Theorem sqabsadd 11481

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 11310	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A + B ) ) = ( ( * ` A ) + ( * ` B ) ) )
2	1	oveq2d 5983	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) x. ( * ` ( A + B ) ) ) = ( ( A + B ) x. ( ( * ` A ) + ( * ` B ) ) ) )
3		cjcl 11274	. . . . 5 |- ( A e. CC -> ( * ` A ) e. CC )
4		cjcl 11274	. . . . 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 8491	. . . 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 2240	. 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 8085	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
10		absvalsq 11479	. . 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 11479	. . . 4 |- ( A e. CC -> ( ( abs ` A ) ^ 2 ) = ( A x. ( * ` A ) ) )
13		absvalsq 11479	. . . . 5 |- ( B e. CC -> ( ( abs ` B ) ^ 2 ) = ( B x. ( * ` B ) ) )
14		mulcom 8089	. . . . . 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 2240	. . . 4 |- ( B e. CC -> ( ( abs ` B ) ^ 2 ) = ( ( * ` B ) x. B ) )
17	12, 16	oveqan12d 5986	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( abs ` A ) ^ 2 ) + ( ( abs ` B ) ^ 2 ) ) = ( ( A x. ( * ` A ) ) + ( ( * ` B ) x. B ) ) )
18		mulcl 8087	. . . . . 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 11383	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. ( * ` B ) ) + ( * ` ( A x. ( * ` B ) ) ) ) = ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) )
21		cjmul 11311	. . . . . . 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 11309	. . . . . . . 8 |- ( B e. CC -> ( * ` ( * ` B ) ) = B )
24	23	adantl 277	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( * ` B ) ) = B )
25	24	oveq2d 5983	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( ( * ` A ) x. ( * ` ( * ` B ) ) ) = ( ( * ` A ) x. B ) )
26	22, 25	eqtrd 2240	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( * ` ( A x. ( * ` B ) ) ) = ( ( * ` A ) x. B ) )
27	26	oveq2d 5983	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. ( * ` B ) ) + ( * ` ( A x. ( * ` B ) ) ) ) = ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) )
28	20, 27	eqtr3d 2242	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( Re ` ( A x. ( * ` B ) ) ) ) = ( ( A x. ( * ` B ) ) + ( ( * ` A ) x. B ) ) )
29	17, 28	oveq12d 5985	. 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 2250	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 1373   e. wcel 2178   ` cfv 5290  (class class class)co 5967   CCcc 7958   + caddc 7963   x. cmul 7965   2c2 9122   ^cexp 10720   *ccj 11265   Recre 11266   abscabs 11423

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-rp 9811  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425

This theorem is referenced by:  abstri  11530  sqabsaddi  11578