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Theorem sqabsadd 10079
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 9909 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  +  B )
)  =  ( ( * `  A )  +  ( * `  B ) ) )
21oveq2d 5559 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  x.  (
* `  ( A  +  B ) ) )  =  ( ( A  +  B )  x.  ( ( * `  A )  +  ( * `  B ) ) ) )
3 cjcl 9873 . . . . 5  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
4 cjcl 9873 . . . . 5  |-  ( B  e.  CC  ->  (
* `  B )  e.  CC )
53, 4anim12i 331 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  A )  e.  CC  /\  ( * `  B
)  e.  CC ) )
6 muladd 7555 . . . 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 412 . . 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 2114 . 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 7160 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
10 absvalsq 10077 . . 3  |-  ( ( A  +  B )  e.  CC  ->  (
( abs `  ( A  +  B )
) ^ 2 )  =  ( ( A  +  B )  x.  ( * `  ( A  +  B )
) ) )
119, 10syl 14 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  +  B )
) ^ 2 )  =  ( ( A  +  B )  x.  ( * `  ( A  +  B )
) ) )
12 absvalsq 10077 . . . 4  |-  ( A  e.  CC  ->  (
( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
13 absvalsq 10077 . . . . 5  |-  ( B  e.  CC  ->  (
( abs `  B
) ^ 2 )  =  ( B  x.  ( * `  B
) ) )
14 mulcom 7164 . . . . . 6  |-  ( ( B  e.  CC  /\  ( * `  B
)  e.  CC )  ->  ( B  x.  ( * `  B
) )  =  ( ( * `  B
)  x.  B ) )
154, 14mpdan 412 . . . . 5  |-  ( B  e.  CC  ->  ( B  x.  ( * `  B ) )  =  ( ( * `  B )  x.  B
) )
1613, 15eqtrd 2114 . . . 4  |-  ( B  e.  CC  ->  (
( abs `  B
) ^ 2 )  =  ( ( * `
 B )  x.  B ) )
1712, 16oveqan12d 5562 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( abs `  A ) ^ 2 )  +  ( ( abs `  B ) ^ 2 ) )  =  ( ( A  x.  ( * `  A ) )  +  ( ( * `  B )  x.  B
) ) )
18 mulcl 7162 . . . . . 6  |-  ( ( A  e.  CC  /\  ( * `  B
)  e.  CC )  ->  ( A  x.  ( * `  B
) )  e.  CC )
194, 18sylan2 280 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  (
* `  B )
)  e.  CC )
2019addcjd 9982 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  ( * `  B
) )  +  ( * `  ( A  x.  ( * `  B ) ) ) )  =  ( 2  x.  ( Re `  ( A  x.  (
* `  B )
) ) ) )
21 cjmul 9910 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( * `  B
)  e.  CC )  ->  ( * `  ( A  x.  (
* `  B )
) )  =  ( ( * `  A
)  x.  ( * `
 ( * `  B ) ) ) )
224, 21sylan2 280 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  x.  ( * `  B ) ) )  =  ( ( * `
 A )  x.  ( * `  (
* `  B )
) ) )
23 cjcj 9908 . . . . . . . 8  |-  ( B  e.  CC  ->  (
* `  ( * `  B ) )  =  B )
2423adantl 271 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  (
* `  B )
)  =  B )
2524oveq2d 5559 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  A )  x.  (
* `  ( * `  B ) ) )  =  ( ( * `
 A )  x.  B ) )
2622, 25eqtrd 2114 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  x.  ( * `  B ) ) )  =  ( ( * `
 A )  x.  B ) )
2726oveq2d 5559 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  ( * `  B
) )  +  ( * `  ( A  x.  ( * `  B ) ) ) )  =  ( ( A  x.  ( * `
 B ) )  +  ( ( * `
 A )  x.  B ) ) )
2820, 27eqtr3d 2116 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  (
Re `  ( A  x.  ( * `  B
) ) ) )  =  ( ( A  x.  ( * `  B ) )  +  ( ( * `  A )  x.  B
) ) )
2917, 28oveq12d 5561 . 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 2124 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 4    /\ wa 102    = wceq 1285    e. wcel 1434   ` cfv 4932  (class class class)co 5543   CCcc 7041    + caddc 7046    x. cmul 7048   2c2 8156   ^cexp 9572   *ccj 9864   Recre 9865   abscabs 10021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-ltirr 7150  ax-pre-ltwlin 7151  ax-pre-lttrn 7152  ax-pre-apti 7153  ax-pre-ltadd 7154  ax-pre-mulgt0 7155  ax-pre-mulext 7156  ax-arch 7157  ax-caucvg 7158
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-if 3360  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-ilim 4132  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-frec 6040  df-pnf 7217  df-mnf 7218  df-xr 7219  df-ltxr 7220  df-le 7221  df-sub 7348  df-neg 7349  df-reap 7742  df-ap 7749  df-div 7828  df-inn 8107  df-2 8165  df-3 8166  df-4 8167  df-n0 8356  df-z 8433  df-uz 8701  df-rp 8816  df-iseq 9522  df-iexp 9573  df-cj 9867  df-re 9868  df-im 9869  df-rsqrt 10022  df-abs 10023
This theorem is referenced by:  abstri  10128  sqabsaddi  10176
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