Theorem cjreim 11463

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 8164	. . 3 |- ( A e. RR -> A e. CC )
2		ax-icn 8126	. . . 4 |- _i e. CC
3		recn 8164	. . . 4 |- ( B e. RR -> B e. CC )
4		mulcl 8158	. . . 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 11444	. . 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 11442	. . 3 |- ( A e. RR -> ( * ` A ) = A )
9		cjmul 11445	. . . . 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 11462	. . . . . 6 |- ( * ` _i ) = -u _i
12	11	a1i 9	. . . . 5 |- ( B e. RR -> ( * ` _i ) = -u _i )
13		cjre 11442	. . . . 5 |- ( B e. RR -> ( * ` B ) = B )
14	12, 13	oveq12d 6035	. . . 4 |- ( B e. RR -> ( ( * ` _i ) x. ( * ` B ) ) = ( -u _i x. B ) )
15		mulneg1 8573	. . . . 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 2268	. . 3 |- ( B e. RR -> ( * ` ( _i x. B ) ) = -u ( _i x. B ) )
18	8, 17	oveqan12d 6036	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( * ` A ) + ( * ` ( _i x. B ) ) ) = ( A + -u ( _i x. B ) ) )
19		negsub 8426	. . 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 2268	1 |- ( ( A e. RR /\ B e. RR ) -> ( * ` ( A + ( _i x. B ) ) ) = ( A - ( _i x. B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   CCcc 8029   RRcr 8030   _ici 8033   + caddc 8034   x. cmul 8036   - cmin 8349   -ucneg 8350   *ccj 11399

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-2 9201  df-cj 11402  df-re 11403  df-im 11404

This theorem is referenced by:  cjreim2  11464  cjap  11466