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Theorem sqabsadd 11732
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 11591 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  +  B )
)  =  ( ( * `  A )  +  ( * `  B ) ) )
21oveq2d 5808 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  x.  (
* `  ( A  +  B ) ) )  =  ( ( A  +  B )  x.  ( ( * `  A )  +  ( * `  B ) ) ) )
3 cjcl 11555 . . . . 5  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
4 cjcl 11555 . . . . 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 9180 . . . 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 652 . . 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 2290 . 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 8787 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  e.  CC )
10 absvalsq 11730 . . 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 11730 . . . 4  |-  ( A  e.  CC  ->  (
( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
13 absvalsq 11730 . . . . 5  |-  ( B  e.  CC  ->  (
( abs `  B
) ^ 2 )  =  ( B  x.  ( * `  B
) ) )
14 mulcom 8791 . . . . . 6  |-  ( ( B  e.  CC  /\  ( * `  B
)  e.  CC )  ->  ( B  x.  ( * `  B
) )  =  ( ( * `  B
)  x.  B ) )
154, 14mpdan 652 . . . . 5  |-  ( B  e.  CC  ->  ( B  x.  ( * `  B ) )  =  ( ( * `  B )  x.  B
) )
1613, 15eqtrd 2290 . . . 4  |-  ( B  e.  CC  ->  (
( abs `  B
) ^ 2 )  =  ( ( * `
 B )  x.  B ) )
1712, 16oveqan12d 5811 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( abs `  A ) ^ 2 )  +  ( ( abs `  B ) ^ 2 ) )  =  ( ( A  x.  ( * `  A ) )  +  ( ( * `  B )  x.  B
) ) )
18 mulcl 8789 . . . . . 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 11662 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  ( * `  B
) )  +  ( * `  ( A  x.  ( * `  B ) ) ) )  =  ( 2  x.  ( Re `  ( A  x.  (
* `  B )
) ) ) )
21 cjmul 11592 . . . . . . 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 11590 . . . . . . . 8  |-  ( B  e.  CC  ->  (
* `  ( * `  B ) )  =  B )
2423adantl 454 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  (
* `  B )
)  =  B )
2524oveq2d 5808 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  A )  x.  (
* `  ( * `  B ) ) )  =  ( ( * `
 A )  x.  B ) )
2622, 25eqtrd 2290 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  x.  ( * `  B ) ) )  =  ( ( * `
 A )  x.  B ) )
2726oveq2d 5808 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  ( * `  B
) )  +  ( * `  ( A  x.  ( * `  B ) ) ) )  =  ( ( A  x.  ( * `
 B ) )  +  ( ( * `
 A )  x.  B ) ) )
2820, 27eqtr3d 2292 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  (
Re `  ( A  x.  ( * `  B
) ) ) )  =  ( ( A  x.  ( * `  B ) )  +  ( ( * `  A )  x.  B
) ) )
2917, 28oveq12d 5810 . 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 2300 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 1619    e. wcel 1621   ` cfv 4673  (class class class)co 5792   CCcc 8703    + caddc 8708    x. cmul 8710   2c2 9763   ^cexp 11070   *ccj 11546   Recre 11547   abscabs 11684
This theorem is referenced by:  abstri  11779  sqabsaddi  11853  cncph  21357
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 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-sup 7162  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-n0 9933  df-z 9992  df-uz 10198  df-rp 10322  df-seq 11013  df-exp 11071  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-abs 11686
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