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Theorem isclmi 18575
Description: Reverse direction of isclm 18562. (Contributed by Mario Carneiro, 30-Oct-2015.)
Hypothesis
Ref Expression
clm0.f  |-  F  =  (Scalar `  W )
Assertion
Ref Expression
isclmi  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  W  e. CMod )

Proof of Theorem isclmi
StepHypRef Expression
1 simp1 955 . 2  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  W  e.  LMod )
2 simp2 956 . . 3  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  F  =  (flds  K ) )
3 eqid 2283 . . . . . . 7  |-  (flds  K )  =  (flds  K )
43subrgbas 15554 . . . . . 6  |-  ( K  e.  (SubRing ` fld )  ->  K  =  ( Base `  (flds  K )
) )
543ad2ant3 978 . . . . 5  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  K  =  ( Base `  (flds  K )
) )
62fveq2d 5529 . . . . 5  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  ( Base `  F )  =  ( Base `  (flds  K )
) )
75, 6eqtr4d 2318 . . . 4  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  K  =  ( Base `  F
) )
87oveq2d 5874 . . 3  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  (flds  K )  =  (flds  (
Base `  F )
) )
92, 8eqtrd 2315 . 2  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  F  =  (flds  (
Base `  F )
) )
10 simp3 957 . . 3  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  K  e.  (SubRing ` fld ) )
117, 10eqeltrrd 2358 . 2  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  ( Base `  F )  e.  (SubRing ` fld ) )
12 clm0.f . . 3  |-  F  =  (Scalar `  W )
13 eqid 2283 . . 3  |-  ( Base `  F )  =  (
Base `  F )
1412, 13isclm 18562 . 2  |-  ( W  e. CMod 
<->  ( W  e.  LMod  /\  F  =  (flds  ( Base `  F
) )  /\  ( Base `  F )  e.  (SubRing ` fld ) ) )
151, 9, 11, 14syl3anbrc 1136 1  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  W  e. CMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149  Scalarcsca 13211  SubRingcsubrg 15541   LModclmod 15627  ℂfldccnfld 16377  CModcclm 18560
This theorem is referenced by:  zlmclm  18593
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-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-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
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-ral 2548  df-rex 2549  df-reu 2550  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-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-recs 6388  df-rdg 6423  df-nn 9747  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-subg 14618  df-rng 15340  df-subrg 15543  df-clm 18561
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