MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnsubrg Unicode version

Theorem cnsubrg 16448
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 3526 . . . 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 3209 . . . . 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 3477 . . . 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 16399 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
109subrgss 15562 . . . . . . . . . 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 11617 . . . . . . . . . . . . 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 11611 . . . . . . . . . . . . . . 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 3194 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Re `  y )  e.  R )
19 ax-icn 8812 . . . . . . . . . . . . . . . . . . 19  |-  _i  e.  CC
2019a1i 10 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  _i  e.  CC )
21 eldifi 3311 . . . . . . . . . . . . . . . . . . . . . 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 3194 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  x  e.  CC )
24 imcl 11612 . . . . . . . . . . . . . . . . . . . 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 8877 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Im `  x )  e.  CC )
27 eldifn 3312 . . . . . . . . . . . . . . . . . . . 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 11620 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  CC  ->  (
x  e.  RR  <->  ( Im `  x )  =  0 ) )
3029necon3bbid 2493 . . . . . . . . . . . . . . . . . . . 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 9558 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( (
_i  x.  ( Im `  x ) )  / 
( Im `  x
) )  =  _i )
34 mulcl 8837 . . . . . . . . . . . . . . . . . . 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 9555 . . . . . . . . . . . . . . . . 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 2330 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  _i  =  ( ( _i  x.  ( Im `  x ) )  x.  ( 1  /  ( Im `  x ) ) ) )
3823recld 11695 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Re `  x )  e.  RR )
3938recnd 8877 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Re `  x )  e.  CC )
4023, 39negsubd 9179 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( x  +  -u ( Re `  x ) )  =  ( x  -  (
Re `  x )
) )
41 replim 11617 . . . . . . . . . . . . . . . . . . . . 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 5889 . . . . . . . . . . . . . . . . . . 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 9175 . . . . . . . . . . . . . . . . . . 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 2332 . . . . . . . . . . . . . . . . . 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 9226 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  -u ( Re
`  x )  e.  RR )
4846, 47sseldd 3194 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  -u ( Re
`  x )  e.  R )
49 cnfldadd 16400 . . . . . . . . . . . . . . . . . . . 20  |-  +  =  ( +g  ` fld )
5049subrgacl 15572 . . . . . . . . . . . . . . . . . . 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 2371 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( _i  x.  ( Im `  x
) )  e.  R
)
5325, 32rereccld 9603 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( 1  /  ( Im `  x ) )  e.  RR )
5446, 53sseldd 3194 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( 1  /  ( Im `  x ) )  e.  R )
55 cnfldmul 16401 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
5655subrgmcl 15573 . . . . . . . . . . . . . . . . 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 2370 . . . . . . . . . . . . . . 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 11612 . . . . . . . . . . . . . . . 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 3194 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Im `  y )  e.  R )
6355subrgmcl 15573 . . . . . . . . . . . . . 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 15572 . . . . . . . . . . . . 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 2370 . . . . . . . . . . 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 3198 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  CC  C_  R
)
7011, 69eqssd 3209 . . . . . . . 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 1626 . . . . 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 2537 . 2  |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  ( R  =  RR  \/  R  =  CC )
)
77 elprg 3670 . . 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 1531    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    C_ wss 3165   (/)c0 3468   {cpr 3654   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754   _ici 8755    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   Recre 11598   Imcim 11599  SubRingcsubrg 15557  ℂfldccnfld 16393
This theorem is referenced by:  cncdrg  18792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-cj 11600  df-re 11601  df-im 11602  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-tset 13243  df-ple 13244  df-ds 13246  df-mnd 14383  df-grp 14505  df-subg 14634  df-mgp 15342  df-rng 15356  df-subrg 15559  df-cnfld 16394
  Copyright terms: Public domain W3C validator