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Theorem lmhmclm 22933
Description: The domain of a linear operator is a subcomplex module iff the range is. (Contributed by Mario Carneiro, 21-Oct-2015.)
Assertion
Ref Expression
lmhmclm (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ ℂMod ↔ 𝑇 ∈ ℂMod))

Proof of Theorem lmhmclm
StepHypRef Expression
1 lmhmlmod1 19081 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2 lmhmlmod2 19080 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
31, 22thd 255 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LMod ↔ 𝑇 ∈ LMod))
4 eqid 2651 . . . . . 6 (Scalar‘𝑆) = (Scalar‘𝑆)
5 eqid 2651 . . . . . 6 (Scalar‘𝑇) = (Scalar‘𝑇)
64, 5lmhmsca 19078 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
76eqcomd 2657 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑆) = (Scalar‘𝑇))
87fveq2d 6233 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑇)))
98oveq2d 6706 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (ℂflds (Base‘(Scalar‘𝑆))) = (ℂflds (Base‘(Scalar‘𝑇))))
107, 9eqeq12d 2666 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((Scalar‘𝑆) = (ℂflds (Base‘(Scalar‘𝑆))) ↔ (Scalar‘𝑇) = (ℂflds (Base‘(Scalar‘𝑇)))))
118eleq1d 2715 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((Base‘(Scalar‘𝑆)) ∈ (SubRing‘ℂfld) ↔ (Base‘(Scalar‘𝑇)) ∈ (SubRing‘ℂfld)))
123, 10, 113anbi123d 1439 . 2 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ (Scalar‘𝑆) = (ℂflds (Base‘(Scalar‘𝑆))) ∧ (Base‘(Scalar‘𝑆)) ∈ (SubRing‘ℂfld)) ↔ (𝑇 ∈ LMod ∧ (Scalar‘𝑇) = (ℂflds (Base‘(Scalar‘𝑇))) ∧ (Base‘(Scalar‘𝑇)) ∈ (SubRing‘ℂfld))))
13 eqid 2651 . . 3 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
144, 13isclm 22910 . 2 (𝑆 ∈ ℂMod ↔ (𝑆 ∈ LMod ∧ (Scalar‘𝑆) = (ℂflds (Base‘(Scalar‘𝑆))) ∧ (Base‘(Scalar‘𝑆)) ∈ (SubRing‘ℂfld)))
15 eqid 2651 . . 3 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
165, 15isclm 22910 . 2 (𝑇 ∈ ℂMod ↔ (𝑇 ∈ LMod ∧ (Scalar‘𝑇) = (ℂflds (Base‘(Scalar‘𝑇))) ∧ (Base‘(Scalar‘𝑇)) ∈ (SubRing‘ℂfld)))
1712, 14, 163bitr4g 303 1 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ ℂMod ↔ 𝑇 ∈ ℂMod))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1054   = wceq 1523  wcel 2030  cfv 5926  (class class class)co 6690  Basecbs 15904  s cress 15905  Scalarcsca 15991  SubRingcsubrg 18824  LModclmod 18911   LMHom clmhm 19067  fldccnfld 19794  ℂModcclm 22908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-lmhm 19070  df-clm 22909
This theorem is referenced by: (None)
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