Theorem rimul 8612

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 8611	. . 3 |- -. _i e. RR
2		recexre 8605	. . . . . 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 8055	. . . . . . . 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 8055	. . . . . . . 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 7974	. . . . . . . . 9 |- _i e. CC
9		mulass 8010	. . . . . . . . 9 |- ( ( _i e. CC /\ A e. CC /\ x e. CC ) -> ( ( _i x. A ) x. x ) = ( _i x. ( A x. x ) ) )
10	8, 9	mp3an1 1335	. . . . . . . 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 5930	. . . . . . . . 9 |- ( ( A x. x ) = 1 -> ( _i x. ( A x. x ) ) = ( _i x. 1 ) )
13	8	mulridi 8028	. . . . . . . . 9 |- ( _i x. 1 ) = _i
14	12, 13	eqtrdi 2245	. . . . . . . 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 2229	. . . . . 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 8057	. . . . . 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 2274	. . . . 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 2619	. . . 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 8026	. . . 4 |- 0 e. RR
24		reapti 8606	. . . 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 1364   e. wcel 2167   E.wrex 2476   class class class wbr 4033  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   1c1 7880   _ici 7881   x. cmul 7884   #_ℝ creap 8601

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-sub 8199  df-neg 8200  df-reap 8602

This theorem is referenced by:  rereim  8613  cru  8629  cju  8988  crre  11022