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Theorem cnsubrg 16432
Description: There are no subrings of the complexes strictly between  RR and  CC. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
cnsubrg  |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  R  e.  { RR ,  CC } )

Proof of Theorem cnsubrg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssdif0 3513 . . . 4  |-  ( R 
C_  RR  <->  ( R  \  RR )  =  (/) )
2 simpr 447 . . . . . 6  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  R  C_  RR )  ->  R  C_  RR )
3 simplr 731 . . . . . 6  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  R  C_  RR )  ->  RR  C_  R
)
42, 3eqssd 3196 . . . . 5  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  R  C_  RR )  ->  R  =  RR )
54orcd 381 . . . 4  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  R  C_  RR )  ->  ( R  =  RR  \/  R  =  CC ) )
61, 5sylan2br 462 . . 3  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( R  \  RR )  =  (/) )  ->  ( R  =  RR  \/  R  =  CC ) )
7 n0 3464 . . . 4  |-  ( ( R  \  RR )  =/=  (/)  <->  E. x  x  e.  ( R  \  RR ) )
8 simpll 730 . . . . . . . . . 10  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  R  e.  (SubRing ` fld ) )
9 cnfldbas 16383 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
109subrgss 15546 . . . . . . . . . 10  |-  ( R  e.  (SubRing ` fld )  ->  R  C_  CC )
118, 10syl 15 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  R  C_  CC )
12 replim 11601 . . . . . . . . . . . . 13  |-  ( y  e.  CC  ->  y  =  ( ( Re
`  y )  +  ( _i  x.  (
Im `  y )
) ) )
1312ad2antll 709 . . . . . . . . . . . 12  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  y  =  ( ( Re
`  y )  +  ( _i  x.  (
Im `  y )
) ) )
14 simpll 730 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  R  e.  (SubRing ` fld ) )
15 simplr 731 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  RR  C_  R )
16 recl 11595 . . . . . . . . . . . . . . 15  |-  ( y  e.  CC  ->  (
Re `  y )  e.  RR )
1716ad2antll 709 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Re `  y )  e.  RR )
1815, 17sseldd 3181 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Re `  y )  e.  R )
19 ax-icn 8796 . . . . . . . . . . . . . . . . . . 19  |-  _i  e.  CC
2019a1i 10 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  _i  e.  CC )
21 eldifi 3298 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  e.  ( R  \  RR )  ->  x  e.  R )
2221adantl 452 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  x  e.  R )
2311, 22sseldd 3181 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  x  e.  CC )
24 imcl 11596 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  CC  ->  (
Im `  x )  e.  RR )
2523, 24syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Im `  x )  e.  RR )
2625recnd 8861 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Im `  x )  e.  CC )
27 eldifn 3299 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( R  \  RR )  ->  -.  x  e.  RR )
2827adantl 452 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  -.  x  e.  RR )
29 reim0b 11604 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  CC  ->  (
x  e.  RR  <->  ( Im `  x )  =  0 ) )
3029necon3bbid 2480 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  CC  ->  ( -.  x  e.  RR  <->  ( Im `  x )  =/=  0 ) )
3123, 30syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( -.  x  e.  RR  <->  ( Im `  x )  =/=  0
) )
3228, 31mpbid 201 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Im `  x )  =/=  0
)
3320, 26, 32divcan4d 9542 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( (
_i  x.  ( Im `  x ) )  / 
( Im `  x
) )  =  _i )
34 mulcl 8821 . . . . . . . . . . . . . . . . . . 19  |-  ( ( _i  e.  CC  /\  ( Im `  x )  e.  CC )  -> 
( _i  x.  (
Im `  x )
)  e.  CC )
3519, 26, 34sylancr 644 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( _i  x.  ( Im `  x
) )  e.  CC )
3635, 26, 32divrecd 9539 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( (
_i  x.  ( Im `  x ) )  / 
( Im `  x
) )  =  ( ( _i  x.  (
Im `  x )
)  x.  ( 1  /  ( Im `  x ) ) ) )
3733, 36eqtr3d 2317 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  _i  =  ( ( _i  x.  ( Im `  x ) )  x.  ( 1  /  ( Im `  x ) ) ) )
3823recld 11679 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Re `  x )  e.  RR )
3938recnd 8861 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Re `  x )  e.  CC )
4023, 39negsubd 9163 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( x  +  -u ( Re `  x ) )  =  ( x  -  (
Re `  x )
) )
41 replim 11601 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  CC  ->  x  =  ( ( Re
`  x )  +  ( _i  x.  (
Im `  x )
) ) )
4223, 41syl 15 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  x  =  ( ( Re `  x )  +  ( _i  x.  ( Im
`  x ) ) ) )
4342oveq1d 5873 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( x  -  ( Re `  x ) )  =  ( ( ( Re
`  x )  +  ( _i  x.  (
Im `  x )
) )  -  (
Re `  x )
) )
4439, 35pncan2d 9159 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( (
( Re `  x
)  +  ( _i  x.  ( Im `  x ) ) )  -  ( Re `  x ) )  =  ( _i  x.  (
Im `  x )
) )
4540, 43, 443eqtrd 2319 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( x  +  -u ( Re `  x ) )  =  ( _i  x.  (
Im `  x )
) )
46 simplr 731 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  RR  C_  R
)
4738renegcld 9210 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  -u ( Re
`  x )  e.  RR )
4846, 47sseldd 3181 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  -u ( Re
`  x )  e.  R )
49 cnfldadd 16384 . . . . . . . . . . . . . . . . . . . 20  |-  +  =  ( +g  ` fld )
5049subrgacl 15556 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R  e.  (SubRing ` fld )  /\  x  e.  R  /\  -u (
Re `  x )  e.  R )  ->  (
x  +  -u (
Re `  x )
)  e.  R )
518, 22, 48, 50syl3anc 1182 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( x  +  -u ( Re `  x ) )  e.  R )
5245, 51eqeltrrd 2358 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( _i  x.  ( Im `  x
) )  e.  R
)
5325, 32rereccld 9587 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( 1  /  ( Im `  x ) )  e.  RR )
5446, 53sseldd 3181 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( 1  /  ( Im `  x ) )  e.  R )
55 cnfldmul 16385 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
5655subrgmcl 15557 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  (SubRing ` fld )  /\  (
_i  x.  ( Im `  x ) )  e.  R  /\  ( 1  /  ( Im `  x ) )  e.  R )  ->  (
( _i  x.  (
Im `  x )
)  x.  ( 1  /  ( Im `  x ) ) )  e.  R )
578, 52, 54, 56syl3anc 1182 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( (
_i  x.  ( Im `  x ) )  x.  ( 1  /  (
Im `  x )
) )  e.  R
)
5837, 57eqeltrd 2357 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  _i  e.  R )
5958adantrr 697 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  _i  e.  R )
60 imcl 11596 . . . . . . . . . . . . . . . 16  |-  ( y  e.  CC  ->  (
Im `  y )  e.  RR )
6160ad2antll 709 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Im `  y )  e.  RR )
6215, 61sseldd 3181 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Im `  y )  e.  R )
6355subrgmcl 15557 . . . . . . . . . . . . . 14  |-  ( ( R  e.  (SubRing ` fld )  /\  _i  e.  R  /\  ( Im `  y )  e.  R
)  ->  ( _i  x.  ( Im `  y
) )  e.  R
)
6414, 59, 62, 63syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
_i  x.  ( Im `  y ) )  e.  R )
6549subrgacl 15556 . . . . . . . . . . . . 13  |-  ( ( R  e.  (SubRing ` fld )  /\  (
Re `  y )  e.  R  /\  (
_i  x.  ( Im `  y ) )  e.  R )  ->  (
( Re `  y
)  +  ( _i  x.  ( Im `  y ) ) )  e.  R )
6614, 18, 64, 65syl3anc 1182 . . . . . . . . . . . 12  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
( Re `  y
)  +  ( _i  x.  ( Im `  y ) ) )  e.  R )
6713, 66eqeltrd 2357 . . . . . . . . . . 11  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  y  e.  R )
6867expr 598 . . . . . . . . . 10  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( y  e.  CC  ->  y  e.  R ) )
6968ssrdv 3185 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  CC  C_  R
)
7011, 69eqssd 3196 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  R  =  CC )
7170olcd 382 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( R  =  RR  \/  R  =  CC ) )
7271ex 423 . . . . . 6  |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  (
x  e.  ( R 
\  RR )  -> 
( R  =  RR  \/  R  =  CC ) ) )
7372exlimdv 1664 . . . . 5  |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  ( E. x  x  e.  ( R  \  RR )  ->  ( R  =  RR  \/  R  =  CC ) ) )
7473imp 418 . . . 4  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  E. x  x  e.  ( R  \  RR ) )  -> 
( R  =  RR  \/  R  =  CC ) )
757, 74sylan2b 461 . . 3  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( R  \  RR )  =/=  (/) )  -> 
( R  =  RR  \/  R  =  CC ) )
766, 75pm2.61dane 2524 . 2  |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  ( R  =  RR  \/  R  =  CC )
)
77 elprg 3657 . . 3  |-  ( R  e.  (SubRing ` fld )  ->  ( R  e.  { RR ,  CC }  <->  ( R  =  RR  \/  R  =  CC ) ) )
7877adantr 451 . 2  |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  ( R  e.  { RR ,  CC }  <->  ( R  =  RR  \/  R  =  CC ) ) )
7976, 78mpbird 223 1  |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  R  e.  { RR ,  CC } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152   (/)c0 3455   {cpr 3641   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   Recre 11582   Imcim 11583  SubRingcsubrg 15541  ℂfldccnfld 16377
This theorem is referenced by:  cncdrg  18776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-cj 11584  df-re 11585  df-im 11586  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-mnd 14367  df-grp 14489  df-subg 14618  df-mgp 15326  df-rng 15340  df-subrg 15543  df-cnfld 16378
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