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Theorem conjmul 9472
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 732 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  ->  P  e.  CC )
2 simprl 734 . . . . . . 7  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  ->  Q  e.  CC )
3 reccl 9426 . . . . . . . 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 9017 . . . . . 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 9433 . . . . . . . 8  |-  ( ( P  e.  CC  /\  P  =/=  0 )  -> 
( P  x.  (
1  /  P ) )  =  1 )
76oveq1d 5834 . . . . . . 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 8831 . . . . . . 7  |-  ( Q  e.  CC  ->  (
1  x.  Q )  =  Q )
109ad2antrl 710 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( 1  x.  Q
)  =  Q )
115, 8, 103eqtrd 2320 . . . . 5  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  (
1  /  P ) )  =  Q )
12 reccl 9426 . . . . . . . 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 8853 . . . . . 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 9433 . . . . . . . 8  |-  ( ( Q  e.  CC  /\  Q  =/=  0 )  -> 
( Q  x.  (
1  /  Q ) )  =  1 )
1615oveq2d 5835 . . . . . . 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 8830 . . . . . . 7  |-  ( P  e.  CC  ->  ( P  x.  1 )  =  P )
1918ad2antrr 708 . . . . . 6  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( P  x.  1 )  =  P )
2014, 17, 193eqtrd 2320 . . . . 5  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  (
1  /  Q ) )  =  P )
2111, 20oveq12d 5837 . . . 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 8816 . . . . . 6  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( P  x.  Q
)  e.  CC )
2322ad2ant2r 729 . . . . 5  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( P  x.  Q
)  e.  CC )
2423, 4, 13adddid 8854 . . . 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 8993 . . . . 5  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( P  +  Q
)  =  ( Q  +  P ) )
2625ad2ant2r 729 . . . 4  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( P  +  Q
)  =  ( Q  +  P ) )
2721, 24, 263eqtr4d 2326 . . 3  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  (
( 1  /  P
)  +  ( 1  /  Q ) ) )  =  ( P  +  Q ) )
2822mulid1d 8847 . . . 4  |-  ( ( P  e.  CC  /\  Q  e.  CC )  ->  ( ( P  x.  Q )  x.  1 )  =  ( P  x.  Q ) )
2928ad2ant2r 729 . . 3  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( ( P  x.  Q )  x.  1 )  =  ( P  x.  Q ) )
3027, 29eqeq12d 2298 . 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 8814 . . . 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 9405 . . 3  |-  ( ( ( P  e.  CC  /\  P  =/=  0 )  /\  ( Q  e.  CC  /\  Q  =/=  0 ) )  -> 
( P  x.  Q
)  =/=  0 )
34 ax-1cn 8790 . . . 4  |-  1  e.  CC
35 mulcan 9400 . . . 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 2286 . . . 4  |-  ( ( P  +  Q )  =  ( P  x.  Q )  <->  ( P  x.  Q )  =  ( P  +  Q ) )
39 muleqadd 9407 . . . 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 729 . 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 1624    e. wcel 1685    =/= wne 2447  (class class class)co 5819   CCcc 8730   0cc0 8732   1c1 8733    + caddc 8735    x. cmul 8737    - cmin 9032    / cdiv 9418
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-po 4313  df-so 4314  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-iota 6252  df-riota 6299  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419
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