Theorem rereim 8372

Description: Decomposition of a real number into real part (itself) and imaginary part (zero). (Contributed by Jim Kingdon, 30-Jan-2020.)

Assertion

Ref	Expression
rereim	|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( B = A /\ C = 0 ) )

Proof of Theorem rereim

Step	Hyp	Ref	Expression
1		simpll 519	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> A e. RR )
2	1	recnd 7818	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> A e. CC )
3		simplr 520	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> B e. RR )
4	3	recnd 7818	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> B e. CC )
5		simprr 522	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> A = ( B + ( _i x. C ) ) )
6	5	eqcomd 2146	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( B + ( _i x. C ) ) = A )
7		ax-icn 7739	. . . . . . . . 9 |- _i e. CC
8	7	a1i 9	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> _i e. CC )
9		simprl 521	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> C e. RR )
10	9	recnd 7818	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> C e. CC )
11	8, 10	mulcld 7810	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( _i x. C ) e. CC )
12	2, 4, 11	subaddd 8115	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( ( A - B ) = ( _i x. C ) <-> ( B + ( _i x. C ) ) = A ) )
13	6, 12	mpbird 166	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( A - B ) = ( _i x. C ) )
14	1, 3	resubcld 8167	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( A - B ) e. RR )
15	13, 14	eqeltrrd 2218	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( _i x. C ) e. RR )
16		rimul 8371	. . . . . . . 8 |- ( ( C e. RR /\ ( _i x. C ) e. RR ) -> C = 0 )
17	9, 15, 16	syl2anc 409	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> C = 0 )
18	17	oveq2d 5798	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( _i x. C ) = ( _i x. 0 ) )
19	7	mul01i 8177	. . . . . 6 |- ( _i x. 0 ) = 0
20	18, 19	eqtrdi 2189	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( _i x. C ) = 0 )
21	13, 20	eqtrd 2173	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( A - B ) = 0 )
22	2, 4, 21	subeq0d 8105	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> A = B )
23	22	eqcomd 2146	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> B = A )
24	23, 17	jca 304	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( B = A /\ C = 0 ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481  (class class class)co 5782   CCcc 7642   RRcr 7643   0cc0 7644   _ici 7646   + caddc 7647   x. cmul 7649   - cmin 7957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-ltxr 7829  df-sub 7959  df-neg 7960  df-reap 8361

This theorem is referenced by:  apreap  8373