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

Theorem conjmul 9724
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 9678 . . . . . . . 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 9269 . . . . . 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 9685 . . . . . . . 8  |-  ( ( P  e.  CC  /\  P  =/=  0 )  -> 
( P  x.  (
1  /  P ) )  =  1 )
76oveq1d 6089 . . . . . . 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 9082 . . . . . . 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 2472 . . . . 5  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  (
1  /  P ) )  =  Q )
12 reccl 9678 . . . . . . . 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 9104 . . . . . 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 9685 . . . . . . . 8  |-  ( ( Q  e.  CC  /\  Q  =/=  0 )  -> 
( Q  x.  (
1  /  Q ) )  =  1 )
1615oveq2d 6090 . . . . . . 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 9081 . . . . . . 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 2472 . . . . 5  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  (
1  /  Q ) )  =  P )
2111, 20oveq12d 6092 . . . 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 9067 . . . . . 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 9105 . . . 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 9245 . . . . 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 2478 . . 3  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  (
( 1  /  P
)  +  ( 1  /  Q ) ) )  =  ( P  +  Q ) )
2822mulid1d 9098 . . . 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 2450 . 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 9065 . . . 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 9657 . . 3  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( P  x.  Q
)  =/=  0 )
34 ax-1cn 9041 . . . 4  |-  1  e.  CC
35 mulcan 9652 . . . 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 2438 . . . 4  |-  ( ( P  +  Q )  =  ( P  x.  Q )  <->  ( P  x.  Q )  =  ( P  +  Q ) )
39 muleqadd 9659 . . . 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 1652    e. wcel 1725    =/= wne 2599  (class class class)co 6074   CCcc 8981   0cc0 8983   1c1 8984    + caddc 8986    x. cmul 8988    - cmin 9284    / cdiv 9670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-po 4496  df-so 4497  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-riota 6542  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-div 9671
  Copyright terms: Public domain W3C validator