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Theorem cphsubrglem 19011
Description: Lemma for cphsubrg 19014. (Contributed by Mario Carneiro, 9-Oct-2015.)
Hypotheses
Ref Expression
cphsubrglem.k  |-  K  =  ( Base `  F
)
cphsubrglem.1  |-  ( ph  ->  F  =  (flds  A ) )
cphsubrglem.2  |-  ( ph  ->  F  e.  DivRing )
Assertion
Ref Expression
cphsubrglem  |-  ( ph  ->  ( F  =  (flds  K )  /\  K  =  ( A  i^i  CC )  /\  K  e.  (SubRing ` fld ) ) )

Proof of Theorem cphsubrglem
StepHypRef Expression
1 cphsubrglem.1 . . 3  |-  ( ph  ->  F  =  (flds  A ) )
2 cphsubrglem.k . . . . . 6  |-  K  =  ( Base `  F
)
31fveq2d 5672 . . . . . . 7  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (flds  A ) ) )
4 cphsubrglem.2 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  DivRing )
5 drngrng 15769 . . . . . . . . . . . 12  |-  ( F  e.  DivRing  ->  F  e.  Ring )
64, 5syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  Ring )
71, 6eqeltrrd 2462 . . . . . . . . . 10  |-  ( ph  ->  (flds  A )  e.  Ring )
8 eqid 2387 . . . . . . . . . . 11  |-  ( Base `  (flds  A ) )  =  (
Base `  (flds  A ) )
9 eqid 2387 . . . . . . . . . . 11  |-  ( 0g
`  (flds  A ) )  =  ( 0g `  (flds  A ) )
108, 9rng0cl 15612 . . . . . . . . . 10  |-  ( (flds  A )  e.  Ring  ->  ( 0g
`  (flds  A ) )  e.  (
Base `  (flds  A ) ) )
11 reldmress 13442 . . . . . . . . . . 11  |-  Rel  doms
12 eqid 2387 . . . . . . . . . . 11  |-  (flds  A )  =  (flds  A )
1311, 12, 8elbasov 13440 . . . . . . . . . 10  |-  ( ( 0g `  (flds  A ) )  e.  ( Base `  (flds  A )
)  ->  (fld  e.  _V  /\  A  e.  _V )
)
147, 10, 133syl 19 . . . . . . . . 9  |-  ( ph  ->  (fld  e.  _V  /\  A  e.  _V ) )
1514simprd 450 . . . . . . . 8  |-  ( ph  ->  A  e.  _V )
16 cnfldbas 16630 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
1712, 16ressbas 13446 . . . . . . . 8  |-  ( A  e.  _V  ->  ( A  i^i  CC )  =  ( Base `  (flds  A )
) )
1815, 17syl 16 . . . . . . 7  |-  ( ph  ->  ( A  i^i  CC )  =  ( Base `  (flds  A ) ) )
193, 18eqtr4d 2422 . . . . . 6  |-  ( ph  ->  ( Base `  F
)  =  ( A  i^i  CC ) )
202, 19syl5eq 2431 . . . . 5  |-  ( ph  ->  K  =  ( A  i^i  CC ) )
2120oveq2d 6036 . . . 4  |-  ( ph  ->  (flds  K )  =  (flds  ( A  i^i  CC ) ) )
2216ressinbas 13452 . . . . 5  |-  ( A  e.  _V  ->  (flds  A )  =  (flds  ( A  i^i  CC ) ) )
2315, 22syl 16 . . . 4  |-  ( ph  ->  (flds  A )  =  (flds  ( A  i^i  CC ) ) )
2421, 23eqtr4d 2422 . . 3  |-  ( ph  ->  (flds  K )  =  (flds  A ) )
251, 24eqtr4d 2422 . 2  |-  ( ph  ->  F  =  (flds  K ) )
2625, 6eqeltrrd 2462 . . . 4  |-  ( ph  ->  (flds  K )  e.  Ring )
27 cnrng 16646 . . . 4  |-fld  e.  Ring
2826, 27jctil 524 . . 3  |-  ( ph  ->  (fld  e.  Ring  /\  (flds  K )  e.  Ring ) )
2912, 16ressbasss 13448 . . . . . 6  |-  ( Base `  (flds  A ) )  C_  CC
303, 29syl6eqss 3341 . . . . 5  |-  ( ph  ->  ( Base `  F
)  C_  CC )
312, 30syl5eqss 3335 . . . 4  |-  ( ph  ->  K  C_  CC )
32 eqid 2387 . . . . . . . . . 10  |-  ( 0g
`  F )  =  ( 0g `  F
)
33 eqid 2387 . . . . . . . . . 10  |-  ( 1r
`  F )  =  ( 1r `  F
)
3432, 33drngunz 15777 . . . . . . . . 9  |-  ( F  e.  DivRing  ->  ( 1r `  F )  =/=  ( 0g `  F ) )
354, 34syl 16 . . . . . . . 8  |-  ( ph  ->  ( 1r `  F
)  =/=  ( 0g
`  F ) )
3625fveq2d 5672 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  F
)  =  ( 0g
`  (flds  K ) ) )
37 rnggrp 15596 . . . . . . . . . . . 12  |-  (fld  e.  Ring  ->fld  e.  Grp )
3827, 37mp1i 12 . . . . . . . . . . 11  |-  ( ph  ->fld  e. 
Grp )
39 rnggrp 15596 . . . . . . . . . . . 12  |-  ( (flds  K )  e.  Ring  ->  (flds  K )  e.  Grp )
4026, 39syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (flds  K )  e.  Grp )
4116issubg 14871 . . . . . . . . . . 11  |-  ( K  e.  (SubGrp ` fld )  <->  (fld  e.  Grp  /\  K  C_  CC  /\  (flds  K )  e.  Grp ) )
4238, 31, 40, 41syl3anbrc 1138 . . . . . . . . . 10  |-  ( ph  ->  K  e.  (SubGrp ` fld )
)
43 eqid 2387 . . . . . . . . . . 11  |-  (flds  K )  =  (flds  K )
44 cnfld0 16648 . . . . . . . . . . 11  |-  0  =  ( 0g ` fld )
4543, 44subg0 14877 . . . . . . . . . 10  |-  ( K  e.  (SubGrp ` fld )  ->  0  =  ( 0g `  (flds  K )
) )
4642, 45syl 16 . . . . . . . . 9  |-  ( ph  ->  0  =  ( 0g
`  (flds  K ) ) )
4736, 46eqtr4d 2422 . . . . . . . 8  |-  ( ph  ->  ( 0g `  F
)  =  0 )
4835, 47neeqtrd 2572 . . . . . . 7  |-  ( ph  ->  ( 1r `  F
)  =/=  0 )
4948neneqd 2566 . . . . . 6  |-  ( ph  ->  -.  ( 1r `  F )  =  0 )
502, 33rngidcl 15611 . . . . . . . . . . . 12  |-  ( F  e.  Ring  ->  ( 1r
`  F )  e.  K )
516, 50syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( 1r `  F
)  e.  K )
5231, 51sseldd 3292 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  e.  CC )
5352sqvald 11447 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  F ) ^ 2 )  =  ( ( 1r `  F )  x.  ( 1r `  F ) ) )
5425fveq2d 5672 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  =  ( 1r
`  (flds  K ) ) )
5554oveq1d 6035 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  F )  x.  ( 1r `  F ) )  =  ( ( 1r
`  (flds  K ) )  x.  ( 1r `  F ) ) )
5625fveq2d 5672 . . . . . . . . . . . 12  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (flds  K ) ) )
572, 56syl5eq 2431 . . . . . . . . . . 11  |-  ( ph  ->  K  =  ( Base `  (flds  K ) ) )
5851, 57eleqtrd 2463 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  e.  ( Base `  (flds  K ) ) )
59 eqid 2387 . . . . . . . . . . 11  |-  ( Base `  (flds  K ) )  =  (
Base `  (flds  K ) )
60 fvex 5682 . . . . . . . . . . . . 13  |-  ( Base `  F )  e.  _V
612, 60eqeltri 2457 . . . . . . . . . . . 12  |-  K  e. 
_V
62 cnfldmul 16632 . . . . . . . . . . . . 13  |-  x.  =  ( .r ` fld )
6343, 62ressmulr 13509 . . . . . . . . . . . 12  |-  ( K  e.  _V  ->  x.  =  ( .r `  (flds  K
) ) )
6461, 63ax-mp 8 . . . . . . . . . . 11  |-  x.  =  ( .r `  (flds  K ) )
65 eqid 2387 . . . . . . . . . . 11  |-  ( 1r
`  (flds  K ) )  =  ( 1r `  (flds  K ) )
6659, 64, 65rnglidm 15614 . . . . . . . . . 10  |-  ( ( (flds  K )  e.  Ring  /\  ( 1r `  F )  e.  ( Base `  (flds  K )
) )  ->  (
( 1r `  (flds  K )
)  x.  ( 1r
`  F ) )  =  ( 1r `  F ) )
6726, 58, 66syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  (flds  K
) )  x.  ( 1r `  F ) )  =  ( 1r `  F ) )
6853, 55, 673eqtrd 2423 . . . . . . . 8  |-  ( ph  ->  ( ( 1r `  F ) ^ 2 )  =  ( 1r
`  F ) )
69 sq01 11428 . . . . . . . . 9  |-  ( ( 1r `  F )  e.  CC  ->  (
( ( 1r `  F ) ^ 2 )  =  ( 1r
`  F )  <->  ( ( 1r `  F )  =  0  \/  ( 1r
`  F )  =  1 ) ) )
7052, 69syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( ( 1r
`  F ) ^
2 )  =  ( 1r `  F )  <-> 
( ( 1r `  F )  =  0  \/  ( 1r `  F )  =  1 ) ) )
7168, 70mpbid 202 . . . . . . 7  |-  ( ph  ->  ( ( 1r `  F )  =  0  \/  ( 1r `  F )  =  1 ) )
7271ord 367 . . . . . 6  |-  ( ph  ->  ( -.  ( 1r
`  F )  =  0  ->  ( 1r `  F )  =  1 ) )
7349, 72mpd 15 . . . . 5  |-  ( ph  ->  ( 1r `  F
)  =  1 )
7473, 51eqeltrrd 2462 . . . 4  |-  ( ph  ->  1  e.  K )
7531, 74jca 519 . . 3  |-  ( ph  ->  ( K  C_  CC  /\  1  e.  K ) )
76 cnfld1 16649 . . . 4  |-  1  =  ( 1r ` fld )
7716, 76issubrg 15795 . . 3  |-  ( K  e.  (SubRing ` fld )  <->  ( (fld  e.  Ring  /\  (flds  K )  e.  Ring )  /\  ( K  C_  CC  /\  1  e.  K ) ) )
7828, 75, 77sylanbrc 646 . 2  |-  ( ph  ->  K  e.  (SubRing ` fld ) )
7925, 20, 783jca 1134 1  |-  ( ph  ->  ( F  =  (flds  K )  /\  K  =  ( A  i^i  CC )  /\  K  e.  (SubRing ` fld ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   _Vcvv 2899    i^i cin 3262    C_ wss 3263   ` cfv 5394  (class class class)co 6020   CCcc 8921   0cc0 8923   1c1 8924    x. cmul 8928   2c2 9981   ^cexp 11309   Basecbs 13396   ↾s cress 13397   .rcmulr 13457   0gc0g 13650   Grpcgrp 14612  SubGrpcsubg 14865   Ringcrg 15587   1rcur 15589   DivRingcdr 15762  SubRingcsubrg 15791  ℂfldccnfld 16626
This theorem is referenced by:  cphreccllem  19012  cphsubrg  19014  tchclm  19060  tchcph  19065
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-addf 9002  ax-mulf 9003
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-tpos 6415  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-fz 10976  df-seq 11251  df-exp 11310  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-0g 13654  df-mnd 14617  df-grp 14739  df-subg 14868  df-cmn 15341  df-mgp 15576  df-rng 15590  df-cring 15591  df-ur 15592  df-oppr 15655  df-dvdsr 15673  df-unit 15674  df-drng 15764  df-subrg 15793  df-cnfld 16627
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