MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  conjmul Unicode version

Theorem conjmul 9665
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 9619 . . . . . . . 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 9210 . . . . . 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 9626 . . . . . . . 8  |-  ( ( P  e.  CC  /\  P  =/=  0 )  -> 
( P  x.  (
1  /  P ) )  =  1 )
76oveq1d 6037 . . . . . . 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 9024 . . . . . . 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 2425 . . . . 5  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  (
1  /  P ) )  =  Q )
12 reccl 9619 . . . . . . . 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 9046 . . . . . 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 9626 . . . . . . . 8  |-  ( ( Q  e.  CC  /\  Q  =/=  0 )  -> 
( Q  x.  (
1  /  Q ) )  =  1 )
1615oveq2d 6038 . . . . . . 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 9023 . . . . . . 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 2425 . . . . 5  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  (
1  /  Q ) )  =  P )
2111, 20oveq12d 6040 . . . 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 9009 . . . . . 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 9047 . . . 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 9186 . . . . 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 2431 . . 3  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  (
( 1  /  P
)  +  ( 1  /  Q ) ) )  =  ( P  +  Q ) )
2822mulid1d 9040 . . . 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 2403 . 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 9007 . . . 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 9598 . . 3  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( P  x.  Q
)  =/=  0 )
34 ax-1cn 8983 . . . 4  |-  1  e.  CC
35 mulcan 9593 . . . 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 2391 . . . 4  |-  ( ( P  +  Q )  =  ( P  x.  Q )  <->  ( P  x.  Q )  =  ( P  +  Q ) )
39 muleqadd 9600 . . . 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 1717    =/= wne 2552  (class class class)co 6022   CCcc 8923   0cc0 8925   1c1 8926    + caddc 8928    x. cmul 8930    - cmin 9225    / cdiv 9611
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-po 4446  df-so 4447  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612
  Copyright terms: Public domain W3C validator