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Theorem conjmulap 8402
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 501 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  P  e.  CC )
2 simprl 503 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  Q  e.  CC )
3 recclap 8352 . . . . . . . 8  |-  ( ( P  e.  CC  /\  P #  0 )  ->  (
1  /  P )  e.  CC )
43adantr 272 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( 1  /  P )  e.  CC )
51, 2, 4mul32d 7838 . . . . . 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 8359 . . . . . . . 8  |-  ( ( P  e.  CC  /\  P #  0 )  ->  ( P  x.  ( 1  /  P ) )  =  1 )
76oveq1d 5743 . . . . . . 7  |-  ( ( P  e.  CC  /\  P #  0 )  ->  (
( P  x.  (
1  /  P ) )  x.  Q )  =  ( 1  x.  Q ) )
87adantr 272 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  ( 1  /  P ) )  x.  Q )  =  ( 1  x.  Q ) )
9 mulid2 7688 . . . . . . 7  |-  ( Q  e.  CC  ->  (
1  x.  Q )  =  Q )
109ad2antrl 479 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( 1  x.  Q )  =  Q )
115, 8, 103eqtrd 2151 . . . . 5  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  ( 1  /  P
) )  =  Q )
12 recclap 8352 . . . . . . . 8  |-  ( ( Q  e.  CC  /\  Q #  0 )  ->  (
1  /  Q )  e.  CC )
1312adantl 273 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( 1  /  Q )  e.  CC )
141, 2, 13mulassd 7713 . . . . . 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 8359 . . . . . . . 8  |-  ( ( Q  e.  CC  /\  Q #  0 )  ->  ( Q  x.  ( 1  /  Q ) )  =  1 )
1615oveq2d 5744 . . . . . . 7  |-  ( ( Q  e.  CC  /\  Q #  0 )  ->  ( P  x.  ( Q  x.  ( 1  /  Q
) ) )  =  ( P  x.  1 ) )
1716adantl 273 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  x.  ( Q  x.  (
1  /  Q ) ) )  =  ( P  x.  1 ) )
18 mulid1 7687 . . . . . . 7  |-  ( P  e.  CC  ->  ( P  x.  1 )  =  P )
1918ad2antrr 477 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  x.  1 )  =  P )
2014, 17, 193eqtrd 2151 . . . . 5  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  ( 1  /  Q
) )  =  P )
2111, 20oveq12d 5746 . . . 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 7671 . . . . . 6  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( P  x.  Q
)  e.  CC )
2322ad2ant2r 498 . . . . 5  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  x.  Q )  e.  CC )
2423, 4, 13adddid 7714 . . . 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 7822 . . . . 5  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( P  +  Q
)  =  ( Q  +  P ) )
2625ad2ant2r 498 . . . 4  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  +  Q )  =  ( Q  +  P ) )
2721, 24, 263eqtr4d 2157 . . 3  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  ( ( 1  /  P )  +  ( 1  /  Q ) ) )  =  ( P  +  Q ) )
2822mulid1d 7707 . . . 4  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  x.  Q )  x.  1 )  =  ( P  x.  Q ) )
2928ad2ant2r 498 . . 3  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  1 )  =  ( P  x.  Q ) )
3027, 29eqeq12d 2129 . 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 7669 . . . 4  |-  ( ( ( 1  /  P
)  e.  CC  /\  ( 1  /  Q
)  e.  CC )  ->  ( ( 1  /  P )  +  ( 1  /  Q
) )  e.  CC )
323, 12, 31syl2an 285 . . 3  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( 1  /  P )  +  ( 1  /  Q
) )  e.  CC )
33 mulap0 8328 . . 3  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  x.  Q ) #  0 )
34 ax-1cn 7638 . . . 4  |-  1  e.  CC
35 mulcanap 8339 . . . 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 1286 . . 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 1197 . 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 2117 . . . 4  |-  ( ( P  +  Q )  =  ( P  x.  Q )  <->  ( P  x.  Q )  =  ( P  +  Q ) )
39 muleqadd 8342 . . . 4  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  x.  Q )  =  ( P  +  Q )  <-> 
( ( P  - 
1 )  x.  ( Q  -  1 ) )  =  1 ) )
4038, 39syl5bb 191 . . 3  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  +  Q )  =  ( P  x.  Q )  <-> 
( ( P  - 
1 )  x.  ( Q  -  1 ) )  =  1 ) )
4140ad2ant2r 498 . 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 217 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 103    <-> wb 104    = wceq 1314    e. wcel 1463   class class class wbr 3895  (class class class)co 5728   CCcc 7545   0cc0 7547   1c1 7548    + caddc 7550    x. cmul 7552    - cmin 7856   # cap 8261    / cdiv 8345
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-id 4175  df-po 4178  df-iso 4179  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346
This theorem is referenced by: (None)
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