
2 simprl 529 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  Q  e.  CC )
3 recclap 8653 . . . . . . . 8  |-  ( ( P  e.  CC  /\  P #  0 )  ->  (
1  /  P )  e.  CC )
43adantr 276 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( 1  /  P )  e.  CC )
51, 2, 4mul32d 8127 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  ( 1  /  P
) )  =  ( ( P  x.  (
1  /  P ) )  x.  Q ) )
6 recidap 8660 . . . . . . . 8  |-  ( ( P  e.  CC  /\  P #  0 )  ->  ( P  x.  ( 1  /  P ) )  =  1 )
76oveq1d 5905 . . . . . . 7  |-  ( ( P  e.  CC  /\  P #  0 )  ->  (
( P  x.  (
1  /  P ) )  x.  Q )  =  ( 1  x.  Q ) )
87adantr 276 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  ( 1  /  P ) )  x.  Q )  =  ( 1  x.  Q ) )
9 mullid 7972 . . . . . . 7  |-  ( Q  e.  CC  ->  (
1  x.  Q )  =  Q )
109ad2antrl 490 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( 1  x.  Q )  =  Q )
115, 8, 103eqtrd 2225 . . . . 5  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  ( 1  /  P
) )  =  Q )
12 recclap 8653 . . . . . . . 8  |-  ( ( Q  e.  CC  /\  Q #  0 )  ->  (
1  /  Q )  e.  CC )
1312adantl 277 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( 1  /  Q )  e.  CC )
141, 2, 13mulassd 7998 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  ( 1  /  Q
) )  =  ( P  x.  ( Q  x.  ( 1  /  Q ) ) ) )
15 recidap 8660 . . . . . . . 8  |-  ( ( Q  e.  CC  /\  Q #  0 )  ->  ( Q  x.  ( 1  /  Q ) )  =  1 )
1615oveq2d 5906 . . . . . . 7  |-  ( ( Q  e.  CC  /\  Q #  0 )  ->  ( P  x.  ( Q  x.  ( 1  /  Q
) ) )  =  ( P  x.  1 ) )
1716adantl 277 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  x.  ( Q  x.  (
1  /  Q ) ) )  =  ( P  x.  1 ) )
18 mulrid 7971 . . . . . . 7  |-  ( P  e.  CC  ->  ( P  x.  1 )  =  P )
1918ad2antrr 488 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  x.  1 )  =  P )
2014, 17, 193eqtrd 2225 . . . . 5  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  ( 1  /  Q
) )  =  P )
2111, 20oveq12d 5908 . . . 4  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( ( P  x.  Q )  x.  ( 1  /  P ) )  +  ( ( P  x.  Q )  x.  (
1  /  Q ) ) )  =  ( Q  +  P ) )
22 mulcl 7955 . . . . . 6  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( P  x.  Q
)  e.  CC )
2322ad2ant2r 509 . . . . 5  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  x.  Q )  e.  CC )
2423, 4, 13adddid 7999 . . . 4  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  ( ( 1  /  P )  +  ( 1  /  Q ) ) )  =  ( ( ( P  x.  Q )  x.  (
1  /  P ) )  +  ( ( P  x.  Q )  x.  ( 1  /  Q ) ) ) )
25 addcom 8111 . . . . 5  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( P  +  Q
)  =  ( Q  +  P ) )
2625ad2ant2r 509 . . . 4  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  +  Q )  =  ( Q  +  P ) )
2721, 24, 263eqtr4d 2231 . . 3  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  ( ( 1  /  P )  +  ( 1  /  Q ) ) )  =  ( P  +  Q ) )
2822mulridd 7991 . . . 4  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  x.  Q )  x.  1 )  =  ( P  x.  Q ) )
2928ad2ant2r 509 . . 3  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  1 )  =  ( P  x.  Q ) )
3027, 29eqeq12d 2203 . 2  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( ( P  x.  Q )  x.  ( ( 1  /  P )  +  ( 1  /  Q
) ) )  =  ( ( P  x.  Q )  x.  1 )  <->  ( P  +  Q )  =  ( P  x.  Q ) ) )
31 addcl 7953 . . . 4  |-  ( ( ( 1  /  P
)  e.  CC  /\  ( 1  /  Q
)  e.  CC )  ->  ( ( 1  /  P )  +  ( 1  /  Q
) )  e.  CC )
323, 12, 31syl2an 289 . . 3  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( 1  /  P )  +  ( 1  /  Q
) )  e.  CC )
33 mulap0 8628 . . 3  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  x.  Q ) #  0 )
34 ax-1cn 7921 . . . 4  |-  1  e.  CC
35 mulcanap 8639 . . . 4  |-  ( ( ( ( 1  /  P )  +  ( 1  /  Q ) )  e.  CC  /\  1  e.  CC  /\  (
( P  x.  Q
)  e.  CC  /\  ( P  x.  Q
) #  0 ) )  ->  ( ( ( P  x.  Q )  x.  ( ( 1  /  P )  +  ( 1  /  Q
) ) )  =  ( ( P  x.  Q )  x.  1 )  <->  ( ( 1  /  P )  +  ( 1  /  Q
) )  =  1 ) )
3634, 35mp3an2 1335 . . 3  |-  ( ( ( ( 1  /  P )  +  ( 1  /  Q ) )  e.  CC  /\  ( ( P  x.  Q )  e.  CC  /\  ( P  x.  Q
) #  0 ) )  ->  ( ( ( P  x.  Q )  x.  ( ( 1  /  P )  +  ( 1  /  Q
) ) )  =  ( ( P  x.  Q )  x.  1 )  <->  ( ( 1  /  P )  +  ( 1  /  Q
) )  =  1 ) )
3732, 23, 33, 36syl12anc 1246 . 2  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( ( P  x.  Q )  x.  ( ( 1  /  P )  +  ( 1  /  Q
) ) )  =  ( ( P  x.  Q )  x.  1 )  <->  ( ( 1  /  P )  +  ( 1  /  Q
) )  =  1 ) )
38 eqcom 2190 . . . 4  |-  ( ( P  +  Q )  =  ( P  x.  Q )  <->  ( P  x.  Q )  =  ( P  +  Q ) )
39 muleqadd 8642 . . . 4  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  x.  Q )  =  ( P  +  Q )  <-> 
( ( P  - 
1 )  x.  ( Q  -  1 ) )  =  1 ) )
4038, 39bitrid 192 . . 3  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  +  Q )  =  ( P  x.  Q )  <-> 
( ( P  - 
1 )  x.  ( Q  -  1 ) )  =  1 ) )
4140ad2ant2r 509 . 2  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  +  Q )  =  ( P  x.  Q
)  <->  ( ( P  -  1 )  x.  ( Q  -  1 ) )  =  1 ) )
4230, 37, 413bitr3d 218 1  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( ( 1  /  P )  +  ( 1  /  Q ) )  =  1  <->  ( ( P  -  1 )  x.  ( Q  -  1 ) )  =  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1363    e. wcel 2159   class class class wbr 4017  (class class class)co 5890   CCcc 7826   0cc0 7828   1c1 7829    + caddc 7831    x. cmul 7833    - cmin 8145   # cap 8555    / cdiv 8646
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-opab 4079  df-id 4307  df-po 4310  df-iso 4311  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-iota 5192  df-fun 5232  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647
This theorem is referenced by: (None)
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