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

Step	Hyp	Ref	Expression
1		cjadd 9909	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A + B ) ) = ( ( * ` A ) + ( * ` B ) ) )
2	1	oveq2d 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 )
5	3, 4	anim12i 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 ) ) ) )
7	5, 6	mpdan 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 ) ) ) )
8	2, 7	eqtrd 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 ) ) ) )
11	9, 10	syl 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 ) )
15	4, 14	mpdan 412	. . . . 5 |- ( B e. CC -> ( B x. ( * ` B ) ) = ( ( * ` B ) x. B ) )
16	13, 15	eqtrd 2114	. . . 4 |- ( B e. CC -> ( ( abs ` B ) ^ 2 ) = ( ( * ` B ) x. B ) )
17	12, 16	oveqan12d 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 )
19	4, 18	sylan2 280	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. ( * ` B ) ) e. CC )
20	19	addcjd 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 ) ) ) )
22	4, 21	sylan2 280	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A x. ( * ` B ) ) ) = ( ( * ` A ) x. ( * ` ( * ` B ) ) ) )
23		cjcj 9908	. . . . . . . 8 |- ( B e. CC -> ( * ` ( * ` B ) ) = B )
24	23	adantl 271	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( * ` B ) ) = B )
25	24	oveq2d 5559	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( * ` A ) x. ( * ` ( * ` B ) ) ) = ( ( * ` A ) x. B ) )
26	22, 25	eqtrd 2114	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A x. ( * ` B ) ) ) = ( ( * ` A ) x. B ) )
27	26	oveq2d 5559	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. ( * ` B ) ) + ( * ` ( A x. ( * ` B ) ) ) ) = ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) )
28	20, 27	eqtr3d 2116	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) = ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) )
29	17, 28	oveq12d 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 ) ) ) )
30	8, 11, 29	3eqtr4d 2124	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( abs ` ( A + B ) ) ^ 2 ) = ( ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) + ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) ) )

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