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Theorem cnsubrg 17773
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 3734 . . . 4  |-  ( R 
C_  RR  <->  ( R  \  RR )  =  (/) )
2 simpr 458 . . . . . 6  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  R  C_  RR )  ->  R  C_  RR )
3 simplr 749 . . . . . 6  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  R  C_  RR )  ->  RR  C_  R
)
42, 3eqssd 3370 . . . . 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 473 . . 3  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( R  \  RR )  =  (/) )  ->  ( R  =  RR  \/  R  =  CC ) )
7 n0 3643 . . . 4  |-  ( ( R  \  RR )  =/=  (/)  <->  E. x  x  e.  ( R  \  RR ) )
8 simpll 748 . . . . . . . . . 10  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  R  e.  (SubRing ` fld ) )
9 cnfldbas 17722 . . . . . . . . . . 11  |-  CC  =  ( Base ` fld )
109subrgss 16846 . . . . . . . . . 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 12601 . . . . . . . . . . . . 13  |-  ( y  e.  CC  ->  y  =  ( ( Re
`  y )  +  ( _i  x.  (
Im `  y )
) ) )
1312ad2antll 723 . . . . . . . . . . . 12  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  y  =  ( ( Re
`  y )  +  ( _i  x.  (
Im `  y )
) ) )
14 simpll 748 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  R  e.  (SubRing ` fld ) )
15 simplr 749 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  RR  C_  R )
16 recl 12595 . . . . . . . . . . . . . . 15  |-  ( y  e.  CC  ->  (
Re `  y )  e.  RR )
1716ad2antll 723 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Re `  y )  e.  RR )
1815, 17sseldd 3354 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Re `  y )  e.  R )
19 ax-icn 9337 . . . . . . . . . . . . . . . . . . 19  |-  _i  e.  CC
2019a1i 11 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  _i  e.  CC )
21 eldifi 3475 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  e.  ( R  \  RR )  ->  x  e.  R )
2221adantl 463 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  x  e.  R )
2311, 22sseldd 3354 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  x  e.  CC )
24 imcl 12596 . . . . . . . . . . . . . . . . . . . 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 9408 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Im `  x )  e.  CC )
27 eldifn 3476 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( R  \  RR )  ->  -.  x  e.  RR )
2827adantl 463 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  -.  x  e.  RR )
29 reim0b 12604 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  CC  ->  (
x  e.  RR  <->  ( Im `  x )  =  0 ) )
3029necon3bbid 2640 . . . . . . . . . . . . . . . . . . . 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 10109 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( (
_i  x.  ( Im `  x ) )  / 
( Im `  x
) )  =  _i )
34 mulcl 9362 . . . . . . . . . . . . . . . . . . 19  |-  ( ( _i  e.  CC  /\  ( Im `  x )  e.  CC )  -> 
( _i  x.  (
Im `  x )
)  e.  CC )
3519, 26, 34sylancr 658 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( _i  x.  ( Im `  x
) )  e.  CC )
3635, 26, 32divrecd 10106 . . . . . . . . . . . . . . . . 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 2475 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  _i  =  ( ( _i  x.  ( Im `  x ) )  x.  ( 1  /  ( Im `  x ) ) ) )
3823recld 12679 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Re `  x )  e.  RR )
3938recnd 9408 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( Re `  x )  e.  CC )
4023, 39negsubd 9721 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( x  +  -u ( Re `  x ) )  =  ( x  -  (
Re `  x )
) )
41 replim 12601 . . . . . . . . . . . . . . . . . . . . 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 6105 . . . . . . . . . . . . . . . . . . 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 9717 . . . . . . . . . . . . . . . . . . 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 2477 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( x  +  -u ( Re `  x ) )  =  ( _i  x.  (
Im `  x )
) )
46 simplr 749 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  RR  C_  R
)
4738renegcld 9771 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  -u ( Re
`  x )  e.  RR )
4846, 47sseldd 3354 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  -u ( Re
`  x )  e.  R )
49 cnfldadd 17723 . . . . . . . . . . . . . . . . . . . 20  |-  +  =  ( +g  ` fld )
5049subrgacl 16856 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R  e.  (SubRing ` fld )  /\  x  e.  R  /\  -u (
Re `  x )  e.  R )  ->  (
x  +  -u (
Re `  x )
)  e.  R )
518, 22, 48, 50syl3anc 1213 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( x  +  -u ( Re `  x ) )  e.  R )
5245, 51eqeltrrd 2516 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( _i  x.  ( Im `  x
) )  e.  R
)
5325, 32rereccld 10154 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( 1  /  ( Im `  x ) )  e.  RR )
5446, 53sseldd 3354 . . . . . . . . . . . . . . . . 17  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( 1  /  ( Im `  x ) )  e.  R )
55 cnfldmul 17724 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
5655subrgmcl 16857 . . . . . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( (
_i  x.  ( Im `  x ) )  x.  ( 1  /  (
Im `  x )
) )  e.  R
)
5837, 57eqeltrd 2515 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  _i  e.  R )
5958adantrr 711 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  _i  e.  R )
60 imcl 12596 . . . . . . . . . . . . . . . 16  |-  ( y  e.  CC  ->  (
Im `  y )  e.  RR )
6160ad2antll 723 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Im `  y )  e.  RR )
6215, 61sseldd 3354 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
Im `  y )  e.  R )
6355subrgmcl 16857 . . . . . . . . . . . . . 14  |-  ( ( R  e.  (SubRing ` fld )  /\  _i  e.  R  /\  ( Im `  y )  e.  R
)  ->  ( _i  x.  ( Im `  y
) )  e.  R
)
6414, 59, 62, 63syl3anc 1213 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  (
_i  x.  ( Im `  y ) )  e.  R )
6549subrgacl 16856 . . . . . . . . . . . . 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 1213 . . . . . . . . . . . 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 2515 . . . . . . . . . . 11  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( x  e.  ( R  \  RR )  /\  y  e.  CC ) )  ->  y  e.  R )
6867expr 612 . . . . . . . . . 10  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  ( y  e.  CC  ->  y  e.  R ) )
6968ssrdv 3359 . . . . . . . . 9  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  x  e.  ( R  \  RR ) )  ->  CC  C_  R
)
7011, 69eqssd 3370 . . . . . . . 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 1695 . . . . 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 472 . . 3  |-  ( ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R
)  /\  ( R  \  RR )  =/=  (/) )  -> 
( R  =  RR  \/  R  =  CC ) )
766, 75pm2.61dane 2687 . 2  |-  ( ( R  e.  (SubRing ` fld )  /\  RR  C_  R )  ->  ( R  =  RR  \/  R  =  CC )
)
77 elprg 3890 . . 3  |-  ( R  e.  (SubRing ` fld )  ->  ( R  e.  { RR ,  CC }  <->  ( R  =  RR  \/  R  =  CC ) ) )
7877adantr 462 . 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 1364   E.wex 1591    e. wcel 1761    =/= wne 2604    \ cdif 3322    C_ wss 3325   (/)c0 3634   {cpr 3876   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279   _ici 9280    + caddc 9281    x. cmul 9283    - cmin 9591   -ucneg 9592    / cdiv 9989   Recre 12582   Imcim 12583  SubRingcsubrg 16841  ℂfldccnfld 17718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-cj 12584  df-re 12585  df-im 12586  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-mnd 15411  df-grp 15538  df-subg 15671  df-mgp 16582  df-rng 16637  df-subrg 16843  df-cnfld 17719
This theorem is referenced by:  cncdrg  20771
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