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Theorem cphreccllem 18614
Description: Lemma for cphreccl 18617. (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 18613 . . . . . . 7  |-  ( ph  ->  ( F  =  (flds  K )  /\  K  =  ( A  i^i  CC )  /\  K  e.  (SubRing ` fld ) ) )
54simp3d 969 . . . . . 6  |-  ( ph  ->  K  e.  (SubRing ` fld ) )
653ad2ant1 976 . . . . 5  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  K  e.  (SubRing ` fld ) )
7 cnfldbas 16383 . . . . . 6  |-  CC  =  ( Base ` fld )
87subrgss 15546 . . . . 5  |-  ( K  e.  (SubRing ` fld )  ->  K  C_  CC )
96, 8syl 15 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  K  C_  CC )
10 simp2 956 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  K )
119, 10sseldd 3181 . . 3  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  CC )
12 simp3 957 . . 3  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  =/=  0 )
13 cnfldinv 16405 . . 3  |-  ( ( X  e.  CC  /\  X  =/=  0 )  -> 
( ( invr ` fld ) `  X )  =  ( 1  /  X ) )
1411, 12, 13syl2anc 642 . 2  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( ( invr ` fld ) `  X )  =  ( 1  /  X ) )
15 eqid 2283 . . . . . . . . . 10  |-  (flds  K )  =  (flds  K )
16 cnfld0 16398 . . . . . . . . . 10  |-  0  =  ( 0g ` fld )
1715, 16subrg0 15552 . . . . . . . . 9  |-  ( K  e.  (SubRing ` fld )  ->  0  =  ( 0g `  (flds  K )
) )
186, 17syl 15 . . . . . . . 8  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  0  =  ( 0g `  (flds  K ) ) )
194simp1d 967 . . . . . . . . . 10  |-  ( ph  ->  F  =  (flds  K ) )
20193ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  F  =  (flds  K
) )
2120fveq2d 5529 . . . . . . . 8  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( 0g `  F )  =  ( 0g `  (flds  K ) ) )
2218, 21eqtr4d 2318 . . . . . . 7  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  0  =  ( 0g `  F ) )
2312, 22neeqtrd 2468 . . . . . 6  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  =/=  ( 0g `  F ) )
24 eldifsn 3749 . . . . . 6  |-  ( X  e.  ( K  \  { ( 0g `  F ) } )  <-> 
( X  e.  K  /\  X  =/=  ( 0g `  F ) ) )
2510, 23, 24sylanbrc 645 . . . . 5  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  ( K  \  { ( 0g `  F ) } ) )
2633ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  F  e.  DivRing )
27 eqid 2283 . . . . . . . . 9  |-  (Unit `  F )  =  (Unit `  F )
28 eqid 2283 . . . . . . . . 9  |-  ( 0g
`  F )  =  ( 0g `  F
)
291, 27, 28isdrng 15516 . . . . . . . 8  |-  ( F  e.  DivRing 
<->  ( F  e.  Ring  /\  (Unit `  F )  =  ( K  \  { ( 0g `  F ) } ) ) )
3029simprbi 450 . . . . . . 7  |-  ( F  e.  DivRing  ->  (Unit `  F
)  =  ( K 
\  { ( 0g
`  F ) } ) )
3126, 30syl 15 . . . . . 6  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  (Unit `  F
)  =  ( K 
\  { ( 0g
`  F ) } ) )
3220fveq2d 5529 . . . . . 6  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  (Unit `  F
)  =  (Unit `  (flds  K
) ) )
3331, 32eqtr3d 2317 . . . . 5  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( K  \  { ( 0g `  F ) } )  =  (Unit `  (flds  K )
) )
3425, 33eleqtrd 2359 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  (Unit `  (flds  K ) ) )
35 eqid 2283 . . . . . 6  |-  (Unit ` fld )  =  (Unit ` fld )
36 eqid 2283 . . . . . 6  |-  (Unit `  (flds  K
) )  =  (Unit `  (flds  K ) )
37 eqid 2283 . . . . . 6  |-  ( invr ` fld )  =  ( invr ` fld )
3815, 35, 36, 37subrgunit 15563 . . . . 5  |-  ( K  e.  (SubRing ` fld )  ->  ( X  e.  (Unit `  (flds  K )
)  <->  ( X  e.  (Unit ` fld )  /\  X  e.  K  /\  ( (
invr ` fld ) `  X )  e.  K ) ) )
396, 38syl 15 . . . 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 201 . . 3  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( X  e.  (Unit ` fld )  /\  X  e.  K  /\  ( (
invr ` fld ) `  X )  e.  K ) )
4140simp3d 969 . 2  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( ( invr ` fld ) `  X )  e.  K )
4214, 41eqeltrrd 2358 1  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( 1  /  X )  e.  K )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    i^i cin 3151    C_ wss 3152   {csn 3640   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    / cdiv 9423   Basecbs 13148   ↾s cress 13149   0gc0g 13400   Ringcrg 15337  Unitcui 15421   invrcinvr 15453   DivRingcdr 15512  SubRingcsubrg 15541  ℂfldccnfld 16377
This theorem is referenced by:  cphreccl  18617  ipcau2  18664
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-minusg 14490  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-invr 15454  df-dvr 15465  df-drng 15514  df-subrg 15543  df-cnfld 16378
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