Theorem sqabsadd 11581

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 11410	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A + B ) ) = ( ( * ` A ) + ( * ` B ) ) )
2	1	oveq2d 6023	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. ( * ` ( A + B ) ) ) = ( ( A + B ) x. ( ( * ` A ) + ( * ` B ) ) ) )
3		cjcl 11374	. . . . 5 |- ( A e. CC -> ( * ` A ) e. CC )
4		cjcl 11374	. . . . 5 |- ( B e. CC -> ( * ` B ) e. CC )
5	3, 4	anim12i 338	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( * ` A ) e. CC /\ ( * ` B ) e. CC ) )
6		muladd 8541	. . . 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 421	. . 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 2262	. 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 8135	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
10		absvalsq 11579	. . 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 11579	. . . 4 |- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( A x. ( * ` A ) ) )
13		absvalsq 11579	. . . . 5 |- ( B e. CC -> ( ( abs ` B ) ^ 2 ) = ( B x. ( * ` B ) ) )
14		mulcom 8139	. . . . . 6 |- ( ( B e. CC /\ ( * ` B ) e. CC ) -> ( B x. ( * ` B ) ) = ( ( * ` B ) x. B ) )
15	4, 14	mpdan 421	. . . . 5 |- ( B e. CC -> ( B x. ( * ` B ) ) = ( ( * ` B ) x. B ) )
16	13, 15	eqtrd 2262	. . . 4 |- ( B e. CC -> ( ( abs ` B ) ^ 2 ) = ( ( * ` B ) x. B ) )
17	12, 16	oveqan12d 6026	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) = ( ( A x. ( * ` A ) ) + ( ( * ` B ) x. B ) ) )
18		mulcl 8137	. . . . . 6 |- ( ( A e. CC /\ ( * ` B ) e. CC ) -> ( A x. ( * ` B ) ) e. CC )
19	4, 18	sylan2 286	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. ( * ` B ) ) e. CC )
20	19	addcjd 11483	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. ( * ` B ) ) + ( * ` ( A x. ( * ` B ) ) ) ) = ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) )
21		cjmul 11411	. . . . . . 7 |- ( ( A e. CC /\ ( * ` B ) e. CC ) -> ( * ` ( A x. ( * ` B ) ) ) = ( ( * ` A ) x. ( * ` ( * ` B ) ) ) )
22	4, 21	sylan2 286	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A x. ( * ` B ) ) ) = ( ( * ` A ) x. ( * ` ( * ` B ) ) ) )
23		cjcj 11409	. . . . . . . 8 |- ( B e. CC -> ( * ` ( * ` B ) ) = B )
24	23	adantl 277	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( * ` B ) ) = B )
25	24	oveq2d 6023	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( * ` A ) x. ( * ` ( * ` B ) ) ) = ( ( * ` A ) x. B ) )
26	22, 25	eqtrd 2262	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A x. ( * ` B ) ) ) = ( ( * ` A ) x. B ) )
27	26	oveq2d 6023	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. ( * ` B ) ) + ( * ` ( A x. ( * ` B ) ) ) ) = ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) )
28	20, 27	eqtr3d 2264	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) = ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) )
29	17, 28	oveq12d 6025	. 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 2272	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 104   = wceq 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6007   CCcc 8008   + caddc 8013   x. cmul 8015   2c2 9172   ^cexp 10772   *ccj 11365   Recre 11366   abscabs 11523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-rp 9862  df-seqfrec 10682  df-exp 10773  df-cj 11368  df-re 11369  df-im 11370  df-rsqrt 11524  df-abs 11525

This theorem is referenced by:  abstri  11630  sqabsaddi  11678