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Theorem cphsubrglem 19093
Description: Lemma for cphsubrg 19096. (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 5691 . . . . . . 7  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (flds  A ) ) )
4 cphsubrglem.2 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  DivRing )
5 drngrng 15797 . . . . . . . . . . . 12  |-  ( F  e.  DivRing  ->  F  e.  Ring )
64, 5syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  Ring )
71, 6eqeltrrd 2479 . . . . . . . . . 10  |-  ( ph  ->  (flds  A )  e.  Ring )
8 eqid 2404 . . . . . . . . . . 11  |-  ( Base `  (flds  A ) )  =  (
Base `  (flds  A ) )
9 eqid 2404 . . . . . . . . . . 11  |-  ( 0g
`  (flds  A ) )  =  ( 0g `  (flds  A ) )
108, 9rng0cl 15640 . . . . . . . . . 10  |-  ( (flds  A )  e.  Ring  ->  ( 0g
`  (flds  A ) )  e.  (
Base `  (flds  A ) ) )
11 reldmress 13470 . . . . . . . . . . 11  |-  Rel  doms
12 eqid 2404 . . . . . . . . . . 11  |-  (flds  A )  =  (flds  A )
1311, 12, 8elbasov 13468 . . . . . . . . . 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 16662 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
1712, 16ressbas 13474 . . . . . . . 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 2439 . . . . . 6  |-  ( ph  ->  ( Base `  F
)  =  ( A  i^i  CC ) )
202, 19syl5eq 2448 . . . . 5  |-  ( ph  ->  K  =  ( A  i^i  CC ) )
2120oveq2d 6056 . . . 4  |-  ( ph  ->  (flds  K )  =  (flds  ( A  i^i  CC ) ) )
2216ressinbas 13480 . . . . 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 2439 . . 3  |-  ( ph  ->  (flds  K )  =  (flds  A ) )
251, 24eqtr4d 2439 . 2  |-  ( ph  ->  F  =  (flds  K ) )
2625, 6eqeltrrd 2479 . . . 4  |-  ( ph  ->  (flds  K )  e.  Ring )
27 cnrng 16678 . . . 4  |-fld  e.  Ring
2826, 27jctil 524 . . 3  |-  ( ph  ->  (fld  e.  Ring  /\  (flds  K )  e.  Ring ) )
2912, 16ressbasss 13476 . . . . . 6  |-  ( Base `  (flds  A ) )  C_  CC
303, 29syl6eqss 3358 . . . . 5  |-  ( ph  ->  ( Base `  F
)  C_  CC )
312, 30syl5eqss 3352 . . . 4  |-  ( ph  ->  K  C_  CC )
32 eqid 2404 . . . . . . . . . 10  |-  ( 0g
`  F )  =  ( 0g `  F
)
33 eqid 2404 . . . . . . . . . 10  |-  ( 1r
`  F )  =  ( 1r `  F
)
3432, 33drngunz 15805 . . . . . . . . 9  |-  ( F  e.  DivRing  ->  ( 1r `  F )  =/=  ( 0g `  F ) )
354, 34syl 16 . . . . . . . 8  |-  ( ph  ->  ( 1r `  F
)  =/=  ( 0g
`  F ) )
3625fveq2d 5691 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  F
)  =  ( 0g
`  (flds  K ) ) )
37 rnggrp 15624 . . . . . . . . . . . 12  |-  (fld  e.  Ring  ->fld  e.  Grp )
3827, 37mp1i 12 . . . . . . . . . . 11  |-  ( ph  ->fld  e. 
Grp )
39 rnggrp 15624 . . . . . . . . . . . 12  |-  ( (flds  K )  e.  Ring  ->  (flds  K )  e.  Grp )
4026, 39syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (flds  K )  e.  Grp )
4116issubg 14899 . . . . . . . . . . 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 2404 . . . . . . . . . . 11  |-  (flds  K )  =  (flds  K )
44 cnfld0 16680 . . . . . . . . . . 11  |-  0  =  ( 0g ` fld )
4543, 44subg0 14905 . . . . . . . . . 10  |-  ( K  e.  (SubGrp ` fld )  ->  0  =  ( 0g `  (flds  K )
) )
4642, 45syl 16 . . . . . . . . 9  |-  ( ph  ->  0  =  ( 0g
`  (flds  K ) ) )
4736, 46eqtr4d 2439 . . . . . . . 8  |-  ( ph  ->  ( 0g `  F
)  =  0 )
4835, 47neeqtrd 2589 . . . . . . 7  |-  ( ph  ->  ( 1r `  F
)  =/=  0 )
4948neneqd 2583 . . . . . 6  |-  ( ph  ->  -.  ( 1r `  F )  =  0 )
502, 33rngidcl 15639 . . . . . . . . . . . 12  |-  ( F  e.  Ring  ->  ( 1r
`  F )  e.  K )
516, 50syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( 1r `  F
)  e.  K )
5231, 51sseldd 3309 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  e.  CC )
5352sqvald 11475 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  F ) ^ 2 )  =  ( ( 1r `  F )  x.  ( 1r `  F ) ) )
5425fveq2d 5691 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  =  ( 1r
`  (flds  K ) ) )
5554oveq1d 6055 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  F )  x.  ( 1r `  F ) )  =  ( ( 1r
`  (flds  K ) )  x.  ( 1r `  F ) ) )
5625fveq2d 5691 . . . . . . . . . . . 12  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (flds  K ) ) )
572, 56syl5eq 2448 . . . . . . . . . . 11  |-  ( ph  ->  K  =  ( Base `  (flds  K ) ) )
5851, 57eleqtrd 2480 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  e.  ( Base `  (flds  K ) ) )
59 eqid 2404 . . . . . . . . . . 11  |-  ( Base `  (flds  K ) )  =  (
Base `  (flds  K ) )
60 fvex 5701 . . . . . . . . . . . . 13  |-  ( Base `  F )  e.  _V
612, 60eqeltri 2474 . . . . . . . . . . . 12  |-  K  e. 
_V
62 cnfldmul 16664 . . . . . . . . . . . . 13  |-  x.  =  ( .r ` fld )
6343, 62ressmulr 13537 . . . . . . . . . . . 12  |-  ( K  e.  _V  ->  x.  =  ( .r `  (flds  K
) ) )
6461, 63ax-mp 8 . . . . . . . . . . 11  |-  x.  =  ( .r `  (flds  K ) )
65 eqid 2404 . . . . . . . . . . 11  |-  ( 1r
`  (flds  K ) )  =  ( 1r `  (flds  K ) )
6659, 64, 65rnglidm 15642 . . . . . . . . . 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 2440 . . . . . . . 8  |-  ( ph  ->  ( ( 1r `  F ) ^ 2 )  =  ( 1r
`  F ) )
69 sq01 11456 . . . . . . . . 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 2479 . . . 4  |-  ( ph  ->  1  e.  K )
7531, 74jca 519 . . 3  |-  ( ph  ->  ( K  C_  CC  /\  1  e.  K ) )
76 cnfld1 16681 . . . 4  |-  1  =  ( 1r ` fld )
7716, 76issubrg 15823 . . 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 1721    =/= wne 2567   _Vcvv 2916    i^i cin 3279    C_ wss 3280   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    x. cmul 8951   2c2 10005   ^cexp 11337   Basecbs 13424   ↾s cress 13425   .rcmulr 13485   0gc0g 13678   Grpcgrp 14640  SubGrpcsubg 14893   Ringcrg 15615   1rcur 15617   DivRingcdr 15790  SubRingcsubrg 15819  ℂfldccnfld 16658
This theorem is referenced by:  cphreccllem  19094  cphsubrg  19096  tchclm  19142  tchcph  19147
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-addf 9025  ax-mulf 9026
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-seq 11279  df-exp 11338  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13682  df-mnd 14645  df-grp 14767  df-subg 14896  df-cmn 15369  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-drng 15792  df-subrg 15821  df-cnfld 16659
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