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Theorem cphreccllem 22205
Description: Lemma for cphreccl 22208. (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 22204 . . . . . . 7  |-  ( ph  ->  ( F  =  (flds  K )  /\  K  =  ( A  i^i  CC )  /\  K  e.  (SubRing ` fld ) ) )
54simp3d 1028 . . . . . 6  |-  ( ph  ->  K  e.  (SubRing ` fld ) )
653ad2ant1 1035 . . . . 5  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  K  e.  (SubRing ` fld ) )
7 cnfldbas 19023 . . . . . 6  |-  CC  =  ( Base ` fld )
87subrgss 18058 . . . . 5  |-  ( K  e.  (SubRing ` fld )  ->  K  C_  CC )
96, 8syl 17 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  K  C_  CC )
10 simp2 1015 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  K )
119, 10sseldd 3445 . . 3  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  CC )
12 simp3 1016 . . 3  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  =/=  0 )
13 cnfldinv 19048 . . 3  |-  ( ( X  e.  CC  /\  X  =/=  0 )  -> 
( ( invr ` fld ) `  X )  =  ( 1  /  X ) )
1411, 12, 13syl2anc 671 . 2  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( ( invr ` fld ) `  X )  =  ( 1  /  X ) )
15 eqid 2462 . . . . . . . . . 10  |-  (flds  K )  =  (flds  K )
16 cnfld0 19041 . . . . . . . . . 10  |-  0  =  ( 0g ` fld )
1715, 16subrg0 18064 . . . . . . . . 9  |-  ( K  e.  (SubRing ` fld )  ->  0  =  ( 0g `  (flds  K )
) )
186, 17syl 17 . . . . . . . 8  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  0  =  ( 0g `  (flds  K ) ) )
194simp1d 1026 . . . . . . . . . 10  |-  ( ph  ->  F  =  (flds  K ) )
20193ad2ant1 1035 . . . . . . . . 9  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  F  =  (flds  K
) )
2120fveq2d 5892 . . . . . . . 8  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( 0g `  F )  =  ( 0g `  (flds  K ) ) )
2218, 21eqtr4d 2499 . . . . . . 7  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  0  =  ( 0g `  F ) )
2312, 22neeqtrd 2705 . . . . . 6  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  =/=  ( 0g `  F ) )
24 eldifsn 4110 . . . . . 6  |-  ( X  e.  ( K  \  { ( 0g `  F ) } )  <-> 
( X  e.  K  /\  X  =/=  ( 0g `  F ) ) )
2510, 23, 24sylanbrc 675 . . . . 5  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  ( K  \  { ( 0g `  F ) } ) )
2633ad2ant1 1035 . . . . . . 7  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  F  e.  DivRing )
27 eqid 2462 . . . . . . . . 9  |-  (Unit `  F )  =  (Unit `  F )
28 eqid 2462 . . . . . . . . 9  |-  ( 0g
`  F )  =  ( 0g `  F
)
291, 27, 28isdrng 18028 . . . . . . . 8  |-  ( F  e.  DivRing 
<->  ( F  e.  Ring  /\  (Unit `  F )  =  ( K  \  { ( 0g `  F ) } ) ) )
3029simprbi 470 . . . . . . 7  |-  ( F  e.  DivRing  ->  (Unit `  F
)  =  ( K 
\  { ( 0g
`  F ) } ) )
3126, 30syl 17 . . . . . 6  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  (Unit `  F
)  =  ( K 
\  { ( 0g
`  F ) } ) )
3220fveq2d 5892 . . . . . 6  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  (Unit `  F
)  =  (Unit `  (flds  K
) ) )
3331, 32eqtr3d 2498 . . . . 5  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( K  \  { ( 0g `  F ) } )  =  (Unit `  (flds  K )
) )
3425, 33eleqtrd 2542 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  (Unit `  (flds  K ) ) )
35 eqid 2462 . . . . . 6  |-  (Unit ` fld )  =  (Unit ` fld )
36 eqid 2462 . . . . . 6  |-  (Unit `  (flds  K
) )  =  (Unit `  (flds  K ) )
37 eqid 2462 . . . . . 6  |-  ( invr ` fld )  =  ( invr ` fld )
3815, 35, 36, 37subrgunit 18075 . . . . 5  |-  ( K  e.  (SubRing ` fld )  ->  ( X  e.  (Unit `  (flds  K )
)  <->  ( X  e.  (Unit ` fld )  /\  X  e.  K  /\  ( (
invr ` fld ) `  X )  e.  K ) ) )
396, 38syl 17 . . . 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 215 . . 3  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( X  e.  (Unit ` fld )  /\  X  e.  K  /\  ( (
invr ` fld ) `  X )  e.  K ) )
4140simp3d 1028 . 2  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( ( invr ` fld ) `  X )  e.  K )
4214, 41eqeltrrd 2541 1  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( 1  /  X )  e.  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633    \ cdif 3413    i^i cin 3415    C_ wss 3416   {csn 3980   ` cfv 5601  (class class class)co 6315   CCcc 9563   0cc0 9565   1c1 9566    / cdiv 10297   Basecbs 15170   ↾s cress 15171   0gc0g 15387   Ringcrg 17829  Unitcui 17916   invrcinvr 17948   DivRingcdr 18024  SubRingcsubrg 18053  ℂfldccnfld 19019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-addf 9644  ax-mulf 9645
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-tpos 6999  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-4 10698  df-5 10699  df-6 10700  df-7 10701  df-8 10702  df-9 10703  df-10 10704  df-n0 10899  df-z 10967  df-dec 11081  df-uz 11189  df-fz 11814  df-seq 12246  df-exp 12305  df-struct 15172  df-ndx 15173  df-slot 15174  df-base 15175  df-sets 15176  df-ress 15177  df-plusg 15252  df-mulr 15253  df-starv 15254  df-tset 15258  df-ple 15259  df-ds 15261  df-unif 15262  df-0g 15389  df-mgm 16537  df-sgrp 16576  df-mnd 16586  df-grp 16722  df-minusg 16723  df-subg 16863  df-cmn 17481  df-mgp 17773  df-ur 17785  df-ring 17831  df-cring 17832  df-oppr 17900  df-dvdsr 17918  df-unit 17919  df-invr 17949  df-dvr 17960  df-drng 18026  df-subrg 18055  df-cnfld 19020
This theorem is referenced by:  cphreccl  22208  ipcau2  22257
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