MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cphreccllem Structured version   Unicode version

Theorem cphreccllem 20810
Description: Lemma for cphreccl 20813. (Contributed by Mario Carneiro, 8-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
cphreccllem  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( 1  /  X )  e.  K )

Proof of Theorem cphreccllem
StepHypRef Expression
1 cphsubrglem.k . . . . . . . 8  |-  K  =  ( Base `  F
)
2 cphsubrglem.1 . . . . . . . 8  |-  ( ph  ->  F  =  (flds  A ) )
3 cphsubrglem.2 . . . . . . . 8  |-  ( ph  ->  F  e.  DivRing )
41, 2, 3cphsubrglem 20809 . . . . . . 7  |-  ( ph  ->  ( F  =  (flds  K )  /\  K  =  ( A  i^i  CC )  /\  K  e.  (SubRing ` fld ) ) )
54simp3d 1002 . . . . . 6  |-  ( ph  ->  K  e.  (SubRing ` fld ) )
653ad2ant1 1009 . . . . 5  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  K  e.  (SubRing ` fld ) )
7 cnfldbas 17928 . . . . . 6  |-  CC  =  ( Base ` fld )
87subrgss 16969 . . . . 5  |-  ( K  e.  (SubRing ` fld )  ->  K  C_  CC )
96, 8syl 16 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  K  C_  CC )
10 simp2 989 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  K )
119, 10sseldd 3452 . . 3  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  CC )
12 simp3 990 . . 3  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  =/=  0 )
13 cnfldinv 17953 . . 3  |-  ( ( X  e.  CC  /\  X  =/=  0 )  -> 
( ( invr ` fld ) `  X )  =  ( 1  /  X ) )
1411, 12, 13syl2anc 661 . 2  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( ( invr ` fld ) `  X )  =  ( 1  /  X ) )
15 eqid 2451 . . . . . . . . . 10  |-  (flds  K )  =  (flds  K )
16 cnfld0 17946 . . . . . . . . . 10  |-  0  =  ( 0g ` fld )
1715, 16subrg0 16975 . . . . . . . . 9  |-  ( K  e.  (SubRing ` fld )  ->  0  =  ( 0g `  (flds  K )
) )
186, 17syl 16 . . . . . . . 8  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  0  =  ( 0g `  (flds  K ) ) )
194simp1d 1000 . . . . . . . . . 10  |-  ( ph  ->  F  =  (flds  K ) )
20193ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  F  =  (flds  K
) )
2120fveq2d 5790 . . . . . . . 8  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( 0g `  F )  =  ( 0g `  (flds  K ) ) )
2218, 21eqtr4d 2494 . . . . . . 7  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  0  =  ( 0g `  F ) )
2312, 22neeqtrd 2741 . . . . . 6  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  =/=  ( 0g `  F ) )
24 eldifsn 4095 . . . . . 6  |-  ( X  e.  ( K  \  { ( 0g `  F ) } )  <-> 
( X  e.  K  /\  X  =/=  ( 0g `  F ) ) )
2510, 23, 24sylanbrc 664 . . . . 5  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  ( K  \  { ( 0g `  F ) } ) )
2633ad2ant1 1009 . . . . . . 7  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  F  e.  DivRing )
27 eqid 2451 . . . . . . . . 9  |-  (Unit `  F )  =  (Unit `  F )
28 eqid 2451 . . . . . . . . 9  |-  ( 0g
`  F )  =  ( 0g `  F
)
291, 27, 28isdrng 16939 . . . . . . . 8  |-  ( F  e.  DivRing 
<->  ( F  e.  Ring  /\  (Unit `  F )  =  ( K  \  { ( 0g `  F ) } ) ) )
3029simprbi 464 . . . . . . 7  |-  ( F  e.  DivRing  ->  (Unit `  F
)  =  ( K 
\  { ( 0g
`  F ) } ) )
3126, 30syl 16 . . . . . 6  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  (Unit `  F
)  =  ( K 
\  { ( 0g
`  F ) } ) )
3220fveq2d 5790 . . . . . 6  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  (Unit `  F
)  =  (Unit `  (flds  K
) ) )
3331, 32eqtr3d 2493 . . . . 5  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( K  \  { ( 0g `  F ) } )  =  (Unit `  (flds  K )
) )
3425, 33eleqtrd 2539 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  (Unit `  (flds  K ) ) )
35 eqid 2451 . . . . . 6  |-  (Unit ` fld )  =  (Unit ` fld )
36 eqid 2451 . . . . . 6  |-  (Unit `  (flds  K
) )  =  (Unit `  (flds  K ) )
37 eqid 2451 . . . . . 6  |-  ( invr ` fld )  =  ( invr ` fld )
3815, 35, 36, 37subrgunit 16986 . . . . 5  |-  ( K  e.  (SubRing ` fld )  ->  ( X  e.  (Unit `  (flds  K )
)  <->  ( X  e.  (Unit ` fld )  /\  X  e.  K  /\  ( (
invr ` fld ) `  X )  e.  K ) ) )
396, 38syl 16 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( X  e.  (Unit `  (flds  K ) )  <->  ( X  e.  (Unit ` fld )  /\  X  e.  K  /\  ( (
invr ` fld ) `  X )  e.  K ) ) )
4034, 39mpbid 210 . . 3  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( X  e.  (Unit ` fld )  /\  X  e.  K  /\  ( (
invr ` fld ) `  X )  e.  K ) )
4140simp3d 1002 . 2  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( ( invr ` fld ) `  X )  e.  K )
4214, 41eqeltrrd 2538 1  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( 1  /  X )  e.  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2642    \ cdif 3420    i^i cin 3422    C_ wss 3423   {csn 3972   ` cfv 5513  (class class class)co 6187   CCcc 9378   0cc0 9380   1c1 9381    / cdiv 10091   Basecbs 14273   ↾s cress 14274   0gc0g 14477   Ringcrg 16748  Unitcui 16834   invrcinvr 16866   DivRingcdr 16935  SubRingcsubrg 16964  ℂfldccnfld 17924
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 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457  ax-addf 9459  ax-mulf 9460
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-1st 6674  df-2nd 6675  df-tpos 6842  df-recs 6929  df-rdg 6963  df-1o 7017  df-oadd 7021  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-div 10092  df-nn 10421  df-2 10478  df-3 10479  df-4 10480  df-5 10481  df-6 10482  df-7 10483  df-8 10484  df-9 10485  df-10 10486  df-n0 10678  df-z 10745  df-dec 10854  df-uz 10960  df-fz 11536  df-seq 11905  df-exp 11964  df-struct 14275  df-ndx 14276  df-slot 14277  df-base 14278  df-sets 14279  df-ress 14280  df-plusg 14350  df-mulr 14351  df-starv 14352  df-tset 14356  df-ple 14357  df-ds 14359  df-unif 14360  df-0g 14479  df-mnd 15514  df-grp 15644  df-minusg 15645  df-subg 15777  df-cmn 16380  df-mgp 16694  df-ur 16706  df-rng 16750  df-cring 16751  df-oppr 16818  df-dvdsr 16836  df-unit 16837  df-invr 16867  df-dvr 16878  df-drng 16937  df-subrg 16966  df-cnfld 17925
This theorem is referenced by:  cphreccl  20813  ipcau2  20862
  Copyright terms: Public domain W3C validator