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Theorem cphsubrglem 18613
Description: Lemma for cphsubrg 18616. (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 5529 . . . . . . 7  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (flds  A ) ) )
4 cphsubrglem.2 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  DivRing )
5 drngrng 15519 . . . . . . . . . . . 12  |-  ( F  e.  DivRing  ->  F  e.  Ring )
64, 5syl 15 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  Ring )
71, 6eqeltrrd 2358 . . . . . . . . . 10  |-  ( ph  ->  (flds  A )  e.  Ring )
8 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  (flds  A ) )  =  (
Base `  (flds  A ) )
9 eqid 2283 . . . . . . . . . . 11  |-  ( 0g
`  (flds  A ) )  =  ( 0g `  (flds  A ) )
108, 9rng0cl 15362 . . . . . . . . . 10  |-  ( (flds  A )  e.  Ring  ->  ( 0g
`  (flds  A ) )  e.  (
Base `  (flds  A ) ) )
11 reldmress 13194 . . . . . . . . . . 11  |-  Rel  doms
12 eqid 2283 . . . . . . . . . . 11  |-  (flds  A )  =  (flds  A )
1311, 12, 8elbasov 13192 . . . . . . . . . 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 449 . . . . . . . 8  |-  ( ph  ->  A  e.  _V )
16 cnfldbas 16383 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
1712, 16ressbas 13198 . . . . . . . 8  |-  ( A  e.  _V  ->  ( A  i^i  CC )  =  ( Base `  (flds  A )
) )
1815, 17syl 15 . . . . . . 7  |-  ( ph  ->  ( A  i^i  CC )  =  ( Base `  (flds  A ) ) )
193, 18eqtr4d 2318 . . . . . 6  |-  ( ph  ->  ( Base `  F
)  =  ( A  i^i  CC ) )
202, 19syl5eq 2327 . . . . 5  |-  ( ph  ->  K  =  ( A  i^i  CC ) )
2120oveq2d 5874 . . . 4  |-  ( ph  ->  (flds  K )  =  (flds  ( A  i^i  CC ) ) )
2216ressinbas 13204 . . . . 5  |-  ( A  e.  _V  ->  (flds  A )  =  (flds  ( A  i^i  CC ) ) )
2315, 22syl 15 . . . 4  |-  ( ph  ->  (flds  A )  =  (flds  ( A  i^i  CC ) ) )
2421, 23eqtr4d 2318 . . 3  |-  ( ph  ->  (flds  K )  =  (flds  A ) )
251, 24eqtr4d 2318 . 2  |-  ( ph  ->  F  =  (flds  K ) )
2625, 6eqeltrrd 2358 . . . 4  |-  ( ph  ->  (flds  K )  e.  Ring )
27 cnrng 16396 . . . 4  |-fld  e.  Ring
2826, 27jctil 523 . . 3  |-  ( ph  ->  (fld  e.  Ring  /\  (flds  K )  e.  Ring ) )
2912, 16ressbasss 13200 . . . . . 6  |-  ( Base `  (flds  A ) )  C_  CC
303sseq1d 3205 . . . . . 6  |-  ( ph  ->  ( ( Base `  F
)  C_  CC  <->  ( Base `  (flds  A ) )  C_  CC ) )
3129, 30mpbiri 224 . . . . 5  |-  ( ph  ->  ( Base `  F
)  C_  CC )
322, 31syl5eqss 3222 . . . 4  |-  ( ph  ->  K  C_  CC )
33 eqid 2283 . . . . . . . . . 10  |-  ( 0g
`  F )  =  ( 0g `  F
)
34 eqid 2283 . . . . . . . . . 10  |-  ( 1r
`  F )  =  ( 1r `  F
)
3533, 34drngunz 15527 . . . . . . . . 9  |-  ( F  e.  DivRing  ->  ( 1r `  F )  =/=  ( 0g `  F ) )
364, 35syl 15 . . . . . . . 8  |-  ( ph  ->  ( 1r `  F
)  =/=  ( 0g
`  F ) )
3725fveq2d 5529 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  F
)  =  ( 0g
`  (flds  K ) ) )
38 rnggrp 15346 . . . . . . . . . . . 12  |-  (fld  e.  Ring  ->fld  e.  Grp )
3927, 38mp1i 11 . . . . . . . . . . 11  |-  ( ph  ->fld  e. 
Grp )
40 rnggrp 15346 . . . . . . . . . . . 12  |-  ( (flds  K )  e.  Ring  ->  (flds  K )  e.  Grp )
4126, 40syl 15 . . . . . . . . . . 11  |-  ( ph  ->  (flds  K )  e.  Grp )
4216issubg 14621 . . . . . . . . . . 11  |-  ( K  e.  (SubGrp ` fld )  <->  (fld  e.  Grp  /\  K  C_  CC  /\  (flds  K )  e.  Grp ) )
4339, 32, 41, 42syl3anbrc 1136 . . . . . . . . . 10  |-  ( ph  ->  K  e.  (SubGrp ` fld )
)
44 eqid 2283 . . . . . . . . . . 11  |-  (flds  K )  =  (flds  K )
45 cnfld0 16398 . . . . . . . . . . 11  |-  0  =  ( 0g ` fld )
4644, 45subg0 14627 . . . . . . . . . 10  |-  ( K  e.  (SubGrp ` fld )  ->  0  =  ( 0g `  (flds  K )
) )
4743, 46syl 15 . . . . . . . . 9  |-  ( ph  ->  0  =  ( 0g
`  (flds  K ) ) )
4837, 47eqtr4d 2318 . . . . . . . 8  |-  ( ph  ->  ( 0g `  F
)  =  0 )
4936, 48neeqtrd 2468 . . . . . . 7  |-  ( ph  ->  ( 1r `  F
)  =/=  0 )
5049neneqd 2462 . . . . . 6  |-  ( ph  ->  -.  ( 1r `  F )  =  0 )
512, 34rngidcl 15361 . . . . . . . . . . . 12  |-  ( F  e.  Ring  ->  ( 1r
`  F )  e.  K )
526, 51syl 15 . . . . . . . . . . 11  |-  ( ph  ->  ( 1r `  F
)  e.  K )
5332, 52sseldd 3181 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  e.  CC )
5453sqvald 11242 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  F ) ^ 2 )  =  ( ( 1r `  F )  x.  ( 1r `  F ) ) )
5525fveq2d 5529 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  =  ( 1r
`  (flds  K ) ) )
5655oveq1d 5873 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  F )  x.  ( 1r `  F ) )  =  ( ( 1r
`  (flds  K ) )  x.  ( 1r `  F ) ) )
5725fveq2d 5529 . . . . . . . . . . . 12  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (flds  K ) ) )
582, 57syl5eq 2327 . . . . . . . . . . 11  |-  ( ph  ->  K  =  ( Base `  (flds  K ) ) )
5952, 58eleqtrd 2359 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  e.  ( Base `  (flds  K ) ) )
60 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  (flds  K ) )  =  (
Base `  (flds  K ) )
61 fvex 5539 . . . . . . . . . . . . 13  |-  ( Base `  F )  e.  _V
622, 61eqeltri 2353 . . . . . . . . . . . 12  |-  K  e. 
_V
63 cnfldmul 16385 . . . . . . . . . . . . 13  |-  x.  =  ( .r ` fld )
6444, 63ressmulr 13261 . . . . . . . . . . . 12  |-  ( K  e.  _V  ->  x.  =  ( .r `  (flds  K
) ) )
6562, 64ax-mp 8 . . . . . . . . . . 11  |-  x.  =  ( .r `  (flds  K ) )
66 eqid 2283 . . . . . . . . . . 11  |-  ( 1r
`  (flds  K ) )  =  ( 1r `  (flds  K ) )
6760, 65, 66rnglidm 15364 . . . . . . . . . 10  |-  ( ( (flds  K )  e.  Ring  /\  ( 1r `  F )  e.  ( Base `  (flds  K )
) )  ->  (
( 1r `  (flds  K )
)  x.  ( 1r
`  F ) )  =  ( 1r `  F ) )
6826, 59, 67syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  (flds  K
) )  x.  ( 1r `  F ) )  =  ( 1r `  F ) )
6954, 56, 683eqtrd 2319 . . . . . . . 8  |-  ( ph  ->  ( ( 1r `  F ) ^ 2 )  =  ( 1r
`  F ) )
70 sq01 11223 . . . . . . . . 9  |-  ( ( 1r `  F )  e.  CC  ->  (
( ( 1r `  F ) ^ 2 )  =  ( 1r
`  F )  <->  ( ( 1r `  F )  =  0  \/  ( 1r
`  F )  =  1 ) ) )
7153, 70syl 15 . . . . . . . 8  |-  ( ph  ->  ( ( ( 1r
`  F ) ^
2 )  =  ( 1r `  F )  <-> 
( ( 1r `  F )  =  0  \/  ( 1r `  F )  =  1 ) ) )
7269, 71mpbid 201 . . . . . . 7  |-  ( ph  ->  ( ( 1r `  F )  =  0  \/  ( 1r `  F )  =  1 ) )
7372ord 366 . . . . . 6  |-  ( ph  ->  ( -.  ( 1r
`  F )  =  0  ->  ( 1r `  F )  =  1 ) )
7450, 73mpd 14 . . . . 5  |-  ( ph  ->  ( 1r `  F
)  =  1 )
7574, 52eqeltrrd 2358 . . . 4  |-  ( ph  ->  1  e.  K )
7632, 75jca 518 . . 3  |-  ( ph  ->  ( K  C_  CC  /\  1  e.  K ) )
77 cnfld1 16399 . . . 4  |-  1  =  ( 1r ` fld )
7816, 77issubrg 15545 . . 3  |-  ( K  e.  (SubRing ` fld )  <->  ( (fld  e.  Ring  /\  (flds  K )  e.  Ring )  /\  ( K  C_  CC  /\  1  e.  K ) ) )
7928, 76, 78sylanbrc 645 . 2  |-  ( ph  ->  K  e.  (SubRing ` fld ) )
8025, 20, 793jca 1132 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    x. cmul 8742   2c2 9795   ^cexp 11104   Basecbs 13148   ↾s cress 13149   .rcmulr 13209   0gc0g 13400   Grpcgrp 14362  SubGrpcsubg 14615   Ringcrg 15337   1rcur 15339   DivRingcdr 15512  SubRingcsubrg 15541  ℂfldccnfld 16377
This theorem is referenced by:  cphreccllem  18614  cphsubrg  18616  tchclm  18662  tchcph  18667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-seq 11047  df-exp 11105  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-mnd 14367  df-grp 14489  df-subg 14618  df-cmn 15091  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-drng 15514  df-subrg 15543  df-cnfld 16378
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