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Theorem cnsubrg 16675
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 3622 . . . 4  |-  ( R 
C_  RR  <->  ( R  \  RR )  =  (/) )
2 simpr 448 . . . . . 6  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  R  C_  RR )  ->  R  C_  RR )
3 simplr 732 . . . . . 6  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  R  C_  RR )  ->  RR  C_  R
)
42, 3eqssd 3301 . . . . 5  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  R  C_  RR )  ->  R  =  RR )
54orcd 382 . . . 4  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  R  C_  RR )  ->  ( R  =  RR  \/  R  =  CC ) )
61, 5sylan2br 463 . . 3  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( R  \  RR )  =  (/) )  ->  ( R  =  RR  \/  R  =  CC ) )
7 n0 3573 . . . 4  |-  ( ( R  \  RR )  =/=  (/)  <->  E. x  x  e.  ( R  \  RR ) )
8 simpll 731 . . . . . . . . . 10  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  R  e.  (SubRing ` fld ) )
9 cnfldbas 16623 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
109subrgss 15789 . . . . . . . . . 10  |-  ( R  e.  (SubRing ` fld )  ->  R  C_  CC )
118, 10syl 16 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  R  C_  CC )
12 replim 11841 . . . . . . . . . . . . 13  |-  ( y  e.  CC  ->  y  =  ( ( Re
`  y )  +  ( _i  x.  (
Im `  y )
) ) )
1312ad2antll 710 . . . . . . . . . . . 12  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  y  =  ( ( Re
`  y )  +  ( _i  x.  (
Im `  y )
) ) )
14 simpll 731 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  R  e.  (SubRing ` fld ) )
15 simplr 732 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  RR  C_  R )
16 recl 11835 . . . . . . . . . . . . . . 15  |-  ( y  e.  CC  ->  (
Re `  y )  e.  RR )
1716ad2antll 710 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Re `  y )  e.  RR )
1815, 17sseldd 3285 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Re `  y )  e.  R )
19 ax-icn 8975 . . . . . . . . . . . . . . . . . . 19  |-  _i  e.  CC
2019a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  _i  e.  CC )
21 eldifi 3405 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  e.  ( R  \  RR )  ->  x  e.  R )
2221adantl 453 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  x  e.  R )
2311, 22sseldd 3285 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  x  e.  CC )
24 imcl 11836 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  CC  ->  (
Im `  x )  e.  RR )
2523, 24syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Im `  x )  e.  RR )
2625recnd 9040 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Im `  x )  e.  CC )
27 eldifn 3406 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( R  \  RR )  ->  -.  x  e.  RR )
2827adantl 453 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  -.  x  e.  RR )
29 reim0b 11844 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  CC  ->  (
x  e.  RR  <->  ( Im `  x )  =  0 ) )
3029necon3bbid 2577 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  CC  ->  ( -.  x  e.  RR  <->  ( Im `  x )  =/=  0 ) )
3123, 30syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( -.  x  e.  RR  <->  ( Im `  x )  =/=  0
) )
3228, 31mpbid 202 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Im `  x )  =/=  0
)
3320, 26, 32divcan4d 9721 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( (
_i  x.  ( Im `  x ) )  / 
( Im `  x
) )  =  _i )
34 mulcl 9000 . . . . . . . . . . . . . . . . . . 19  |-  ( ( _i  e.  CC  /\  ( Im `  x )  e.  CC )  -> 
( _i  x.  (
Im `  x )
)  e.  CC )
3519, 26, 34sylancr 645 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( _i  x.  ( Im `  x
) )  e.  CC )
3635, 26, 32divrecd 9718 . . . . . . . . . . . . . . . . 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 2414 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  _i  =  ( ( _i  x.  ( Im `  x ) )  x.  ( 1  /  ( Im `  x ) ) ) )
3823recld 11919 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Re `  x )  e.  RR )
3938recnd 9040 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Re `  x )  e.  CC )
4023, 39negsubd 9342 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( x  +  -u ( Re `  x ) )  =  ( x  -  (
Re `  x )
) )
41 replim 11841 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  CC  ->  x  =  ( ( Re
`  x )  +  ( _i  x.  (
Im `  x )
) ) )
4223, 41syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  x  =  ( ( Re `  x )  +  ( _i  x.  ( Im
`  x ) ) ) )
4342oveq1d 6028 . . . . . . . . . . . . . . . . . . 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 9338 . . . . . . . . . . . . . . . . . . 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 2416 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( x  +  -u ( Re `  x ) )  =  ( _i  x.  (
Im `  x )
) )
46 simplr 732 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  RR  C_  R
)
4738renegcld 9389 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  -u ( Re
`  x )  e.  RR )
4846, 47sseldd 3285 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  -u ( Re
`  x )  e.  R )
49 cnfldadd 16624 . . . . . . . . . . . . . . . . . . . 20  |-  +  =  ( +g  ` fld )
5049subrgacl 15799 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R  e.  (SubRing ` fld )  /\  x  e.  R  /\  -u (
Re `  x )  e.  R )  ->  (
x  +  -u (
Re `  x )
)  e.  R )
518, 22, 48, 50syl3anc 1184 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( x  +  -u ( Re `  x ) )  e.  R )
5245, 51eqeltrrd 2455 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( _i  x.  ( Im `  x
) )  e.  R
)
5325, 32rereccld 9766 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( 1  /  ( Im `  x ) )  e.  RR )
5446, 53sseldd 3285 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( 1  /  ( Im `  x ) )  e.  R )
55 cnfldmul 16625 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
5655subrgmcl 15800 . . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( (
_i  x.  ( Im `  x ) )  x.  ( 1  /  (
Im `  x )
) )  e.  R
)
5837, 57eqeltrd 2454 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  _i  e.  R )
5958adantrr 698 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  _i  e.  R )
60 imcl 11836 . . . . . . . . . . . . . . . 16  |-  ( y  e.  CC  ->  (
Im `  y )  e.  RR )
6160ad2antll 710 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Im `  y )  e.  RR )
6215, 61sseldd 3285 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Im `  y )  e.  R )
6355subrgmcl 15800 . . . . . . . . . . . . . 14  |-  ( ( R  e.  (SubRing ` fld )  /\  _i  e.  R  /\  ( Im `  y )  e.  R
)  ->  ( _i  x.  ( Im `  y
) )  e.  R
)
6414, 59, 62, 63syl3anc 1184 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
_i  x.  ( Im `  y ) )  e.  R )
6549subrgacl 15799 . . . . . . . . . . . . 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 1184 . . . . . . . . . . . 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 2454 . . . . . . . . . . 11  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  y  e.  R )
6867expr 599 . . . . . . . . . 10  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( y  e.  CC  ->  y  e.  R ) )
6968ssrdv 3290 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  CC  C_  R
)
7011, 69eqssd 3301 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  R  =  CC )
7170olcd 383 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( R  =  RR  \/  R  =  CC ) )
7271ex 424 . . . . . 6  |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  (
x  e.  ( R 
\  RR )  -> 
( R  =  RR  \/  R  =  CC ) ) )
7372exlimdv 1643 . . . . 5  |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  ( E. x  x  e.  ( R  \  RR )  ->  ( R  =  RR  \/  R  =  CC ) ) )
7473imp 419 . . . 4  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  E. x  x  e.  ( R  \  RR ) )  -> 
( R  =  RR  \/  R  =  CC ) )
757, 74sylan2b 462 . . 3  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( R  \  RR )  =/=  (/) )  -> 
( R  =  RR  \/  R  =  CC ) )
766, 75pm2.61dane 2621 . 2  |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  ( R  =  RR  \/  R  =  CC )
)
77 elprg 3767 . . 3  |-  ( R  e.  (SubRing ` fld )  ->  ( R  e.  { RR ,  CC }  <->  ( R  =  RR  \/  R  =  CC ) ) )
7877adantr 452 . 2  |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  ( R  e.  { RR ,  CC }  <->  ( R  =  RR  \/  R  =  CC ) ) )
7976, 78mpbird 224 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 177    \/ wo 358    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2543    \ cdif 3253    C_ wss 3256   (/)c0 3564   {cpr 3751   ` cfv 5387  (class class class)co 6013   CCcc 8914   RRcr 8915   0cc0 8916   1c1 8917   _ici 8918    + caddc 8919    x. cmul 8921    - cmin 9216   -ucneg 9217    / cdiv 9602   Recre 11822   Imcim 11823  SubRingcsubrg 15784  ℂfldccnfld 16619
This theorem is referenced by:  cncdrg  19173
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-addf 8995  ax-mulf 8996
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-fz 10969  df-cj 11824  df-re 11825  df-im 11826  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-starv 13464  df-tset 13468  df-ple 13469  df-ds 13471  df-unif 13472  df-mnd 14610  df-grp 14732  df-subg 14861  df-mgp 15569  df-rng 15583  df-subrg 15786  df-cnfld 16620
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