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Theorem isclmi 19059
Description: Reverse direction of isclm 19046. (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 957 . 2  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  W  e.  LMod )
2 simp2 958 . . 3  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  F  =  (flds  K ) )
3 eqid 2408 . . . . . . 7  |-  (flds  K )  =  (flds  K )
43subrgbas 15836 . . . . . 6  |-  ( K  e.  (SubRing ` fld )  ->  K  =  ( Base `  (flds  K )
) )
543ad2ant3 980 . . . . 5  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  K  =  ( Base `  (flds  K )
) )
62fveq2d 5695 . . . . 5  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  ( Base `  F )  =  ( Base `  (flds  K )
) )
75, 6eqtr4d 2443 . . . 4  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  K  =  ( Base `  F
) )
87oveq2d 6060 . . 3  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  (flds  K )  =  (flds  (
Base `  F )
) )
92, 8eqtrd 2440 . 2  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  F  =  (flds  (
Base `  F )
) )
10 simp3 959 . . 3  |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  K  e.  (SubRing ` fld ) )
117, 10eqeltrrd 2483 . 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 2408 . . 3  |-  ( Base `  F )  =  (
Base `  F )
1412, 13isclm 19046 . 2  |-  ( W  e. CMod 
<->  ( W  e.  LMod  /\  F  =  (flds  ( Base `  F
) )  /\  ( Base `  F )  e.  (SubRing ` fld ) ) )
151, 9, 11, 14syl3anbrc 1138 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 936    = wceq 1649    e. wcel 1721   ` cfv 5417  (class class class)co 6044   Basecbs 13428   ↾s cress 13429  Scalarcsca 13491  SubRingcsubrg 15823   LModclmod 15909  ℂfldccnfld 16662  CModcclm 19044
This theorem is referenced by:  zlmclm  19077
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-i2m1 9018  ax-1ne0 9019  ax-rrecex 9022  ax-cnre 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-recs 6596  df-rdg 6631  df-nn 9961  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-subg 14900  df-rng 15622  df-subrg 15825  df-clm 19045
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