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Theorem isclm 19090
 Description: A complex module is a left module over a subring of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
isclm.f Scalar
isclm.k
Assertion
Ref Expression
isclm CMod flds SubRingfld

Proof of Theorem isclm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5743 . . . . 5 Scalar
21a1i 11 . . . 4 Scalar
3 fvex 5743 . . . . . 6
43a1i 11 . . . . 5 Scalar
5 id 21 . . . . . . . . 9 Scalar Scalar
6 fveq2 5729 . . . . . . . . . 10 Scalar Scalar
7 isclm.f . . . . . . . . . 10 Scalar
86, 7syl6eqr 2487 . . . . . . . . 9 Scalar
95, 8sylan9eqr 2491 . . . . . . . 8 Scalar
109adantr 453 . . . . . . 7 Scalar
11 id 21 . . . . . . . . 9
129fveq2d 5733 . . . . . . . . . 10 Scalar
13 isclm.k . . . . . . . . . 10
1412, 13syl6eqr 2487 . . . . . . . . 9 Scalar
1511, 14sylan9eqr 2491 . . . . . . . 8 Scalar
1615oveq2d 6098 . . . . . . 7 Scalar flds flds
1710, 16eqeq12d 2451 . . . . . 6 Scalar flds flds
1815eleq1d 2503 . . . . . 6 Scalar SubRingfld SubRingfld
1917, 18anbi12d 693 . . . . 5 Scalar flds SubRingfld flds SubRingfld
204, 19sbcied 3198 . . . 4 Scalar flds SubRingfld flds SubRingfld
212, 20sbcied 3198 . . 3 Scalar flds SubRingfld flds SubRingfld
22 df-clm 19089 . . 3 CMod Scalar flds SubRingfld
2321, 22elrab2 3095 . 2 CMod flds SubRingfld
24 3anass 941 . 2 flds SubRingfld flds SubRingfld
2523, 24bitr4i 245 1 CMod flds SubRingfld
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   w3a 937   wceq 1653   wcel 1726  cvv 2957  wsbc 3162  cfv 5455  (class class class)co 6082  cbs 13470   ↾s cress 13471  Scalarcsca 13533  SubRingcsubrg 15865  clmod 15951  ℂfldccnfld 16704  CModcclm 19088 This theorem is referenced by:  clmsca  19091  clmsubrg  19092  clmlmod  19093  isclmi  19103  lmhmclm  19112  cphclm  19153  tchclm  19190 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-nul 4339 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-iota 5419  df-fv 5463  df-ov 6085  df-clm 19089
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