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Theorem cphsubrglem 20823
Description: Lemma for cphsubrg 20826. (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 5798 . . . . . . 7  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (flds  A ) ) )
4 cphsubrglem.2 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  DivRing )
5 drngrng 16957 . . . . . . . . . . . 12  |-  ( F  e.  DivRing  ->  F  e.  Ring )
64, 5syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  Ring )
71, 6eqeltrrd 2541 . . . . . . . . . 10  |-  ( ph  ->  (flds  A )  e.  Ring )
8 eqid 2452 . . . . . . . . . . 11  |-  ( Base `  (flds  A ) )  =  (
Base `  (flds  A ) )
9 eqid 2452 . . . . . . . . . . 11  |-  ( 0g
`  (flds  A ) )  =  ( 0g `  (flds  A ) )
108, 9rng0cl 16784 . . . . . . . . . 10  |-  ( (flds  A )  e.  Ring  ->  ( 0g
`  (flds  A ) )  e.  (
Base `  (flds  A ) ) )
11 reldmress 14338 . . . . . . . . . . 11  |-  Rel  doms
12 eqid 2452 . . . . . . . . . . 11  |-  (flds  A )  =  (flds  A )
1311, 12, 8elbasov 14335 . . . . . . . . . 10  |-  ( ( 0g `  (flds  A ) )  e.  ( Base `  (flds  A )
)  ->  (fld  e.  _V  /\  A  e.  _V )
)
147, 10, 133syl 20 . . . . . . . . 9  |-  ( ph  ->  (fld  e.  _V  /\  A  e.  _V ) )
1514simprd 463 . . . . . . . 8  |-  ( ph  ->  A  e.  _V )
16 cnfldbas 17942 . . . . . . . . 9  |-  CC  =  ( Base ` fld )
1712, 16ressbas 14342 . . . . . . . 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 2496 . . . . . 6  |-  ( ph  ->  ( Base `  F
)  =  ( A  i^i  CC ) )
202, 19syl5eq 2505 . . . . 5  |-  ( ph  ->  K  =  ( A  i^i  CC ) )
2120oveq2d 6211 . . . 4  |-  ( ph  ->  (flds  K )  =  (flds  ( A  i^i  CC ) ) )
2216ressinbas 14348 . . . . 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 2496 . . 3  |-  ( ph  ->  (flds  K )  =  (flds  A ) )
251, 24eqtr4d 2496 . 2  |-  ( ph  ->  F  =  (flds  K ) )
2625, 6eqeltrrd 2541 . . . 4  |-  ( ph  ->  (flds  K )  e.  Ring )
27 cnrng 17958 . . . 4  |-fld  e.  Ring
2826, 27jctil 537 . . 3  |-  ( ph  ->  (fld  e.  Ring  /\  (flds  K )  e.  Ring ) )
2912, 16ressbasss 14344 . . . . . 6  |-  ( Base `  (flds  A ) )  C_  CC
303, 29syl6eqss 3509 . . . . 5  |-  ( ph  ->  ( Base `  F
)  C_  CC )
312, 30syl5eqss 3503 . . . 4  |-  ( ph  ->  K  C_  CC )
32 eqid 2452 . . . . . . . . . 10  |-  ( 0g
`  F )  =  ( 0g `  F
)
33 eqid 2452 . . . . . . . . . 10  |-  ( 1r
`  F )  =  ( 1r `  F
)
3432, 33drngunz 16965 . . . . . . . . 9  |-  ( F  e.  DivRing  ->  ( 1r `  F )  =/=  ( 0g `  F ) )
354, 34syl 16 . . . . . . . 8  |-  ( ph  ->  ( 1r `  F
)  =/=  ( 0g
`  F ) )
3625fveq2d 5798 . . . . . . . . 9  |-  ( ph  ->  ( 0g `  F
)  =  ( 0g
`  (flds  K ) ) )
37 rnggrp 16768 . . . . . . . . . . . 12  |-  (fld  e.  Ring  ->fld  e.  Grp )
3827, 37mp1i 12 . . . . . . . . . . 11  |-  ( ph  ->fld  e. 
Grp )
39 rnggrp 16768 . . . . . . . . . . . 12  |-  ( (flds  K )  e.  Ring  ->  (flds  K )  e.  Grp )
4026, 39syl 16 . . . . . . . . . . 11  |-  ( ph  ->  (flds  K )  e.  Grp )
4116issubg 15795 . . . . . . . . . . 11  |-  ( K  e.  (SubGrp ` fld )  <->  (fld  e.  Grp  /\  K  C_  CC  /\  (flds  K )  e.  Grp ) )
4238, 31, 40, 41syl3anbrc 1172 . . . . . . . . . 10  |-  ( ph  ->  K  e.  (SubGrp ` fld )
)
43 eqid 2452 . . . . . . . . . . 11  |-  (flds  K )  =  (flds  K )
44 cnfld0 17960 . . . . . . . . . . 11  |-  0  =  ( 0g ` fld )
4543, 44subg0 15801 . . . . . . . . . 10  |-  ( K  e.  (SubGrp ` fld )  ->  0  =  ( 0g `  (flds  K )
) )
4642, 45syl 16 . . . . . . . . 9  |-  ( ph  ->  0  =  ( 0g
`  (flds  K ) ) )
4736, 46eqtr4d 2496 . . . . . . . 8  |-  ( ph  ->  ( 0g `  F
)  =  0 )
4835, 47neeqtrd 2744 . . . . . . 7  |-  ( ph  ->  ( 1r `  F
)  =/=  0 )
4948neneqd 2652 . . . . . 6  |-  ( ph  ->  -.  ( 1r `  F )  =  0 )
502, 33rngidcl 16783 . . . . . . . . . . . 12  |-  ( F  e.  Ring  ->  ( 1r
`  F )  e.  K )
516, 50syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( 1r `  F
)  e.  K )
5231, 51sseldd 3460 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  e.  CC )
5352sqvald 12117 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  F ) ^ 2 )  =  ( ( 1r `  F )  x.  ( 1r `  F ) ) )
5425fveq2d 5798 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  =  ( 1r
`  (flds  K ) ) )
5554oveq1d 6210 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  F )  x.  ( 1r `  F ) )  =  ( ( 1r
`  (flds  K ) )  x.  ( 1r `  F ) ) )
5625fveq2d 5798 . . . . . . . . . . . 12  |-  ( ph  ->  ( Base `  F
)  =  ( Base `  (flds  K ) ) )
572, 56syl5eq 2505 . . . . . . . . . . 11  |-  ( ph  ->  K  =  ( Base `  (flds  K ) ) )
5851, 57eleqtrd 2542 . . . . . . . . . 10  |-  ( ph  ->  ( 1r `  F
)  e.  ( Base `  (flds  K ) ) )
59 eqid 2452 . . . . . . . . . . 11  |-  ( Base `  (flds  K ) )  =  (
Base `  (flds  K ) )
60 fvex 5804 . . . . . . . . . . . . 13  |-  ( Base `  F )  e.  _V
612, 60eqeltri 2536 . . . . . . . . . . . 12  |-  K  e. 
_V
62 cnfldmul 17944 . . . . . . . . . . . . 13  |-  x.  =  ( .r ` fld )
6343, 62ressmulr 14405 . . . . . . . . . . . 12  |-  ( K  e.  _V  ->  x.  =  ( .r `  (flds  K
) ) )
6461, 63ax-mp 5 . . . . . . . . . . 11  |-  x.  =  ( .r `  (flds  K ) )
65 eqid 2452 . . . . . . . . . . 11  |-  ( 1r
`  (flds  K ) )  =  ( 1r `  (flds  K ) )
6659, 64, 65rnglidm 16786 . . . . . . . . . 10  |-  ( ( (flds  K )  e.  Ring  /\  ( 1r `  F )  e.  ( Base `  (flds  K )
) )  ->  (
( 1r `  (flds  K )
)  x.  ( 1r
`  F ) )  =  ( 1r `  F ) )
6726, 58, 66syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( ( 1r `  (flds  K
) )  x.  ( 1r `  F ) )  =  ( 1r `  F ) )
6853, 55, 673eqtrd 2497 . . . . . . . 8  |-  ( ph  ->  ( ( 1r `  F ) ^ 2 )  =  ( 1r
`  F ) )
69 sq01 12098 . . . . . . . . 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 210 . . . . . . 7  |-  ( ph  ->  ( ( 1r `  F )  =  0  \/  ( 1r `  F )  =  1 ) )
7271ord 377 . . . . . 6  |-  ( ph  ->  ( -.  ( 1r
`  F )  =  0  ->  ( 1r `  F )  =  1 ) )
7349, 72mpd 15 . . . . 5  |-  ( ph  ->  ( 1r `  F
)  =  1 )
7473, 51eqeltrrd 2541 . . . 4  |-  ( ph  ->  1  e.  K )
7531, 74jca 532 . . 3  |-  ( ph  ->  ( K  C_  CC  /\  1  e.  K ) )
76 cnfld1 17961 . . . 4  |-  1  =  ( 1r ` fld )
7716, 76issubrg 16983 . . 3  |-  ( K  e.  (SubRing ` fld )  <->  ( (fld  e.  Ring  /\  (flds  K )  e.  Ring )  /\  ( K  C_  CC  /\  1  e.  K ) ) )
7828, 75, 77sylanbrc 664 . 2  |-  ( ph  ->  K  e.  (SubRing ` fld ) )
7925, 20, 783jca 1168 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 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   _Vcvv 3072    i^i cin 3430    C_ wss 3431   ` cfv 5521  (class class class)co 6195   CCcc 9386   0cc0 9388   1c1 9389    x. cmul 9393   2c2 10477   ^cexp 11977   Basecbs 14287   ↾s cress 14288   .rcmulr 14353   0gc0g 14492   Grpcgrp 15524  SubGrpcsubg 15789   1rcur 16720   Ringcrg 16763   DivRingcdr 16950  SubRingcsubrg 16979  ℂfldccnfld 17938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-addf 9467  ax-mulf 9468
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-tpos 6850  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-z 10753  df-dec 10862  df-uz 10968  df-fz 11550  df-seq 11919  df-exp 11978  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-starv 14367  df-tset 14371  df-ple 14372  df-ds 14374  df-unif 14375  df-0g 14494  df-mnd 15529  df-grp 15659  df-subg 15792  df-cmn 16395  df-mgp 16709  df-ur 16721  df-rng 16765  df-cring 16766  df-oppr 16833  df-dvdsr 16851  df-unit 16852  df-drng 16952  df-subrg 16981  df-cnfld 17939
This theorem is referenced by:  cphreccllem  20824  cphsubrg  20826  tchclm  20874  tchcph  20879
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