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Theorem conjmulap 7936
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 496 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  P  e.  CC )
2 simprl 498 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  Q  e.  CC )
3 recclap 7886 . . . . . . . 8  |-  ( ( P  e.  CC  /\  P #  0 )  ->  (
1  /  P )  e.  CC )
43adantr 270 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( 1  /  P )  e.  CC )
51, 2, 4mul32d 7380 . . . . . 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 7893 . . . . . . . 8  |-  ( ( P  e.  CC  /\  P #  0 )  ->  ( P  x.  ( 1  /  P ) )  =  1 )
76oveq1d 5578 . . . . . . 7  |-  ( ( P  e.  CC  /\  P #  0 )  ->  (
( P  x.  (
1  /  P ) )  x.  Q )  =  ( 1  x.  Q ) )
87adantr 270 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  ( 1  /  P ) )  x.  Q )  =  ( 1  x.  Q ) )
9 mulid2 7231 . . . . . . 7  |-  ( Q  e.  CC  ->  (
1  x.  Q )  =  Q )
109ad2antrl 474 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( 1  x.  Q )  =  Q )
115, 8, 103eqtrd 2119 . . . . 5  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  ( 1  /  P
) )  =  Q )
12 recclap 7886 . . . . . . . 8  |-  ( ( Q  e.  CC  /\  Q #  0 )  ->  (
1  /  Q )  e.  CC )
1312adantl 271 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( 1  /  Q )  e.  CC )
141, 2, 13mulassd 7256 . . . . . 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 7893 . . . . . . . 8  |-  ( ( Q  e.  CC  /\  Q #  0 )  ->  ( Q  x.  ( 1  /  Q ) )  =  1 )
1615oveq2d 5579 . . . . . . 7  |-  ( ( Q  e.  CC  /\  Q #  0 )  ->  ( P  x.  ( Q  x.  ( 1  /  Q
) ) )  =  ( P  x.  1 ) )
1716adantl 271 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  x.  ( Q  x.  (
1  /  Q ) ) )  =  ( P  x.  1 ) )
18 mulid1 7230 . . . . . . 7  |-  ( P  e.  CC  ->  ( P  x.  1 )  =  P )
1918ad2antrr 472 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  x.  1 )  =  P )
2014, 17, 193eqtrd 2119 . . . . 5  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  ( 1  /  Q
) )  =  P )
2111, 20oveq12d 5581 . . . 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 7214 . . . . . 6  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( P  x.  Q
)  e.  CC )
2322ad2ant2r 493 . . . . 5  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  x.  Q )  e.  CC )
2423, 4, 13adddid 7257 . . . 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 7364 . . . . 5  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( P  +  Q
)  =  ( Q  +  P ) )
2625ad2ant2r 493 . . . 4  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  +  Q )  =  ( Q  +  P ) )
2721, 24, 263eqtr4d 2125 . . 3  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  ( ( 1  /  P )  +  ( 1  /  Q ) ) )  =  ( P  +  Q ) )
2822mulid1d 7250 . . . 4  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  x.  Q )  x.  1 )  =  ( P  x.  Q ) )
2928ad2ant2r 493 . . 3  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  1 )  =  ( P  x.  Q ) )
3027, 29eqeq12d 2097 . 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 7212 . . . 4  |-  ( ( ( 1  /  P
)  e.  CC  /\  ( 1  /  Q
)  e.  CC )  ->  ( ( 1  /  P )  +  ( 1  /  Q
) )  e.  CC )
323, 12, 31syl2an 283 . . 3  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( 1  /  P )  +  ( 1  /  Q
) )  e.  CC )
33 mulap0 7863 . . 3  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  x.  Q ) #  0 )
34 ax-1cn 7183 . . . 4  |-  1  e.  CC
35 mulcanap 7874 . . . 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 1257 . . 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 1168 . 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 2085 . . . 4  |-  ( ( P  +  Q )  =  ( P  x.  Q )  <->  ( P  x.  Q )  =  ( P  +  Q ) )
39 muleqadd 7877 . . . 4  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  x.  Q )  =  ( P  +  Q )  <-> 
( ( P  - 
1 )  x.  ( Q  -  1 ) )  =  1 ) )
4038, 39syl5bb 190 . . 3  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  +  Q )  =  ( P  x.  Q )  <-> 
( ( P  - 
1 )  x.  ( Q  -  1 ) )  =  1 ) )
4140ad2ant2r 493 . 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 216 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 102    <-> wb 103    = wceq 1285    e. wcel 1434   class class class wbr 3805  (class class class)co 5563   CCcc 7093   0cc0 7095   1c1 7096    + caddc 7098    x. cmul 7100    - cmin 7398   # cap 7800    / cdiv 7879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-po 4079  df-iso 4080  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880
This theorem is referenced by: (None)
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