Theorem rimul 8604

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 8603	. . 3 |- -. _i e. RR
2		recexre 8597	. . . . . 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 8048	. . . . . . . 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 8048	. . . . . . . 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 7967	. . . . . . . . 9 |- _i e. CC
9		mulass 8003	. . . . . . . . 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 5926	. . . . . . . . 9 |- ( ( A x. x ) = 1 -> ( _i x. ( A x. x ) ) = ( _i x. 1 ) )
13	8	mulid1i 8021	. . . . . . . . 9 |- ( _i x. 1 ) = _i
14	12, 13	eqtrdi 2242	. . . . . . . 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 2226	. . . . . 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 8050	. . . . . 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 2271	. . . . 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 2616	. . . 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 8019	. . . 4 |- 0 e. RR
24		reapti 8598	. . . 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 2164   E.wrex 2473   class class class wbr 4029  (class class class)co 5918   CCcc 7870   RRcr 7871   0cc0 7872   1c1 7873   _ici 7874   x. cmul 7877   #_ℝ creap 8593

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-sub 8192  df-neg 8193  df-reap 8594

This theorem is referenced by:  rereim  8605  cru  8621  cju  8980  crre  11001