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

Theorem conjmul 9431
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 733 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  ->  P  e.  CC )
2 simprl 735 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  ->  Q  e.  CC )
3 reccl 9385 . . . . . . . 8  |-  ( ( P  e.  CC  /\  P  =/=  0 )  -> 
( 1  /  P
)  e.  CC )
43adantr 453 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( 1  /  P
)  e.  CC )
51, 2, 4mul32d 8976 . . . . . 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 9392 . . . . . . . 8  |-  ( ( P  e.  CC  /\  P  =/=  0 )  -> 
( P  x.  (
1  /  P ) )  =  1 )
76oveq1d 5793 . . . . . . 7  |-  ( ( P  e.  CC  /\  P  =/=  0 )  -> 
( ( P  x.  ( 1  /  P
) )  x.  Q
)  =  ( 1  x.  Q ) )
87adantr 453 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  ( 1  /  P
) )  x.  Q
)  =  ( 1  x.  Q ) )
9 mulid2 8789 . . . . . . 7  |-  ( Q  e.  CC  ->  (
1  x.  Q )  =  Q )
109ad2antrl 711 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( 1  x.  Q
)  =  Q )
115, 8, 103eqtrd 2292 . . . . 5  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  (
1  /  P ) )  =  Q )
12 reccl 9385 . . . . . . . 8  |-  ( ( Q  e.  CC  /\  Q  =/=  0 )  -> 
( 1  /  Q
)  e.  CC )
1312adantl 454 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( 1  /  Q
)  e.  CC )
141, 2, 13mulassd 8812 . . . . . 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 9392 . . . . . . . 8  |-  ( ( Q  e.  CC  /\  Q  =/=  0 )  -> 
( Q  x.  (
1  /  Q ) )  =  1 )
1615oveq2d 5794 . . . . . . 7  |-  ( ( Q  e.  CC  /\  Q  =/=  0 )  -> 
( P  x.  ( Q  x.  ( 1  /  Q ) ) )  =  ( P  x.  1 ) )
1716adantl 454 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( P  x.  ( Q  x.  ( 1  /  Q ) ) )  =  ( P  x.  1 ) )
18 mulid1 8788 . . . . . . 7  |-  ( P  e.  CC  ->  ( P  x.  1 )  =  P )
1918ad2antrr 709 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( P  x.  1 )  =  P )
2014, 17, 193eqtrd 2292 . . . . 5  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  (
1  /  Q ) )  =  P )
2111, 20oveq12d 5796 . . . 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 8775 . . . . . 6  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( P  x.  Q
)  e.  CC )
2322ad2ant2r 730 . . . . 5  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( P  x.  Q
)  e.  CC )
2423, 4, 13adddid 8813 . . . 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 8952 . . . . 5  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( P  +  Q
)  =  ( Q  +  P ) )
2625ad2ant2r 730 . . . 4  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( P  +  Q
)  =  ( Q  +  P ) )
2721, 24, 263eqtr4d 2298 . . 3  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  (
( 1  /  P
)  +  ( 1  /  Q ) ) )  =  ( P  +  Q ) )
2822mulid1d 8806 . . . 4  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  x.  Q )  x.  1 )  =  ( P  x.  Q ) )
2928ad2ant2r 730 . . 3  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  1 )  =  ( P  x.  Q ) )
3027, 29eqeq12d 2270 . 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 8773 . . . 4  |-  ( ( ( 1  /  P
)  e.  CC  /\  ( 1  /  Q
)  e.  CC )  ->  ( ( 1  /  P )  +  ( 1  /  Q
) )  e.  CC )
323, 12, 31syl2an 465 . . 3  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( 1  /  P )  +  ( 1  /  Q ) )  e.  CC )
33 mulne0 9364 . . 3  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( P  x.  Q
)  =/=  0 )
34 ax-1cn 8749 . . . 4  |-  1  e.  CC
35 mulcan 9359 . . . 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 1270 . . 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 1185 . 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 2258 . . . 4  |-  ( ( P  +  Q )  =  ( P  x.  Q )  <->  ( P  x.  Q )  =  ( P  +  Q ) )
39 muleqadd 9366 . . . 4  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  x.  Q )  =  ( P  +  Q )  <-> 
( ( P  - 
1 )  x.  ( Q  -  1 ) )  =  1 ) )
4038, 39syl5bb 250 . . 3  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  +  Q )  =  ( P  x.  Q )  <-> 
( ( P  - 
1 )  x.  ( Q  -  1 ) )  =  1 ) )
4140ad2ant2r 730 . 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 276 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419  (class class class)co 5778   CCcc 8689   0cc0 8691   1c1 8692    + caddc 8694    x. cmul 8696    - cmin 8991    / cdiv 9377
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-po 4272  df-so 4273  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-iota 6211  df-riota 6258  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378
  Copyright terms: Public domain W3C validator