Theorem rimul 8556

Description: A real number times the imaginary unit is real only if the number is 0. (Contributed by NM, 28-May-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
rimul	|- ( ( A e. RR /\ ( _i x. A ) e. RR ) -> A = 0 )

Proof of Theorem rimul

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inelr 8555	. . 3 |- -. _i e. RR
2		recexre 8549	. . . . . 6 |- ( ( A e. RR /\ A #ℝ 0 ) -> E. x e. RR ( A x. x ) = 1 )
3	2	adantlr 477	. . . . 5 |- ( ( ( A e. RR /\ ( _i x. A ) e. RR ) /\ A #ℝ 0 ) -> E. x e. RR ( A x. x ) = 1 )
4		simplll 533	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ ( _i x. A ) e. RR ) /\ A #ℝ 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> A e. RR )
5	4	recnd 8000	. . . . . . . 8 |- ( ( ( ( A e. RR /\ ( _i x. A ) e. RR ) /\ A #ℝ 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> A e. CC )
6		simprl 529	. . . . . . . . 9 |- ( ( ( ( A e. RR /\ ( _i x. A ) e. RR ) /\ A #ℝ 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> x e. RR )
7	6	recnd 8000	. . . . . . . 8 |- ( ( ( ( A e. RR /\ ( _i x. A ) e. RR ) /\ A #ℝ 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> x e. CC )
8		ax-icn 7920	. . . . . . . . 9 |- _i e. CC
9		mulass 7956	. . . . . . . . 9 |- ( ( _i e. CC /\ A e. CC /\ x e. CC ) -> ( ( _i x. A ) x. x ) = ( _i x. ( A x. x ) ) )
10	8, 9	mp3an1 1334	. . . . . . . 8 |- ( ( A e. CC /\ x e. CC ) -> ( ( _i x. A ) x. x ) = ( _i x. ( A x. x ) ) )
11	5, 7, 10	syl2anc 411	. . . . . . 7 |- ( ( ( ( A e. RR /\ ( _i x. A ) e. RR ) /\ A #ℝ 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> ( ( _i x. A ) x. x ) = ( _i x. ( A x. x ) ) )
12		oveq2 5896	. . . . . . . . 9 |- ( ( A x. x ) = 1 -> ( _i x. ( A x. x ) ) = ( _i x. 1 ) )
13	8	mulid1i 7973	. . . . . . . . 9 |- ( _i x. 1 ) = _i
14	12, 13	eqtrdi 2236	. . . . . . . 8 |- ( ( A x. x ) = 1 -> ( _i x. ( A x. x ) ) = _i )
15	14	ad2antll 491	. . . . . . 7 |- ( ( ( ( A e. RR /\ ( _i x. A ) e. RR ) /\ A #ℝ 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> ( _i x. ( A x. x ) ) = _i )
16	11, 15	eqtrd 2220	. . . . . 6 |- ( ( ( ( A e. RR /\ ( _i x. A ) e. RR ) /\ A #ℝ 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> ( ( _i x. A ) x. x ) = _i )
17		simpllr 534	. . . . . . 7 |- ( ( ( ( A e. RR /\ ( _i x. A ) e. RR ) /\ A #ℝ 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> ( _i x. A ) e. RR )
18	17, 6	remulcld 8002	. . . . . 6 |- ( ( ( ( A e. RR /\ ( _i x. A ) e. RR ) /\ A #ℝ 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> ( ( _i x. A ) x. x ) e. RR )
19	16, 18	eqeltrrd 2265	. . . . 5 |- ( ( ( ( A e. RR /\ ( _i x. A ) e. RR ) /\ A #ℝ 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> _i e. RR )
20	3, 19	rexlimddv 2609	. . . 4 |- ( ( ( A e. RR /\ ( _i x. A ) e. RR ) /\ A #ℝ 0 ) -> _i e. RR )
21	20	ex 115	. . 3 |- ( ( A e. RR /\ ( _i x. A ) e. RR ) -> ( A #ℝ 0 -> _i e. RR ) )
22	1, 21	mtoi 665	. 2 |- ( ( A e. RR /\ ( _i x. A ) e. RR ) -> -. A #ℝ 0 )
23		0re 7971	. . . 4 |- 0 e. RR
24		reapti 8550	. . . 4 |- ( ( A e. RR /\ 0 e. RR ) -> ( A = 0 <-> -. A #ℝ 0 ) )
25	23, 24	mpan2 425	. . 3 |- ( A e. RR -> ( A = 0 <-> -. A #ℝ 0 ) )
26	25	adantr 276	. 2 |- ( ( A e. RR /\ ( _i x. A ) e. RR ) -> ( A = 0 <-> -. A #ℝ 0 ) )
27	22, 26	mpbird 167	1 |- ( ( A e. RR /\ ( _i x. A ) e. RR ) -> A = 0 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1363   e. wcel 2158   E.wrex 2466   class class class wbr 4015  (class class class)co 5888   CCcc 7823   RRcr 7824   0cc0 7825   1c1 7826   _ici 7827   x. cmul 7830   #_ℝ creap 8545

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-ltxr 8011  df-sub 8144  df-neg 8145  df-reap 8546

This theorem is referenced by:  rereim  8557  cru  8573  cju  8932  crre  10880