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Theorem cnsubrg 18346
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 ` 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 3867 . . . 4  |-  ( R 
C_  RR  <->  ( R  \  RR )  =  (/) )
2 simpr 461 . . . . . 6  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  R  C_  RR )  ->  R  C_  RR )
3 simplr 754 . . . . . 6  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  R  C_  RR )  ->  RR  C_  R
)
42, 3eqssd 3503 . . . . 5  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  R  C_  RR )  ->  R  =  RR )
54orcd 392 . . . 4  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  R  C_  RR )  ->  ( R  =  RR  \/  R  =  CC ) )
61, 5sylan2br 476 . . 3  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( R  \  RR )  =  (/) )  ->  ( R  =  RR  \/  R  =  CC ) )
7 n0 3776 . . . 4  |-  ( ( R  \  RR )  =/=  (/)  <->  E. x  x  e.  ( R  \  RR ) )
8 simpll 753 . . . . . . . . . 10  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  R  e.  (SubRing ` fld ) )
9 cnfldbas 18292 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
109subrgss 17298 . . . . . . . . . 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 12923 . . . . . . . . . . . . 13  |-  ( y  e.  CC  ->  y  =  ( ( Re
`  y )  +  ( _i  x.  (
Im `  y )
) ) )
1312ad2antll 728 . . . . . . . . . . . 12  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  y  =  ( ( Re
`  y )  +  ( _i  x.  (
Im `  y )
) ) )
14 simpll 753 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  R  e.  (SubRing ` fld ) )
15 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  RR  C_  R )
16 recl 12917 . . . . . . . . . . . . . . 15  |-  ( y  e.  CC  ->  (
Re `  y )  e.  RR )
1716ad2antll 728 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Re `  y )  e.  RR )
1815, 17sseldd 3487 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Re `  y )  e.  R )
19 ax-icn 9549 . . . . . . . . . . . . . . . . . . 19  |-  _i  e.  CC
2019a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  _i  e.  CC )
21 eldifi 3608 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  e.  ( R  \  RR )  ->  x  e.  R )
2221adantl 466 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  x  e.  R )
2311, 22sseldd 3487 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  x  e.  CC )
24 imcl 12918 . . . . . . . . . . . . . . . . . . . 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 9620 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Im `  x )  e.  CC )
27 eldifn 3609 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( R  \  RR )  ->  -.  x  e.  RR )
2827adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  -.  x  e.  RR )
29 reim0b 12926 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  CC  ->  (
x  e.  RR  <->  ( Im `  x )  =  0 ) )
3029necon3bbid 2688 . . . . . . . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Im `  x )  =/=  0
)
3320, 26, 32divcan4d 10327 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( (
_i  x.  ( Im `  x ) )  / 
( Im `  x
) )  =  _i )
34 mulcl 9574 . . . . . . . . . . . . . . . . . . 19  |-  ( ( _i  e.  CC  /\  ( Im `  x )  e.  CC )  -> 
( _i  x.  (
Im `  x )
)  e.  CC )
3519, 26, 34sylancr 663 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( _i  x.  ( Im `  x
) )  e.  CC )
3635, 26, 32divrecd 10324 . . . . . . . . . . . . . . . . 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 2484 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  _i  =  ( ( _i  x.  ( Im `  x ) )  x.  ( 1  /  ( Im `  x ) ) ) )
3823recld 13001 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Re `  x )  e.  RR )
3938recnd 9620 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Re `  x )  e.  CC )
4023, 39negsubd 9937 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( x  +  -u ( Re `  x ) )  =  ( x  -  (
Re `  x )
) )
41 replim 12923 . . . . . . . . . . . . . . . . . . . . 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 6292 . . . . . . . . . . . . . . . . . . 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 9933 . . . . . . . . . . . . . . . . . . 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 2486 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( x  +  -u ( Re `  x ) )  =  ( _i  x.  (
Im `  x )
) )
46 simplr 754 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  RR  C_  R
)
4738renegcld 9987 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  -u ( Re
`  x )  e.  RR )
4846, 47sseldd 3487 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  -u ( Re
`  x )  e.  R )
49 cnfldadd 18293 . . . . . . . . . . . . . . . . . . . 20  |-  +  =  ( +g  ` fld )
5049subrgacl 17308 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R  e.  (SubRing ` fld )  /\  x  e.  R  /\  -u (
Re `  x )  e.  R )  ->  (
x  +  -u (
Re `  x )
)  e.  R )
518, 22, 48, 50syl3anc 1227 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( x  +  -u ( Re `  x ) )  e.  R )
5245, 51eqeltrrd 2530 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( _i  x.  ( Im `  x
) )  e.  R
)
5325, 32rereccld 10372 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( 1  /  ( Im `  x ) )  e.  RR )
5446, 53sseldd 3487 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( 1  /  ( Im `  x ) )  e.  R )
55 cnfldmul 18294 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
5655subrgmcl 17309 . . . . . . . . . . . . . . . . 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 1227 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( (
_i  x.  ( Im `  x ) )  x.  ( 1  /  (
Im `  x )
) )  e.  R
)
5837, 57eqeltrd 2529 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  _i  e.  R )
5958adantrr 716 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  _i  e.  R )
60 imcl 12918 . . . . . . . . . . . . . . . 16  |-  ( y  e.  CC  ->  (
Im `  y )  e.  RR )
6160ad2antll 728 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Im `  y )  e.  RR )
6215, 61sseldd 3487 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Im `  y )  e.  R )
6355subrgmcl 17309 . . . . . . . . . . . . . 14  |-  ( ( R  e.  (SubRing ` fld )  /\  _i  e.  R  /\  ( Im `  y )  e.  R
)  ->  ( _i  x.  ( Im `  y
) )  e.  R
)
6414, 59, 62, 63syl3anc 1227 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
_i  x.  ( Im `  y ) )  e.  R )
6549subrgacl 17308 . . . . . . . . . . . . 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 1227 . . . . . . . . . . . 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 2529 . . . . . . . . . . 11  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  y  e.  R )
6867expr 615 . . . . . . . . . 10  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( y  e.  CC  ->  y  e.  R ) )
6968ssrdv 3492 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  CC  C_  R
)
7011, 69eqssd 3503 . . . . . . . 8  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  R  =  CC )
7170olcd 393 . . . . . . 7  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( R  =  RR  \/  R  =  CC ) )
7271ex 434 . . . . . 6  |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  (
x  e.  ( R 
\  RR )  -> 
( R  =  RR  \/  R  =  CC ) ) )
7372exlimdv 1709 . . . . 5  |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  ( E. x  x  e.  ( R  \  RR )  ->  ( R  =  RR  \/  R  =  CC ) ) )
7473imp 429 . . . 4  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  E. x  x  e.  ( R  \  RR ) )  -> 
( R  =  RR  \/  R  =  CC ) )
757, 74sylan2b 475 . . 3  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( R  \  RR )  =/=  (/) )  -> 
( R  =  RR  \/  R  =  CC ) )
766, 75pm2.61dane 2759 . 2  |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  ( R  =  RR  \/  R  =  CC )
)
77 elprg 4026 . . 3  |-  ( R  e.  (SubRing ` fld )  ->  ( R  e.  { RR ,  CC }  <->  ( R  =  RR  \/  R  =  CC ) ) )
7877adantr 465 . 2  |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  ( R  e.  { RR ,  CC }  <->  ( R  =  RR  \/  R  =  CC ) ) )
7976, 78mpbird 232 1  |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  R  e.  { RR ,  CC } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1381   E.wex 1597    e. wcel 1802    =/= wne 2636    \ cdif 3455    C_ wss 3458   (/)c0 3767   {cpr 4012   ` cfv 5574  (class class class)co 6277   CCcc 9488   RRcr 9489   0cc0 9490   1c1 9491   _ici 9492    + caddc 9493    x. cmul 9495    - cmin 9805   -ucneg 9806    / cdiv 10207   Recre 12904   Imcim 12905  SubRingcsubrg 17293  ℂfldccnfld 18288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-addf 9569  ax-mulf 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-fz 11677  df-cj 12906  df-re 12907  df-im 12908  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-starv 14584  df-tset 14588  df-ple 14589  df-ds 14591  df-unif 14592  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-subg 16067  df-mgp 17010  df-ring 17068  df-subrg 17295  df-cnfld 18289
This theorem is referenced by:  cncdrg  21665
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