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Theorem cphreccllem 19133
Description: Lemma for cphreccl 19136. (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 19132 . . . . . . 7  |-  ( ph  ->  ( F  =  (flds  K )  /\  K  =  ( A  i^i  CC )  /\  K  e.  (SubRing ` fld ) ) )
54simp3d 971 . . . . . 6  |-  ( ph  ->  K  e.  (SubRing ` fld ) )
653ad2ant1 978 . . . . 5  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  K  e.  (SubRing ` fld ) )
7 cnfldbas 16699 . . . . . 6  |-  CC  =  ( Base ` fld )
87subrgss 15861 . . . . 5  |-  ( K  e.  (SubRing ` fld )  ->  K  C_  CC )
96, 8syl 16 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  K  C_  CC )
10 simp2 958 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  K )
119, 10sseldd 3341 . . 3  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  CC )
12 simp3 959 . . 3  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  =/=  0 )
13 cnfldinv 16724 . . 3  |-  ( ( X  e.  CC  /\  X  =/=  0 )  -> 
( ( invr ` fld ) `  X )  =  ( 1  /  X ) )
1411, 12, 13syl2anc 643 . 2  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( ( invr ` fld ) `  X )  =  ( 1  /  X ) )
15 eqid 2435 . . . . . . . . . 10  |-  (flds  K )  =  (flds  K )
16 cnfld0 16717 . . . . . . . . . 10  |-  0  =  ( 0g ` fld )
1715, 16subrg0 15867 . . . . . . . . 9  |-  ( K  e.  (SubRing ` fld )  ->  0  =  ( 0g `  (flds  K )
) )
186, 17syl 16 . . . . . . . 8  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  0  =  ( 0g `  (flds  K ) ) )
194simp1d 969 . . . . . . . . . 10  |-  ( ph  ->  F  =  (flds  K ) )
20193ad2ant1 978 . . . . . . . . 9  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  F  =  (flds  K
) )
2120fveq2d 5724 . . . . . . . 8  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( 0g `  F )  =  ( 0g `  (flds  K ) ) )
2218, 21eqtr4d 2470 . . . . . . 7  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  0  =  ( 0g `  F ) )
2312, 22neeqtrd 2620 . . . . . 6  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  =/=  ( 0g `  F ) )
24 eldifsn 3919 . . . . . 6  |-  ( X  e.  ( K  \  { ( 0g `  F ) } )  <-> 
( X  e.  K  /\  X  =/=  ( 0g `  F ) ) )
2510, 23, 24sylanbrc 646 . . . . 5  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  ( K  \  { ( 0g `  F ) } ) )
2633ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  F  e.  DivRing )
27 eqid 2435 . . . . . . . . 9  |-  (Unit `  F )  =  (Unit `  F )
28 eqid 2435 . . . . . . . . 9  |-  ( 0g
`  F )  =  ( 0g `  F
)
291, 27, 28isdrng 15831 . . . . . . . 8  |-  ( F  e.  DivRing 
<->  ( F  e.  Ring  /\  (Unit `  F )  =  ( K  \  { ( 0g `  F ) } ) ) )
3029simprbi 451 . . . . . . 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 5724 . . . . . 6  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  (Unit `  F
)  =  (Unit `  (flds  K
) ) )
3331, 32eqtr3d 2469 . . . . 5  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( K  \  { ( 0g `  F ) } )  =  (Unit `  (flds  K )
) )
3425, 33eleqtrd 2511 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  (Unit `  (flds  K ) ) )
35 eqid 2435 . . . . . 6  |-  (Unit ` fld )  =  (Unit ` fld )
36 eqid 2435 . . . . . 6  |-  (Unit `  (flds  K
) )  =  (Unit `  (flds  K ) )
37 eqid 2435 . . . . . 6  |-  ( invr ` fld )  =  ( invr ` fld )
3815, 35, 36, 37subrgunit 15878 . . . . 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 202 . . 3  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( X  e.  (Unit ` fld )  /\  X  e.  K  /\  ( (
invr ` fld ) `  X )  e.  K ) )
4140simp3d 971 . 2  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( ( invr ` fld ) `  X )  e.  K )
4214, 41eqeltrrd 2510 1  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( 1  /  X )  e.  K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309    i^i cin 3311    C_ wss 3312   {csn 3806   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    / cdiv 9669   Basecbs 13461   ↾s cress 13462   0gc0g 13715   Ringcrg 15652  Unitcui 15736   invrcinvr 15768   DivRingcdr 15827  SubRingcsubrg 15856  ℂfldccnfld 16695
This theorem is referenced by:  cphreccl  19136  ipcau2  19183
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-minusg 14805  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-invr 15769  df-dvr 15780  df-drng 15829  df-subrg 15858  df-cnfld 16696
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