Theorem cjreim 11647

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 8302	. . 3 |- ( A e. RR -> A e. CC )
2		ax-icn 8264	. . . 4 |- _i e. CC
3		recn 8302	. . . 4 |- ( B e. RR -> B e. CC )
4		mulcl 8296	. . . 4 |- ( ( _i e. CC /\ B e. CC ) -> ( _i x. B ) e. CC )
5	2, 3, 4	sylancr 418	. . 3 |- ( B e. RR -> ( _i x. B ) e. CC )
6		cjadd 11627	. . 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 11625	. . 3 |- ( A e. RR -> ( * ` A ) = A )
9		cjmul 11628	. . . . 5 |- ( ( _i e. CC /\ B e. CC ) -> ( * ` ( _i x. B ) ) = ( ( * ` _i ) x. ( * ` B ) ) )
10	2, 3, 9	sylancr 418	. . . 4 |- ( B e. RR -> ( * ` ( _i x. B ) ) = ( ( * ` _i ) x. ( * ` B ) ) )
11		cji 11646	. . . . . 6 |- ( * ` _i ) = -u _i
12	11	a1i 9	. . . . 5 |- ( B e. RR -> ( * ` _i ) = -u _i )
13		cjre 11625	. . . . 5 |- ( B e. RR -> ( * ` B ) = B )
14	12, 13	oveq12d 6093	. . . 4 |- ( B e. RR -> ( ( * ` _i ) x. ( * ` B ) ) = ( -u _i x. B ) )
15		mulneg1 8712	. . . . 5 |- ( ( _i e. CC /\ B e. CC ) -> ( -u _i x. B ) = -u ( _i x. B ) )
16	2, 3, 15	sylancr 418	. . . 4 |- ( B e. RR -> ( -u _i x. B ) = -u ( _i x. B ) )
17	10, 14, 16	3eqtrd 2275	. . 3 |- ( B e. RR -> ( * ` ( _i x. B ) ) = -u ( _i x. B ) )
18	8, 17	oveqan12d 6094	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( * ` A ) + ( * ` ( _i x. B ) ) ) = ( A + -u ( _i x. B ) ) )
19		negsub 8564	. . 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 2275	1 |- ( ( A e. RR /\ B e. RR ) -> ( * ` ( A + ( _i x. B ) ) ) = ( A - ( _i x. B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   ` cfv 5372  (class class class)co 6075   CCcc 8167   RRcr 8168   _ici 8171   + caddc 8172   x. cmul 8174   - cmin 8487   -ucneg 8488   *ccj 11582

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-2 9342  df-cj 11585  df-re 11586  df-im 11587

This theorem is referenced by:  cjreim2  11648  cjap  11650