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Theorem sqabsadd 12892
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 12751 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  +  B )
)  =  ( ( * `  A )  +  ( * `  B ) ) )
21oveq2d 6219 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  x.  (
* `  ( A  +  B ) ) )  =  ( ( A  +  B )  x.  ( ( * `  A )  +  ( * `  B ) ) ) )
3 cjcl 12715 . . . . 5  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
4 cjcl 12715 . . . . 5  |-  ( B  e.  CC  ->  (
* `  B )  e.  CC )
53, 4anim12i 566 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  A )  e.  CC  /\  ( * `  B
)  e.  CC ) )
6 muladd 9891 . . . 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 668 . . 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 2495 . 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 9478 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
10 absvalsq 12890 . . 3  |-  ( ( A  +  B )  e.  CC  ->  (
( abs `  ( A  +  B )
) ^ 2 )  =  ( ( A  +  B )  x.  ( * `  ( A  +  B )
) ) )
119, 10syl 16 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  +  B )
) ^ 2 )  =  ( ( A  +  B )  x.  ( * `  ( A  +  B )
) ) )
12 absvalsq 12890 . . . 4  |-  ( A  e.  CC  ->  (
( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
13 absvalsq 12890 . . . . 5  |-  ( B  e.  CC  ->  (
( abs `  B
) ^ 2 )  =  ( B  x.  ( * `  B
) ) )
14 mulcom 9482 . . . . . 6  |-  ( ( B  e.  CC  /\  ( * `  B
)  e.  CC )  ->  ( B  x.  ( * `  B
) )  =  ( ( * `  B
)  x.  B ) )
154, 14mpdan 668 . . . . 5  |-  ( B  e.  CC  ->  ( B  x.  ( * `  B ) )  =  ( ( * `  B )  x.  B
) )
1613, 15eqtrd 2495 . . . 4  |-  ( B  e.  CC  ->  (
( abs `  B
) ^ 2 )  =  ( ( * `
 B )  x.  B ) )
1712, 16oveqan12d 6222 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( abs `  A ) ^ 2 )  +  ( ( abs `  B ) ^ 2 ) )  =  ( ( A  x.  ( * `  A ) )  +  ( ( * `  B )  x.  B
) ) )
18 mulcl 9480 . . . . . 6  |-  ( ( A  e.  CC  /\  ( * `  B
)  e.  CC )  ->  ( A  x.  ( * `  B
) )  e.  CC )
194, 18sylan2 474 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  (
* `  B )
)  e.  CC )
2019addcjd 12822 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  ( * `  B
) )  +  ( * `  ( A  x.  ( * `  B ) ) ) )  =  ( 2  x.  ( Re `  ( A  x.  (
* `  B )
) ) ) )
21 cjmul 12752 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( * `  B
)  e.  CC )  ->  ( * `  ( A  x.  (
* `  B )
) )  =  ( ( * `  A
)  x.  ( * `
 ( * `  B ) ) ) )
224, 21sylan2 474 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  x.  ( * `  B ) ) )  =  ( ( * `
 A )  x.  ( * `  (
* `  B )
) ) )
23 cjcj 12750 . . . . . . . 8  |-  ( B  e.  CC  ->  (
* `  ( * `  B ) )  =  B )
2423adantl 466 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  (
* `  B )
)  =  B )
2524oveq2d 6219 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  A )  x.  (
* `  ( * `  B ) ) )  =  ( ( * `
 A )  x.  B ) )
2622, 25eqtrd 2495 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  x.  ( * `  B ) ) )  =  ( ( * `
 A )  x.  B ) )
2726oveq2d 6219 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  ( * `  B
) )  +  ( * `  ( A  x.  ( * `  B ) ) ) )  =  ( ( A  x.  ( * `
 B ) )  +  ( ( * `
 A )  x.  B ) ) )
2820, 27eqtr3d 2497 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  (
Re `  ( A  x.  ( * `  B
) ) ) )  =  ( ( A  x.  ( * `  B ) )  +  ( ( * `  A )  x.  B
) ) )
2917, 28oveq12d 6221 . 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 2505 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   ` cfv 5529  (class class class)co 6203   CCcc 9394    + caddc 9399    x. cmul 9401   2c2 10485   ^cexp 11985   *ccj 12706   Recre 12707   abscabs 12844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-seq 11927  df-exp 11986  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846
This theorem is referenced by:  abstri  12939  sqabsaddi  13013  cncph  24391
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