Theorem cjreim 11409

Description: The conjugate of a representation of a complex number in terms of real and imaginary parts. (Contributed by NM, 1-Jul-2005.)

Assertion

Ref	Expression
cjreim	|- ( ( A e. RR /\ B e. RR ) -> ( * ` ( A + ( _i x. B ) ) ) = ( A - ( _i x. B ) ) )

Proof of Theorem cjreim

Step	Hyp	Ref	Expression
1		recn 8128	. . 3 |- ( A e. RR -> A e. CC )
2		ax-icn 8090	. . . 4 |- _i e. CC
3		recn 8128	. . . 4 |- ( B e. RR -> B e. CC )
4		mulcl 8122	. . . 4 |- ( ( _i e. CC /\ B e. CC ) -> ( _i x. B ) e. CC )
5	2, 3, 4	sylancr 414	. . 3 |- ( B e. RR -> ( _i x. B ) e. CC )
6		cjadd 11390	. . 3 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( * ` ( A + ( _i x. B ) ) ) = ( ( * ` A ) + ( * ` ( _i x. B ) ) ) )
7	1, 5, 6	syl2an 289	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( * ` ( A + ( _i x. B ) ) ) = ( ( * ` A ) + ( * ` ( _i x. B ) ) ) )
8		cjre 11388	. . 3 |- ( A e. RR -> ( * ` A ) = A )
9		cjmul 11391	. . . . 5 |- ( ( _i e. CC /\ B e. CC ) -> ( * ` ( _i x. B ) ) = ( ( * ` _i ) x. ( * ` B ) ) )
10	2, 3, 9	sylancr 414	. . . 4 |- ( B e. RR -> ( * ` ( _i x. B ) ) = ( ( * ` _i ) x. ( * ` B ) ) )
11		cji 11408	. . . . . 6 |- ( * ` _i ) = -u _i
12	11	a1i 9	. . . . 5 |- ( B e. RR -> ( * ` _i ) = -u _i )
13		cjre 11388	. . . . 5 |- ( B e. RR -> ( * ` B ) = B )
14	12, 13	oveq12d 6018	. . . 4 |- ( B e. RR -> ( ( * ` _i ) x. ( * ` B ) ) = ( -u _i x. B ) )
15		mulneg1 8537	. . . . 5 |- ( ( _i e. CC /\ B e. CC ) -> ( -u _i x. B ) = -u ( _i x. B ) )
16	2, 3, 15	sylancr 414	. . . 4 |- ( B e. RR -> ( -u _i x. B ) = -u ( _i x. B ) )
17	10, 14, 16	3eqtrd 2266	. . 3 |- ( B e. RR -> ( * ` ( _i x. B ) ) = -u ( _i x. B ) )
18	8, 17	oveqan12d 6019	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( * ` A ) + ( * ` ( _i x. B ) ) ) = ( A + -u ( _i x. B ) ) )
19		negsub 8390	. . 3 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( A + -u ( _i x. B ) ) = ( A - ( _i x. B ) ) )
20	1, 5, 19	syl2an 289	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A + -u ( _i x. B ) ) = ( A - ( _i x. B ) ) )
21	7, 18, 20	3eqtrd 2266	1 |- ( ( A e. RR /\ B e. RR ) -> ( * ` ( A + ( _i x. B ) ) ) = ( A - ( _i x. B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   ` cfv 5317  (class class class)co 6000   CCcc 7993   RRcr 7994   _ici 7997   + caddc 7998   x. cmul 8000   - cmin 8313   -ucneg 8314   *ccj 11345

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-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113

This theorem depends on definitions:  df-bi 117  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-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-po 4386  df-iso 4387  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-2 9165  df-cj 11348  df-re 11349  df-im 11350

This theorem is referenced by:  cjreim2  11410  cjap  11412