MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isclmi Structured version   Visualization version   GIF version

Theorem isclmi 25123
Description: Reverse direction of isclm 25110. (Contributed by Mario Carneiro, 30-Oct-2015.)
Hypothesis
Ref Expression
clm0.f 𝐹 = (Scalar‘𝑊)
Assertion
Ref Expression
isclmi ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)

Proof of Theorem isclmi
StepHypRef Expression
1 simp1 1135 . 2 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ LMod)
2 simp2 1136 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐹 = (ℂflds 𝐾))
3 eqid 2734 . . . . . . 7 (ℂflds 𝐾) = (ℂflds 𝐾)
43subrgbas 20597 . . . . . 6 (𝐾 ∈ (SubRing‘ℂfld) → 𝐾 = (Base‘(ℂflds 𝐾)))
543ad2ant3 1134 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 = (Base‘(ℂflds 𝐾)))
62fveq2d 6910 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (Base‘𝐹) = (Base‘(ℂflds 𝐾)))
75, 6eqtr4d 2777 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 = (Base‘𝐹))
87oveq2d 7446 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (ℂflds 𝐾) = (ℂflds (Base‘𝐹)))
92, 8eqtrd 2774 . 2 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐹 = (ℂflds (Base‘𝐹)))
10 simp3 1137 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 ∈ (SubRing‘ℂfld))
117, 10eqeltrrd 2839 . 2 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (Base‘𝐹) ∈ (SubRing‘ℂfld))
12 clm0.f . . 3 𝐹 = (Scalar‘𝑊)
13 eqid 2734 . . 3 (Base‘𝐹) = (Base‘𝐹)
1412, 13isclm 25110 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘ℂfld)))
151, 9, 11, 14syl3anbrc 1342 1 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1536  wcel 2105  cfv 6562  (class class class)co 7430  Basecbs 17244  s cress 17273  Scalarcsca 17300  SubRingcsubrg 20585  LModclmod 20874  fldccnfld 21381  ℂModcclm 25108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-1cn 11210  ax-addcl 11212
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-nn 12264  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-subg 19153  df-ring 20252  df-subrg 20586  df-clm 25109
This theorem is referenced by:  zlmclm  25158  cnstrcvs  25187  cncvs  25191  recvs  25192  recvsOLD  25193  qcvs  25194  zclmncvs  25195
  Copyright terms: Public domain W3C validator