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Theorem conjmulap 7779
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 489 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  P  e.  CC )
2 simprl 491 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  Q  e.  CC )
3 recclap 7731 . . . . . . . 8  |-  ( ( P  e.  CC  /\  P #  0 )  ->  (
1  /  P )  e.  CC )
43adantr 265 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( 1  /  P )  e.  CC )
51, 2, 4mul32d 7226 . . . . . 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 7738 . . . . . . . 8  |-  ( ( P  e.  CC  /\  P #  0 )  ->  ( P  x.  ( 1  /  P ) )  =  1 )
76oveq1d 5554 . . . . . . 7  |-  ( ( P  e.  CC  /\  P #  0 )  ->  (
( P  x.  (
1  /  P ) )  x.  Q )  =  ( 1  x.  Q ) )
87adantr 265 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  ( 1  /  P ) )  x.  Q )  =  ( 1  x.  Q ) )
9 mulid2 7082 . . . . . . 7  |-  ( Q  e.  CC  ->  (
1  x.  Q )  =  Q )
109ad2antrl 467 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( 1  x.  Q )  =  Q )
115, 8, 103eqtrd 2092 . . . . 5  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  ( 1  /  P
) )  =  Q )
12 recclap 7731 . . . . . . . 8  |-  ( ( Q  e.  CC  /\  Q #  0 )  ->  (
1  /  Q )  e.  CC )
1312adantl 266 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( 1  /  Q )  e.  CC )
141, 2, 13mulassd 7107 . . . . . 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 7738 . . . . . . . 8  |-  ( ( Q  e.  CC  /\  Q #  0 )  ->  ( Q  x.  ( 1  /  Q ) )  =  1 )
1615oveq2d 5555 . . . . . . 7  |-  ( ( Q  e.  CC  /\  Q #  0 )  ->  ( P  x.  ( Q  x.  ( 1  /  Q
) ) )  =  ( P  x.  1 ) )
1716adantl 266 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  x.  ( Q  x.  (
1  /  Q ) ) )  =  ( P  x.  1 ) )
18 mulid1 7081 . . . . . . 7  |-  ( P  e.  CC  ->  ( P  x.  1 )  =  P )
1918ad2antrr 465 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  x.  1 )  =  P )
2014, 17, 193eqtrd 2092 . . . . 5  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  ( 1  /  Q
) )  =  P )
2111, 20oveq12d 5557 . . . 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 7065 . . . . . 6  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( P  x.  Q
)  e.  CC )
2322ad2ant2r 486 . . . . 5  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  x.  Q )  e.  CC )
2423, 4, 13adddid 7108 . . . 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 7210 . . . . 5  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( P  +  Q
)  =  ( Q  +  P ) )
2625ad2ant2r 486 . . . 4  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  +  Q )  =  ( Q  +  P ) )
2721, 24, 263eqtr4d 2098 . . 3  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  ( ( 1  /  P )  +  ( 1  /  Q ) ) )  =  ( P  +  Q ) )
2822mulid1d 7101 . . . 4  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  x.  Q )  x.  1 )  =  ( P  x.  Q ) )
2928ad2ant2r 486 . . 3  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( P  x.  Q )  x.  1 )  =  ( P  x.  Q ) )
3027, 29eqeq12d 2070 . 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 7063 . . . 4  |-  ( ( ( 1  /  P
)  e.  CC  /\  ( 1  /  Q
)  e.  CC )  ->  ( ( 1  /  P )  +  ( 1  /  Q
) )  e.  CC )
323, 12, 31syl2an 277 . . 3  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( ( 1  /  P )  +  ( 1  /  Q
) )  e.  CC )
33 mulap0 7708 . . 3  |-  ( ( ( P  e.  CC  /\  P #  0 )  /\  ( Q  e.  CC  /\  Q #  0 ) )  ->  ( P  x.  Q ) #  0 )
34 ax-1cn 7034 . . . 4  |-  1  e.  CC
35 mulcanap 7719 . . . 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 1231 . . 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 1144 . 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 2058 . . . 4  |-  ( ( P  +  Q )  =  ( P  x.  Q )  <->  ( P  x.  Q )  =  ( P  +  Q ) )
39 muleqadd 7722 . . . 4  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  x.  Q )  =  ( P  +  Q )  <-> 
( ( P  - 
1 )  x.  ( Q  -  1 ) )  =  1 ) )
4038, 39syl5bb 185 . . 3  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  +  Q )  =  ( P  x.  Q )  <-> 
( ( P  - 
1 )  x.  ( Q  -  1 ) )  =  1 ) )
4140ad2ant2r 486 . 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 211 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 101    <-> wb 102    = wceq 1259    e. wcel 1409   class class class wbr 3791  (class class class)co 5539   CCcc 6944   0cc0 6946   1c1 6947    + caddc 6949    x. cmul 6951    - cmin 7244   # cap 7645    / cdiv 7724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338  ax-cnex 7032  ax-resscn 7033  ax-1cn 7034  ax-1re 7035  ax-icn 7036  ax-addcl 7037  ax-addrcl 7038  ax-mulcl 7039  ax-mulrcl 7040  ax-addcom 7041  ax-mulcom 7042  ax-addass 7043  ax-mulass 7044  ax-distr 7045  ax-i2m1 7046  ax-1rid 7048  ax-0id 7049  ax-rnegex 7050  ax-precex 7051  ax-cnre 7052  ax-pre-ltirr 7053  ax-pre-ltwlin 7054  ax-pre-lttrn 7055  ax-pre-apti 7056  ax-pre-ltadd 7057  ax-pre-mulgt0 7058  ax-pre-mulext 7059
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rmo 2331  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-riota 5495  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-i1p 6622  df-iplp 6623  df-iltp 6625  df-enr 6868  df-nr 6869  df-ltr 6872  df-0r 6873  df-1r 6874  df-0 6953  df-1 6954  df-r 6956  df-lt 6959  df-pnf 7120  df-mnf 7121  df-xr 7122  df-ltxr 7123  df-le 7124  df-sub 7246  df-neg 7247  df-reap 7639  df-ap 7646  df-div 7725
This theorem is referenced by: (None)
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