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Theorem cphsubrglem 22048
Description: Lemma for cphsubrg 22051. (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 5885 . . . . . . 7  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (flds  A ) ) )
4 cphsubrglem.2 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  DivRing )
5 drngring 17917 . . . . . . . . . . . 12  |-  ( F  e.  DivRing  ->  F  e.  Ring )
64, 5syl 17 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  Ring )
71, 6eqeltrrd 2518 . . . . . . . . . 10  |-  ( ph  ->  (flds  A )  e.  Ring )
8 eqid 2429 . . . . . . . . . . 11  |-  ( Base `  (flds  A ) )  =  (
Base `  (flds  A ) )
9 eqid 2429 . . . . . . . . . . 11  |-  ( 0g
`  (flds  A ) )  =  ( 0g `  (flds  A ) )
108, 9ring0cl 17737 . . . . . . . . . 10  |-  ( (flds  A )  e.  Ring  ->  ( 0g
`  (flds  A ) )  e.  (
Base `  (flds  A ) ) )
11 reldmress 15137 . . . . . . . . . . 11  |-  Rel  doms
12 eqid 2429 . . . . . . . . . . 11  |-  (flds  A )  =  (flds  A )
1311, 12, 8elbasov 15134 . . . . . . . . . 10  |-  ( ( 0g `  (flds  A ) )  e.  ( Base `  (flds  A )
)  ->  (fld  e.  _V  /\  A  e.  _V )
)
147, 10, 133syl 18 . . . . . . . . 9  |-  ( ph  ->  (fld  e.  _V  /\  A  e.  _V ) )
1514simprd 464 . . . . . . . 8  |-  ( ph  ->  A  e.  _V )
16 cnfldbas 18909 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
1712, 16ressbas 15141 . . . . . . . 8  |-  ( A  e.  _V  ->  ( A  i^i  CC )  =  ( Base `  (flds  A )
) )
1815, 17syl 17 . . . . . . 7  |-  ( ph  ->  ( A  i^i  CC )  =  ( Base `  (flds  A ) ) )
193, 18eqtr4d 2473 . . . . . 6  |-  ( ph  ->  ( Base `  F
)  =  ( A  i^i  CC ) )
202, 19syl5eq 2482 . . . . 5  |-  ( ph  ->  K  =  ( A  i^i  CC ) )
2120oveq2d 6321 . . . 4  |-  ( ph  ->  (flds  K )  =  (flds  ( A  i^i  CC ) ) )
2216ressinbas 15147 . . . . 5  |-  ( A  e.  _V  ->  (flds  A )  =  (flds  ( A  i^i  CC ) ) )
2315, 22syl 17 . . . 4  |-  ( ph  ->  (flds  A )  =  (flds  ( A  i^i  CC ) ) )
2421, 23eqtr4d 2473 . . 3  |-  ( ph  ->  (flds  K )  =  (flds  A ) )
251, 24eqtr4d 2473 . 2  |-  ( ph  ->  F  =  (flds  K ) )
2625, 6eqeltrrd 2518 . . . 4  |-  ( ph  ->  (flds  K )  e.  Ring )
27 cnring 18925 . . . 4  |-fld  e.  Ring
2826, 27jctil 539 . . 3  |-  ( ph  ->  (fld  e.  Ring  /\  (flds  K )  e.  Ring ) )
2912, 16ressbasss 15143 . . . . . 6  |-  ( Base `  (flds  A ) )  C_  CC
303, 29syl6eqss 3520 . . . . 5  |-  ( ph  ->  ( Base `  F
)  C_  CC )
312, 30syl5eqss 3514 . . . 4  |-  ( ph  ->  K  C_  CC )
32 eqid 2429 . . . . . . . . . 10  |-  ( 0g
`  F )  =  ( 0g `  F
)
33 eqid 2429 . . . . . . . . . 10  |-  ( 1r
`  F )  =  ( 1r `  F
)
3432, 33drngunz 17925 . . . . . . . . 9  |-  ( F  e.  DivRing  ->  ( 1r `  F )  =/=  ( 0g `  F ) )
354, 34syl 17 . . . . . . . 8  |-  ( ph  ->  ( 1r `  F
)  =/=  ( 0g
`  F ) )
3625fveq2d 5885 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  F
)  =  ( 0g
`  (flds  K ) ) )
37 ringgrp 17720 . . . . . . . . . . . 12  |-  (fld  e.  Ring  ->fld  e.  Grp )
3827, 37mp1i 13 . . . . . . . . . . 11  |-  ( ph  ->fld  e. 
Grp )
39 ringgrp 17720 . . . . . . . . . . . 12  |-  ( (flds  K )  e.  Ring  ->  (flds  K )  e.  Grp )
4026, 39syl 17 . . . . . . . . . . 11  |-  ( ph  ->  (flds  K )  e.  Grp )
4116issubg 16768 . . . . . . . . . . 11  |-  ( K  e.  (SubGrp ` fld )  <->  (fld  e.  Grp  /\  K  C_  CC  /\  (flds  K )  e.  Grp ) )
4238, 31, 40, 41syl3anbrc 1189 . . . . . . . . . 10  |-  ( ph  ->  K  e.  (SubGrp ` fld )
)
43 eqid 2429 . . . . . . . . . . 11  |-  (flds  K )  =  (flds  K )
44 cnfld0 18927 . . . . . . . . . . 11  |-  0  =  ( 0g ` fld )
4543, 44subg0 16774 . . . . . . . . . 10  |-  ( K  e.  (SubGrp ` fld )  ->  0  =  ( 0g `  (flds  K )
) )
4642, 45syl 17 . . . . . . . . 9  |-  ( ph  ->  0  =  ( 0g
`  (flds  K ) ) )
4736, 46eqtr4d 2473 . . . . . . . 8  |-  ( ph  ->  ( 0g `  F
)  =  0 )
4835, 47neeqtrd 2726 . . . . . . 7  |-  ( ph  ->  ( 1r `  F
)  =/=  0 )
4948neneqd 2632 . . . . . 6  |-  ( ph  ->  -.  ( 1r `  F )  =  0 )
502, 33ringidcl 17736 . . . . . . . . . . . 12  |-  ( F  e.  Ring  ->  ( 1r
`  F )  e.  K )
516, 50syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( 1r `  F
)  e.  K )
5231, 51sseldd 3471 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  e.  CC )
5352sqvald 12410 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  F ) ^ 2 )  =  ( ( 1r `  F )  x.  ( 1r `  F ) ) )
5425fveq2d 5885 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  =  ( 1r
`  (flds  K ) ) )
5554oveq1d 6320 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  F )  x.  ( 1r `  F ) )  =  ( ( 1r
`  (flds  K ) )  x.  ( 1r `  F ) ) )
5625fveq2d 5885 . . . . . . . . . . . 12  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (flds  K ) ) )
572, 56syl5eq 2482 . . . . . . . . . . 11  |-  ( ph  ->  K  =  ( Base `  (flds  K ) ) )
5851, 57eleqtrd 2519 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  e.  ( Base `  (flds  K ) ) )
59 eqid 2429 . . . . . . . . . . 11  |-  ( Base `  (flds  K ) )  =  (
Base `  (flds  K ) )
60 fvex 5891 . . . . . . . . . . . . 13  |-  ( Base `  F )  e.  _V
612, 60eqeltri 2513 . . . . . . . . . . . 12  |-  K  e. 
_V
62 cnfldmul 18911 . . . . . . . . . . . . 13  |-  x.  =  ( .r ` fld )
6343, 62ressmulr 15209 . . . . . . . . . . . 12  |-  ( K  e.  _V  ->  x.  =  ( .r `  (flds  K
) ) )
6461, 63ax-mp 5 . . . . . . . . . . 11  |-  x.  =  ( .r `  (flds  K ) )
65 eqid 2429 . . . . . . . . . . 11  |-  ( 1r
`  (flds  K ) )  =  ( 1r `  (flds  K ) )
6659, 64, 65ringlidm 17739 . . . . . . . . . 10  |-  ( ( (flds  K )  e.  Ring  /\  ( 1r `  F )  e.  ( Base `  (flds  K )
) )  ->  (
( 1r `  (flds  K )
)  x.  ( 1r
`  F ) )  =  ( 1r `  F ) )
6726, 58, 66syl2anc 665 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  (flds  K
) )  x.  ( 1r `  F ) )  =  ( 1r `  F ) )
6853, 55, 673eqtrd 2474 . . . . . . . 8  |-  ( ph  ->  ( ( 1r `  F ) ^ 2 )  =  ( 1r
`  F ) )
69 sq01 12391 . . . . . . . . 9  |-  ( ( 1r `  F )  e.  CC  ->  (
( ( 1r `  F ) ^ 2 )  =  ( 1r
`  F )  <->  ( ( 1r `  F )  =  0  \/  ( 1r
`  F )  =  1 ) ) )
7052, 69syl 17 . . . . . . . 8  |-  ( ph  ->  ( ( ( 1r
`  F ) ^
2 )  =  ( 1r `  F )  <-> 
( ( 1r `  F )  =  0  \/  ( 1r `  F )  =  1 ) ) )
7168, 70mpbid 213 . . . . . . 7  |-  ( ph  ->  ( ( 1r `  F )  =  0  \/  ( 1r `  F )  =  1 ) )
7271ord 378 . . . . . 6  |-  ( ph  ->  ( -.  ( 1r
`  F )  =  0  ->  ( 1r `  F )  =  1 ) )
7349, 72mpd 15 . . . . 5  |-  ( ph  ->  ( 1r `  F
)  =  1 )
7473, 51eqeltrrd 2518 . . . 4  |-  ( ph  ->  1  e.  K )
7531, 74jca 534 . . 3  |-  ( ph  ->  ( K  C_  CC  /\  1  e.  K ) )
76 cnfld1 18928 . . . 4  |-  1  =  ( 1r ` fld )
7716, 76issubrg 17943 . . 3  |-  ( K  e.  (SubRing ` fld )  <->  ( (fld  e.  Ring  /\  (flds  K )  e.  Ring )  /\  ( K  C_  CC  /\  1  e.  K ) ) )
7828, 75, 77sylanbrc 668 . 2  |-  ( ph  ->  K  e.  (SubRing ` fld ) )
7925, 20, 783jca 1185 1  |-  ( ph  ->  ( F  =  (flds  K )  /\  K  =  ( A  i^i  CC )  /\  K  e.  (SubRing ` fld ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   _Vcvv 3087    i^i cin 3441    C_ wss 3442   ` cfv 5601  (class class class)co 6305   CCcc 9536   0cc0 9538   1c1 9539    x. cmul 9543   2c2 10659   ^cexp 12269   Basecbs 15084   ↾s cress 15085   .rcmulr 15153   0gc0g 15297   Grpcgrp 16620  SubGrpcsubg 16762   1rcur 17670   Ringcrg 17715   DivRingcdr 17910  SubRingcsubrg 17939  ℂfldccnfld 18905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-seq 12211  df-exp 12270  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-subg 16765  df-cmn 17367  df-mgp 17659  df-ur 17671  df-ring 17717  df-cring 17718  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-drng 17912  df-subrg 17941  df-cnfld 18906
This theorem is referenced by:  cphreccllem  22049  cphsubrg  22051  tchclm  22099  tchcph  22104
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