Theorem rereim 8765

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 527	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> A e. RR )
2	1	recnd 8207	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> A e. CC )
3		simplr 529	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> B e. RR )
4	3	recnd 8207	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> B e. CC )
5		simprr 533	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> A = ( B + ( _i x. C ) ) )
6	5	eqcomd 2237	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( B + ( _i x. C ) ) = A )
7		ax-icn 8126	. . . . . . . . 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 531	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> C e. RR )
10	9	recnd 8207	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> C e. CC )
11	8, 10	mulcld 8199	. . . . . . 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 8507	. . . . . 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 167	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( A - B ) = ( _i x. C ) )
14	1, 3	resubcld 8559	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( A - B ) e. RR )
15	13, 14	eqeltrrd 2309	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( _i x. C ) e. RR )
16		rimul 8764	. . . . . . . 8 |- ( ( C e. RR /\ ( _i x. C ) e. RR ) -> C = 0 )
17	9, 15, 16	syl2anc 411	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> C = 0 )
18	17	oveq2d 6033	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( _i x. C ) = ( _i x. 0 ) )
19	7	mul01i 8569	. . . . . 6 |- ( _i x. 0 ) = 0
20	18, 19	eqtrdi 2280	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( _i x. C ) = 0 )
21	13, 20	eqtrd 2264	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( A - B ) = 0 )
22	2, 4, 21	subeq0d 8497	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> A = B )
23	22	eqcomd 2237	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> B = A )
24	23, 17	jca 306	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ A = ( B + ( _i x. C ) ) ) ) -> ( B = A /\ C = 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202  (class class class)co 6017   CCcc 8029   RRcr 8030   0cc0 8031   _ici 8033   + caddc 8034   x. cmul 8036   - cmin 8349

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-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148

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-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-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-sub 8351  df-neg 8352  df-reap 8754

This theorem is referenced by:  apreap  8766