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

Theorem sqabsadd 11761
Description: Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.)
Assertion
Ref Expression
sqabsadd  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  +  B )
) ^ 2 )  =  ( ( ( ( abs `  A
) ^ 2 )  +  ( ( abs `  B ) ^ 2 ) )  +  ( 2  x.  ( Re
`  ( A  x.  ( * `  B
) ) ) ) ) )

Proof of Theorem sqabsadd
StepHypRef Expression
1 cjadd 11620 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  +  B )
)  =  ( ( * `  A )  +  ( * `  B ) ) )
21oveq2d 5835 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  x.  (
* `  ( A  +  B ) ) )  =  ( ( A  +  B )  x.  ( ( * `  A )  +  ( * `  B ) ) ) )
3 cjcl 11584 . . . . 5  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
4 cjcl 11584 . . . . 5  |-  ( B  e.  CC  ->  (
* `  B )  e.  CC )
53, 4anim12i 551 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  A )  e.  CC  /\  ( * `  B
)  e.  CC ) )
6 muladd 9207 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( * `
 A )  e.  CC  /\  ( * `
 B )  e.  CC ) )  -> 
( ( A  +  B )  x.  (
( * `  A
)  +  ( * `
 B ) ) )  =  ( ( ( A  x.  (
* `  A )
)  +  ( ( * `  B )  x.  B ) )  +  ( ( A  x.  ( * `  B ) )  +  ( ( * `  A )  x.  B
) ) ) )
75, 6mpdan 651 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  x.  (
( * `  A
)  +  ( * `
 B ) ) )  =  ( ( ( A  x.  (
* `  A )
)  +  ( ( * `  B )  x.  B ) )  +  ( ( A  x.  ( * `  B ) )  +  ( ( * `  A )  x.  B
) ) ) )
82, 7eqtrd 2316 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  x.  (
* `  ( A  +  B ) ) )  =  ( ( ( A  x.  ( * `
 A ) )  +  ( ( * `
 B )  x.  B ) )  +  ( ( A  x.  ( * `  B
) )  +  ( ( * `  A
)  x.  B ) ) ) )
9 addcl 8814 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
10 absvalsq 11759 . . 3  |-  ( ( A  +  B )  e.  CC  ->  (
( abs `  ( A  +  B )
) ^ 2 )  =  ( ( A  +  B )  x.  ( * `  ( A  +  B )
) ) )
119, 10syl 17 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  +  B )
) ^ 2 )  =  ( ( A  +  B )  x.  ( * `  ( A  +  B )
) ) )
12 absvalsq 11759 . . . 4  |-  ( A  e.  CC  ->  (
( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
13 absvalsq 11759 . . . . 5  |-  ( B  e.  CC  ->  (
( abs `  B
) ^ 2 )  =  ( B  x.  ( * `  B
) ) )
14 mulcom 8818 . . . . . 6  |-  ( ( B  e.  CC  /\  ( * `  B
)  e.  CC )  ->  ( B  x.  ( * `  B
) )  =  ( ( * `  B
)  x.  B ) )
154, 14mpdan 651 . . . . 5  |-  ( B  e.  CC  ->  ( B  x.  ( * `  B ) )  =  ( ( * `  B )  x.  B
) )
1613, 15eqtrd 2316 . . . 4  |-  ( B  e.  CC  ->  (
( abs `  B
) ^ 2 )  =  ( ( * `
 B )  x.  B ) )
1712, 16oveqan12d 5838 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( abs `  A ) ^ 2 )  +  ( ( abs `  B ) ^ 2 ) )  =  ( ( A  x.  ( * `  A ) )  +  ( ( * `  B )  x.  B
) ) )
18 mulcl 8816 . . . . . 6  |-  ( ( A  e.  CC  /\  ( * `  B
)  e.  CC )  ->  ( A  x.  ( * `  B
) )  e.  CC )
194, 18sylan2 462 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  (
* `  B )
)  e.  CC )
2019addcjd 11691 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  ( * `  B
) )  +  ( * `  ( A  x.  ( * `  B ) ) ) )  =  ( 2  x.  ( Re `  ( A  x.  (
* `  B )
) ) ) )
21 cjmul 11621 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( * `  B
)  e.  CC )  ->  ( * `  ( A  x.  (
* `  B )
) )  =  ( ( * `  A
)  x.  ( * `
 ( * `  B ) ) ) )
224, 21sylan2 462 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  x.  ( * `  B ) ) )  =  ( ( * `
 A )  x.  ( * `  (
* `  B )
) ) )
23 cjcj 11619 . . . . . . . 8  |-  ( B  e.  CC  ->  (
* `  ( * `  B ) )  =  B )
2423adantl 454 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  (
* `  B )
)  =  B )
2524oveq2d 5835 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  A )  x.  (
* `  ( * `  B ) ) )  =  ( ( * `
 A )  x.  B ) )
2622, 25eqtrd 2316 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  x.  ( * `  B ) ) )  =  ( ( * `
 A )  x.  B ) )
2726oveq2d 5835 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  ( * `  B
) )  +  ( * `  ( A  x.  ( * `  B ) ) ) )  =  ( ( A  x.  ( * `
 B ) )  +  ( ( * `
 A )  x.  B ) ) )
2820, 27eqtr3d 2318 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  (
Re `  ( A  x.  ( * `  B
) ) ) )  =  ( ( A  x.  ( * `  B ) )  +  ( ( * `  A )  x.  B
) ) )
2917, 28oveq12d 5837 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( abs `  A ) ^ 2 )  +  ( ( abs `  B
) ^ 2 ) )  +  ( 2  x.  ( Re `  ( A  x.  (
* `  B )
) ) ) )  =  ( ( ( A  x.  ( * `
 A ) )  +  ( ( * `
 B )  x.  B ) )  +  ( ( A  x.  ( * `  B
) )  +  ( ( * `  A
)  x.  B ) ) ) )
308, 11, 293eqtr4d 2326 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  +  B )
) ^ 2 )  =  ( ( ( ( abs `  A
) ^ 2 )  +  ( ( abs `  B ) ^ 2 ) )  +  ( 2  x.  ( Re
`  ( A  x.  ( * `  B
) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685   ` cfv 5221  (class class class)co 5819   CCcc 8730    + caddc 8735    x. cmul 8737   2c2 9790   ^cexp 11098   *ccj 11575   Recre 11576   abscabs 11713
This theorem is referenced by:  abstri  11808  sqabsaddi  11882  cncph  21389
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-cnex 8788  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  ax-pre-sup 8810
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-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  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-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-sup 7189  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-n0 9961  df-z 10020  df-uz 10226  df-rp 10350  df-seq 11041  df-exp 11099  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715
  Copyright terms: Public domain W3C validator