Theorem rimul 8764

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 8763	. . 3 |- -. _i e. RR
2		recexre 8757	. . . . . 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 535	. . . . . . . . 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 8207	. . . . . . . 8 |- ( ( ( ( A e. RR /\ ( _i x. A ) e. RR ) /\ A #ℝ 0 ) /\ ( x e. RR /\ ( A x. x ) = 1 ) ) -> A e. CC )
6		simprl 531	. . . . . . . . 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 8207	. . . . . . . 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 8126	. . . . . . . . 9 |- _i e. CC
9		mulass 8162	. . . . . . . . 9 |- ( ( _i e. CC /\ A e. CC /\ x e. CC ) -> ( ( _i x. A ) x. x ) = ( _i x. ( A x. x ) ) )
10	8, 9	mp3an1 1360	. . . . . . . 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 6025	. . . . . . . . 9 |- ( ( A x. x ) = 1 -> ( _i x. ( A x. x ) ) = ( _i x. 1 ) )
13	8	mulridi 8180	. . . . . . . . 9 |- ( _i x. 1 ) = _i
14	12, 13	eqtrdi 2280	. . . . . . . 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 2264	. . . . . 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 536	. . . . . . 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 8209	. . . . . 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 2309	. . . . 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 2655	. . . 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 670	. 2 |- ( ( A e. RR /\ ( _i x. A ) e. RR ) -> -. A #ℝ 0 )
23		0re 8178	. . . 4 |- 0 e. RR
24		reapti 8758	. . . 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 1397   e. wcel 2202   E.wrex 2511   class class class wbr 4088  (class class class)co 6017   CCcc 8029   RRcr 8030   0cc0 8031   1c1 8032   _ici 8033   x. cmul 8036   #_ℝ creap 8753

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:  rereim  8765  cru  8781  cju  9140  crre  11417