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

Theorem isclmi 25110
Description: Reverse direction of isclm 25097. (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 1137 . 2 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ LMod)
2 simp2 1138 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐹 = (ℂflds 𝐾))
3 eqid 2737 . . . . . . 7 (ℂflds 𝐾) = (ℂflds 𝐾)
43subrgbas 20581 . . . . . 6 (𝐾 ∈ (SubRing‘ℂfld) → 𝐾 = (Base‘(ℂflds 𝐾)))
543ad2ant3 1136 . . . . 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 2780 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 = (Base‘𝐹))
87oveq2d 7447 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (ℂflds 𝐾) = (ℂflds (Base‘𝐹)))
92, 8eqtrd 2777 . 2 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐹 = (ℂflds (Base‘𝐹)))
10 simp3 1139 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝐾 ∈ (SubRing‘ℂfld))
117, 10eqeltrrd 2842 . 2 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → (Base‘𝐹) ∈ (SubRing‘ℂfld))
12 clm0.f . . 3 𝐹 = (Scalar‘𝑊)
13 eqid 2737 . . 3 (Base‘𝐹) = (Base‘𝐹)
1412, 13isclm 25097 . 2 (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂflds (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘ℂfld)))
151, 9, 11, 14syl3anbrc 1344 1 ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  Scalarcsca 17300  SubRingcsubrg 20569  LModclmod 20858  fldccnfld 21364  ℂModcclm 25095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-subg 19141  df-ring 20232  df-subrg 20570  df-clm 25096
This theorem is referenced by:  zlmclm  25145  cnstrcvs  25174  cncvs  25178  recvs  25179  recvsOLD  25180  qcvs  25181  zclmncvs  25182
  Copyright terms: Public domain W3C validator