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Theorem cphsubrglem 19132
Description: Lemma for cphsubrg 19135. (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 5724 . . . . . . 7  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (flds  A ) ) )
4 cphsubrglem.2 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  DivRing )
5 drngrng 15834 . . . . . . . . . . . 12  |-  ( F  e.  DivRing  ->  F  e.  Ring )
64, 5syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  Ring )
71, 6eqeltrrd 2510 . . . . . . . . . 10  |-  ( ph  ->  (flds  A )  e.  Ring )
8 eqid 2435 . . . . . . . . . . 11  |-  ( Base `  (flds  A ) )  =  (
Base `  (flds  A ) )
9 eqid 2435 . . . . . . . . . . 11  |-  ( 0g
`  (flds  A ) )  =  ( 0g `  (flds  A ) )
108, 9rng0cl 15677 . . . . . . . . . 10  |-  ( (flds  A )  e.  Ring  ->  ( 0g
`  (flds  A ) )  e.  (
Base `  (flds  A ) ) )
11 reldmress 13507 . . . . . . . . . . 11  |-  Rel  doms
12 eqid 2435 . . . . . . . . . . 11  |-  (flds  A )  =  (flds  A )
1311, 12, 8elbasov 13505 . . . . . . . . . 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 16699 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
1712, 16ressbas 13511 . . . . . . . 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 2470 . . . . . 6  |-  ( ph  ->  ( Base `  F
)  =  ( A  i^i  CC ) )
202, 19syl5eq 2479 . . . . 5  |-  ( ph  ->  K  =  ( A  i^i  CC ) )
2120oveq2d 6089 . . . 4  |-  ( ph  ->  (flds  K )  =  (flds  ( A  i^i  CC ) ) )
2216ressinbas 13517 . . . . 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 2470 . . 3  |-  ( ph  ->  (flds  K )  =  (flds  A ) )
251, 24eqtr4d 2470 . 2  |-  ( ph  ->  F  =  (flds  K ) )
2625, 6eqeltrrd 2510 . . . 4  |-  ( ph  ->  (flds  K )  e.  Ring )
27 cnrng 16715 . . . 4  |-fld  e.  Ring
2826, 27jctil 524 . . 3  |-  ( ph  ->  (fld  e.  Ring  /\  (flds  K )  e.  Ring ) )
2912, 16ressbasss 13513 . . . . . 6  |-  ( Base `  (flds  A ) )  C_  CC
303, 29syl6eqss 3390 . . . . 5  |-  ( ph  ->  ( Base `  F
)  C_  CC )
312, 30syl5eqss 3384 . . . 4  |-  ( ph  ->  K  C_  CC )
32 eqid 2435 . . . . . . . . . 10  |-  ( 0g
`  F )  =  ( 0g `  F
)
33 eqid 2435 . . . . . . . . . 10  |-  ( 1r
`  F )  =  ( 1r `  F
)
3432, 33drngunz 15842 . . . . . . . . 9  |-  ( F  e.  DivRing  ->  ( 1r `  F )  =/=  ( 0g `  F ) )
354, 34syl 16 . . . . . . . 8  |-  ( ph  ->  ( 1r `  F
)  =/=  ( 0g
`  F ) )
3625fveq2d 5724 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  F
)  =  ( 0g
`  (flds  K ) ) )
37 rnggrp 15661 . . . . . . . . . . . 12  |-  (fld  e.  Ring  ->fld  e.  Grp )
3827, 37mp1i 12 . . . . . . . . . . 11  |-  ( ph  ->fld  e. 
Grp )
39 rnggrp 15661 . . . . . . . . . . . 12  |-  ( (flds  K )  e.  Ring  ->  (flds  K )  e.  Grp )
4026, 39syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (flds  K )  e.  Grp )
4116issubg 14936 . . . . . . . . . . 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 2435 . . . . . . . . . . 11  |-  (flds  K )  =  (flds  K )
44 cnfld0 16717 . . . . . . . . . . 11  |-  0  =  ( 0g ` fld )
4543, 44subg0 14942 . . . . . . . . . 10  |-  ( K  e.  (SubGrp ` fld )  ->  0  =  ( 0g `  (flds  K )
) )
4642, 45syl 16 . . . . . . . . 9  |-  ( ph  ->  0  =  ( 0g
`  (flds  K ) ) )
4736, 46eqtr4d 2470 . . . . . . . 8  |-  ( ph  ->  ( 0g `  F
)  =  0 )
4835, 47neeqtrd 2620 . . . . . . 7  |-  ( ph  ->  ( 1r `  F
)  =/=  0 )
4948neneqd 2614 . . . . . 6  |-  ( ph  ->  -.  ( 1r `  F )  =  0 )
502, 33rngidcl 15676 . . . . . . . . . . . 12  |-  ( F  e.  Ring  ->  ( 1r
`  F )  e.  K )
516, 50syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( 1r `  F
)  e.  K )
5231, 51sseldd 3341 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  e.  CC )
5352sqvald 11512 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  F ) ^ 2 )  =  ( ( 1r `  F )  x.  ( 1r `  F ) ) )
5425fveq2d 5724 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  =  ( 1r
`  (flds  K ) ) )
5554oveq1d 6088 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  F )  x.  ( 1r `  F ) )  =  ( ( 1r
`  (flds  K ) )  x.  ( 1r `  F ) ) )
5625fveq2d 5724 . . . . . . . . . . . 12  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (flds  K ) ) )
572, 56syl5eq 2479 . . . . . . . . . . 11  |-  ( ph  ->  K  =  ( Base `  (flds  K ) ) )
5851, 57eleqtrd 2511 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  e.  ( Base `  (flds  K ) ) )
59 eqid 2435 . . . . . . . . . . 11  |-  ( Base `  (flds  K ) )  =  (
Base `  (flds  K ) )
60 fvex 5734 . . . . . . . . . . . . 13  |-  ( Base `  F )  e.  _V
612, 60eqeltri 2505 . . . . . . . . . . . 12  |-  K  e. 
_V
62 cnfldmul 16701 . . . . . . . . . . . . 13  |-  x.  =  ( .r ` fld )
6343, 62ressmulr 13574 . . . . . . . . . . . 12  |-  ( K  e.  _V  ->  x.  =  ( .r `  (flds  K
) ) )
6461, 63ax-mp 8 . . . . . . . . . . 11  |-  x.  =  ( .r `  (flds  K ) )
65 eqid 2435 . . . . . . . . . . 11  |-  ( 1r
`  (flds  K ) )  =  ( 1r `  (flds  K ) )
6659, 64, 65rnglidm 15679 . . . . . . . . . 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 2471 . . . . . . . 8  |-  ( ph  ->  ( ( 1r `  F ) ^ 2 )  =  ( 1r
`  F ) )
69 sq01 11493 . . . . . . . . 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 2510 . . . 4  |-  ( ph  ->  1  e.  K )
7531, 74jca 519 . . 3  |-  ( ph  ->  ( K  C_  CC  /\  1  e.  K ) )
76 cnfld1 16718 . . . 4  |-  1  =  ( 1r ` fld )
7716, 76issubrg 15860 . . 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 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    i^i cin 3311    C_ wss 3312   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    x. cmul 8987   2c2 10041   ^cexp 11374   Basecbs 13461   ↾s cress 13462   .rcmulr 13522   0gc0g 13715   Grpcgrp 14677  SubGrpcsubg 14930   Ringcrg 15652   1rcur 15654   DivRingcdr 15827  SubRingcsubrg 15856  ℂfldccnfld 16695
This theorem is referenced by:  cphreccllem  19133  cphsubrg  19135  tchclm  19181  tchcph  19186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-seq 11316  df-exp 11375  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-0g 13719  df-mnd 14682  df-grp 14804  df-subg 14933  df-cmn 15406  df-mgp 15641  df-rng 15655  df-cring 15656  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-drng 15829  df-subrg 15858  df-cnfld 16696
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