Theorem cnsubrg 19806

Description: There are no subrings of the complex numbers strictly between RR and CC. (Contributed by Mario Carneiro, 15-Oct-2015.)

Assertion

Ref	Expression
cnsubrg	|- ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) -> R e. { RR , CC } )

Proof of Theorem cnsubrg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssdif0 3942	. . . 4 |- ( R C_ RR <-> ( R \ RR ) = (/) )
2		simpr 477	. . . . . 6 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ R C_ RR ) -> R C_ RR )
3		simplr 792	. . . . . 6 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ R C_ RR ) -> RR C_ R )
4	2, 3	eqssd 3620	. . . . 5 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ R C_ RR ) -> R = RR )
5	4	orcd 407	. . . 4 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ R C_ RR ) -> ( R = RR \/ R = CC ) )
6	1, 5	sylan2br 493	. . 3 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ ( R \ RR ) = (/) ) -> ( R = RR \/ R = CC ) )
7		n0 3931	. . . 4 |- ( ( R \ RR ) =/= (/) <-> E. x x e. ( R \ RR ) )
8		simpll 790	. . . . . . . . . 10 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> R e. (SubRing ` ℂfld ) )
9		cnfldbas 19750	. . . . . . . . . . 11 |- CC = ( Base ` ℂfld )
10	9	subrgss 18781	. . . . . . . . . 10 |- ( R e. (SubRing ` ℂfld ) -> R C_ CC )
11	8, 10	syl 17	. . . . . . . . 9 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> R C_ CC )
12		replim 13856	. . . . . . . . . . . . 13 |- ( y e. CC -> y = ( ( Re ` y ) + ( _i x. ( Im ` y ) ) ) )
13	12	ad2antll 765	. . . . . . . . . . . 12 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ ( x e. ( R \ RR ) /\ y e. CC ) ) -> y = ( ( Re ` y ) + ( _i x. ( Im ` y ) ) ) )
14		simpll 790	. . . . . . . . . . . . 13 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ ( x e. ( R \ RR ) /\ y e. CC ) ) -> R e. (SubRing ` ℂfld ) )
15		simplr 792	. . . . . . . . . . . . . 14 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ ( x e. ( R \ RR ) /\ y e. CC ) ) -> RR C_ R )
16		recl 13850	. . . . . . . . . . . . . . 15 |- ( y e. CC -> ( Re ` y ) e. RR )
17	16	ad2antll 765	. . . . . . . . . . . . . 14 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ ( x e. ( R \ RR ) /\ y e. CC ) ) -> ( Re ` y ) e. RR )
18	15, 17	sseldd 3604	. . . . . . . . . . . . 13 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ ( x e. ( R \ RR ) /\ y e. CC ) ) -> ( Re ` y ) e. R )
19		ax-icn 9995	. . . . . . . . . . . . . . . . . . 19 |- _i e. CC
20	19	a1i 11	. . . . . . . . . . . . . . . . . 18 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> _i e. CC )
21		eldifi 3732	. . . . . . . . . . . . . . . . . . . . . 22 |- ( x e. ( R \ RR ) -> x e. R )
22	21	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> x e. R )
23	11, 22	sseldd 3604	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> x e. CC )
24		imcl 13851	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. CC -> ( Im ` x ) e. RR )
25	23, 24	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( Im ` x ) e. RR )
26	25	recnd 10068	. . . . . . . . . . . . . . . . . 18 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( Im ` x ) e. CC )
27		eldifn 3733	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. ( R \ RR ) -> -. x e. RR )
28	27	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> -. x e. RR )
29		reim0b 13859	. . . . . . . . . . . . . . . . . . . . 21 |- ( x e. CC -> ( x e. RR <-> ( Im ` x ) = 0 ) )
30	29	necon3bbid 2831	. . . . . . . . . . . . . . . . . . . 20 |- ( x e. CC -> ( -. x e. RR <-> ( Im ` x ) =/= 0 ) )
31	23, 30	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( -. x e. RR <-> ( Im ` x ) =/= 0 ) )
32	28, 31	mpbid 222	. . . . . . . . . . . . . . . . . 18 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( Im ` x ) =/= 0 )
33	20, 26, 32	divcan4d 10807	. . . . . . . . . . . . . . . . 17 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( ( _i x. ( Im ` x ) ) / ( Im ` x ) ) = _i )
34		mulcl 10020	. . . . . . . . . . . . . . . . . . 19 |- ( ( _i e. CC /\ ( Im ` x ) e. CC ) -> ( _i x. ( Im ` x ) ) e. CC )
35	19, 26, 34	sylancr 695	. . . . . . . . . . . . . . . . . 18 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( _i x. ( Im ` x ) ) e. CC )
36	35, 26, 32	divrecd 10804	. . . . . . . . . . . . . . . . 17 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( ( _i x. ( Im ` x ) ) / ( Im ` x ) ) = ( ( _i x. ( Im ` x ) ) x. ( 1 / ( Im ` x ) ) ) )
37	33, 36	eqtr3d 2658	. . . . . . . . . . . . . . . 16 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> _i = ( ( _i x. ( Im ` x ) ) x. ( 1 / ( Im ` x ) ) ) )
38	23	recld 13934	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( Re ` x ) e. RR )
39	38	recnd 10068	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( Re ` x ) e. CC )
40	23, 39	negsubd 10398	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( x + -u ( Re ` x ) ) = ( x - ( Re ` x ) ) )
41		replim 13856	. . . . . . . . . . . . . . . . . . . . 21 |- ( x e. CC -> x = ( ( Re ` x ) + ( _i x. ( Im ` x ) ) ) )
42	23, 41	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> x = ( ( Re ` x ) + ( _i x. ( Im ` x ) ) ) )
43	42	oveq1d 6665	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( x - ( Re ` x ) ) = ( ( ( Re ` x ) + ( _i x. ( Im ` x ) ) ) - ( Re ` x ) ) )
44	39, 35	pncan2d 10394	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( ( ( Re ` x ) + ( _i x. ( Im ` x ) ) ) - ( Re ` x ) ) = ( _i x. ( Im ` x ) ) )
45	40, 43, 44	3eqtrd 2660	. . . . . . . . . . . . . . . . . 18 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( x + -u ( Re ` x ) ) = ( _i x. ( Im ` x ) ) )
46		simplr 792	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> RR C_ R )
47	38	renegcld 10457	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> -u ( Re ` x ) e. RR )
48	46, 47	sseldd 3604	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> -u ( Re ` x ) e. R )
49		cnfldadd 19751	. . . . . . . . . . . . . . . . . . . 20 |- + = ( +g ` ℂfld )
50	49	subrgacl 18791	. . . . . . . . . . . . . . . . . . 19 |- ( ( R e. (SubRing ` ℂfld ) /\ x e. R /\ -u ( Re ` x ) e. R ) -> ( x + -u ( Re ` x ) ) e. R )
51	8, 22, 48, 50	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( x + -u ( Re ` x ) ) e. R )
52	45, 51	eqeltrrd 2702	. . . . . . . . . . . . . . . . 17 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( _i x. ( Im ` x ) ) e. R )
53	25, 32	rereccld 10852	. . . . . . . . . . . . . . . . . 18 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( 1 / ( Im ` x ) ) e. RR )
54	46, 53	sseldd 3604	. . . . . . . . . . . . . . . . 17 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( 1 / ( Im ` x ) ) e. R )
55		cnfldmul 19752	. . . . . . . . . . . . . . . . . 18 |- x. = ( .r ` ℂfld )
56	55	subrgmcl 18792	. . . . . . . . . . . . . . . . 17 |- ( ( R e. (SubRing ` ℂfld ) /\ ( _i x. ( Im ` x ) ) e. R /\ ( 1 / ( Im ` x ) ) e. R ) -> ( ( _i x. ( Im ` x ) ) x. ( 1 / ( Im ` x ) ) ) e. R )
57	8, 52, 54, 56	syl3anc 1326	. . . . . . . . . . . . . . . 16 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( ( _i x. ( Im ` x ) ) x. ( 1 / ( Im ` x ) ) ) e. R )
58	37, 57	eqeltrd 2701	. . . . . . . . . . . . . . 15 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> _i e. R )
59	58	adantrr 753	. . . . . . . . . . . . . 14 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ ( x e. ( R \ RR ) /\ y e. CC ) ) -> _i e. R )
60		imcl 13851	. . . . . . . . . . . . . . . 16 |- ( y e. CC -> ( Im ` y ) e. RR )
61	60	ad2antll 765	. . . . . . . . . . . . . . 15 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ ( x e. ( R \ RR ) /\ y e. CC ) ) -> ( Im ` y ) e. RR )
62	15, 61	sseldd 3604	. . . . . . . . . . . . . 14 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ ( x e. ( R \ RR ) /\ y e. CC ) ) -> ( Im ` y ) e. R )
63	55	subrgmcl 18792	. . . . . . . . . . . . . 14 |- ( ( R e. (SubRing ` ℂfld ) /\ _i e. R /\ ( Im ` y ) e. R ) -> ( _i x. ( Im ` y ) ) e. R )
64	14, 59, 62, 63	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ ( x e. ( R \ RR ) /\ y e. CC ) ) -> ( _i x. ( Im ` y ) ) e. R )
65	49	subrgacl 18791	. . . . . . . . . . . . 13 |- ( ( R e. (SubRing ` ℂfld ) /\ ( Re ` y ) e. R /\ ( _i x. ( Im ` y ) ) e. R ) -> ( ( Re ` y ) + ( _i x. ( Im ` y ) ) ) e. R )
66	14, 18, 64, 65	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ ( x e. ( R \ RR ) /\ y e. CC ) ) -> ( ( Re ` y ) + ( _i x. ( Im ` y ) ) ) e. R )
67	13, 66	eqeltrd 2701	. . . . . . . . . . 11 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ ( x e. ( R \ RR ) /\ y e. CC ) ) -> y e. R )
68	67	expr 643	. . . . . . . . . 10 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( y e. CC -> y e. R ) )
69	68	ssrdv 3609	. . . . . . . . 9 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> CC C_ R )
70	11, 69	eqssd 3620	. . . . . . . 8 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> R = CC )
71	70	olcd 408	. . . . . . 7 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ x e. ( R \ RR ) ) -> ( R = RR \/ R = CC ) )
72	71	ex 450	. . . . . 6 |- ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) -> ( x e. ( R \ RR ) -> ( R = RR \/ R = CC ) ) )
73	72	exlimdv 1861	. . . . 5 |- ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) -> ( E. x x e. ( R \ RR ) -> ( R = RR \/ R = CC ) ) )
74	73	imp 445	. . . 4 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ E. x x e. ( R \ RR ) ) -> ( R = RR \/ R = CC ) )
75	7, 74	sylan2b 492	. . 3 |- ( ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) /\ ( R \ RR ) =/= (/) ) -> ( R = RR \/ R = CC ) )
76	6, 75	pm2.61dane 2881	. 2 |- ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) -> ( R = RR \/ R = CC ) )
77		elprg 4196	. . 3 |- ( R e. (SubRing ` ℂfld ) -> ( R e. { RR , CC } <-> ( R = RR \/ R = CC ) ) )
78	77	adantr 481	. 2 |- ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) -> ( R e. { RR , CC } <-> ( R = RR \/ R = CC ) ) )
79	76, 78	mpbird 247	1 |- ( ( R e. (SubRing ` ℂfld ) /\ RR C_ R ) -> R e. { RR , CC } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   (/)c0 3915   {cpr 4179   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   _ici 9938   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   Recre 13837   Imcim 13838  SubRingcsubrg 18776  ℂ_fldccnfld 19746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-cj 13839  df-re 13840  df-im 13841  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-subg 17591  df-mgp 18490  df-ring 18549  df-subrg 18778  df-cnfld 19747

This theorem is referenced by:  cncdrg  23155