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Theorem conjmul 9687
Description: Two numbers whose reciprocals sum to 1 are called "conjugates" and satisfy this relationship. Equation 5 of [Kreyszig] p. 12. (Contributed by NM, 12-Nov-2006.)
Assertion
Ref Expression
conjmul  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( ( 1  /  P )  +  ( 1  /  Q
) )  =  1  <-> 
( ( P  - 
1 )  x.  ( Q  -  1 ) )  =  1 ) )

Proof of Theorem conjmul
StepHypRef Expression
1 simpll 731 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  ->  P  e.  CC )
2 simprl 733 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  ->  Q  e.  CC )
3 reccl 9641 . . . . . . . 8  |-  ( ( P  e.  CC  /\  P  =/=  0 )  -> 
( 1  /  P
)  e.  CC )
43adantr 452 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( 1  /  P
)  e.  CC )
51, 2, 4mul32d 9232 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  (
1  /  P ) )  =  ( ( P  x.  ( 1  /  P ) )  x.  Q ) )
6 recid 9648 . . . . . . . 8  |-  ( ( P  e.  CC  /\  P  =/=  0 )  -> 
( P  x.  (
1  /  P ) )  =  1 )
76oveq1d 6055 . . . . . . 7  |-  ( ( P  e.  CC  /\  P  =/=  0 )  -> 
( ( P  x.  ( 1  /  P
) )  x.  Q
)  =  ( 1  x.  Q ) )
87adantr 452 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  ( 1  /  P
) )  x.  Q
)  =  ( 1  x.  Q ) )
9 mulid2 9045 . . . . . . 7  |-  ( Q  e.  CC  ->  (
1  x.  Q )  =  Q )
109ad2antrl 709 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( 1  x.  Q
)  =  Q )
115, 8, 103eqtrd 2440 . . . . 5  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  (
1  /  P ) )  =  Q )
12 reccl 9641 . . . . . . . 8  |-  ( ( Q  e.  CC  /\  Q  =/=  0 )  -> 
( 1  /  Q
)  e.  CC )
1312adantl 453 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( 1  /  Q
)  e.  CC )
141, 2, 13mulassd 9067 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  (
1  /  Q ) )  =  ( P  x.  ( Q  x.  ( 1  /  Q
) ) ) )
15 recid 9648 . . . . . . . 8  |-  ( ( Q  e.  CC  /\  Q  =/=  0 )  -> 
( Q  x.  (
1  /  Q ) )  =  1 )
1615oveq2d 6056 . . . . . . 7  |-  ( ( Q  e.  CC  /\  Q  =/=  0 )  -> 
( P  x.  ( Q  x.  ( 1  /  Q ) ) )  =  ( P  x.  1 ) )
1716adantl 453 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( P  x.  ( Q  x.  ( 1  /  Q ) ) )  =  ( P  x.  1 ) )
18 mulid1 9044 . . . . . . 7  |-  ( P  e.  CC  ->  ( P  x.  1 )  =  P )
1918ad2antrr 707 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( P  x.  1 )  =  P )
2014, 17, 193eqtrd 2440 . . . . 5  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  (
1  /  Q ) )  =  P )
2111, 20oveq12d 6058 . . . 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 9030 . . . . . 6  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( P  x.  Q
)  e.  CC )
2322ad2ant2r 728 . . . . 5  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( P  x.  Q
)  e.  CC )
2423, 4, 13adddid 9068 . . . 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 9208 . . . . 5  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( P  +  Q
)  =  ( Q  +  P ) )
2625ad2ant2r 728 . . . 4  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( P  +  Q
)  =  ( Q  +  P ) )
2721, 24, 263eqtr4d 2446 . . 3  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  (
( 1  /  P
)  +  ( 1  /  Q ) ) )  =  ( P  +  Q ) )
2822mulid1d 9061 . . . 4  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  x.  Q )  x.  1 )  =  ( P  x.  Q ) )
2928ad2ant2r 728 . . 3  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  1 )  =  ( P  x.  Q ) )
3027, 29eqeq12d 2418 . 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 9028 . . . 4  |-  ( ( ( 1  /  P
)  e.  CC  /\  ( 1  /  Q
)  e.  CC )  ->  ( ( 1  /  P )  +  ( 1  /  Q
) )  e.  CC )
323, 12, 31syl2an 464 . . 3  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( 1  /  P )  +  ( 1  /  Q ) )  e.  CC )
33 mulne0 9620 . . 3  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( P  x.  Q
)  =/=  0 )
34 ax-1cn 9004 . . . 4  |-  1  e.  CC
35 mulcan 9615 . . . 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 1267 . . 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 1182 . 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 2406 . . . 4  |-  ( ( P  +  Q )  =  ( P  x.  Q )  <->  ( P  x.  Q )  =  ( P  +  Q ) )
39 muleqadd 9622 . . . 4  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  x.  Q )  =  ( P  +  Q )  <-> 
( ( P  - 
1 )  x.  ( Q  -  1 ) )  =  1 ) )
4038, 39syl5bb 249 . . 3  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  +  Q )  =  ( P  x.  Q )  <-> 
( ( P  - 
1 )  x.  ( Q  -  1 ) )  =  1 ) )
4140ad2ant2r 728 . 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 275 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    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247    / cdiv 9633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634
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