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Theorem cphreccllem 21751
Description: Lemma for cphreccl 21754. (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 21750 . . . . . . 7  |-  ( ph  ->  ( F  =  (flds  K )  /\  K  =  ( A  i^i  CC )  /\  K  e.  (SubRing ` fld ) ) )
54simp3d 1010 . . . . . 6  |-  ( ph  ->  K  e.  (SubRing ` fld ) )
653ad2ant1 1017 . . . . 5  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  K  e.  (SubRing ` fld ) )
7 cnfldbas 18551 . . . . . 6  |-  CC  =  ( Base ` fld )
87subrgss 17557 . . . . 5  |-  ( K  e.  (SubRing ` fld )  ->  K  C_  CC )
96, 8syl 16 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  K  C_  CC )
10 simp2 997 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  K )
119, 10sseldd 3500 . . 3  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  CC )
12 simp3 998 . . 3  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  =/=  0 )
13 cnfldinv 18576 . . 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 2457 . . . . . . . . . 10  |-  (flds  K )  =  (flds  K )
16 cnfld0 18569 . . . . . . . . . 10  |-  0  =  ( 0g ` fld )
1715, 16subrg0 17563 . . . . . . . . 9  |-  ( K  e.  (SubRing ` fld )  ->  0  =  ( 0g `  (flds  K )
) )
186, 17syl 16 . . . . . . . 8  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  0  =  ( 0g `  (flds  K ) ) )
194simp1d 1008 . . . . . . . . . 10  |-  ( ph  ->  F  =  (flds  K ) )
20193ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  F  =  (flds  K
) )
2120fveq2d 5876 . . . . . . . 8  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( 0g `  F )  =  ( 0g `  (flds  K ) ) )
2218, 21eqtr4d 2501 . . . . . . 7  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  0  =  ( 0g `  F ) )
2312, 22neeqtrd 2752 . . . . . 6  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  =/=  ( 0g `  F ) )
24 eldifsn 4157 . . . . . 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 1017 . . . . . . 7  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  F  e.  DivRing )
27 eqid 2457 . . . . . . . . 9  |-  (Unit `  F )  =  (Unit `  F )
28 eqid 2457 . . . . . . . . 9  |-  ( 0g
`  F )  =  ( 0g `  F
)
291, 27, 28isdrng 17527 . . . . . . . 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 5876 . . . . . 6  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  (Unit `  F
)  =  (Unit `  (flds  K
) ) )
3331, 32eqtr3d 2500 . . . . 5  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( K  \  { ( 0g `  F ) } )  =  (Unit `  (flds  K )
) )
3425, 33eleqtrd 2547 . . . 4  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  X  e.  (Unit `  (flds  K ) ) )
35 eqid 2457 . . . . . 6  |-  (Unit ` fld )  =  (Unit ` fld )
36 eqid 2457 . . . . . 6  |-  (Unit `  (flds  K
) )  =  (Unit `  (flds  K ) )
37 eqid 2457 . . . . . 6  |-  ( invr ` fld )  =  ( invr ` fld )
3815, 35, 36, 37subrgunit 17574 . . . . 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 1010 . 2  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( ( invr ` fld ) `  X )  e.  K )
4214, 41eqeltrrd 2546 1  |-  ( (
ph  /\  X  e.  K  /\  X  =/=  0
)  ->  ( 1  /  X )  e.  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652    \ cdif 3468    i^i cin 3470    C_ wss 3471   {csn 4032   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    / cdiv 10227   Basecbs 14644   ↾s cress 14645   0gc0g 14857   Ringcrg 17325  Unitcui 17415   invrcinvr 17447   DivRingcdr 17523  SubRingcsubrg 17552  ℂfldccnfld 18547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-seq 12111  df-exp 12170  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-minusg 16185  df-subg 16325  df-cmn 16927  df-mgp 17269  df-ur 17281  df-ring 17327  df-cring 17328  df-oppr 17399  df-dvdsr 17417  df-unit 17418  df-invr 17448  df-dvr 17459  df-drng 17525  df-subrg 17554  df-cnfld 18548
This theorem is referenced by:  cphreccl  21754  ipcau2  21803
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