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Theorem lmhmclm 23670
 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 19780 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2 lmhmlmod2 19779 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
31, 22thd 268 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LMod ↔ 𝑇 ∈ LMod))
4 eqid 2821 . . . . . 6 (Scalar‘𝑆) = (Scalar‘𝑆)
5 eqid 2821 . . . . . 6 (Scalar‘𝑇) = (Scalar‘𝑇)
64, 5lmhmsca 19777 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
76eqcomd 2827 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑆) = (Scalar‘𝑇))
87fveq2d 6647 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑇)))
98oveq2d 7146 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (ℂflds (Base‘(Scalar‘𝑆))) = (ℂflds (Base‘(Scalar‘𝑇))))
107, 9eqeq12d 2837 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((Scalar‘𝑆) = (ℂflds (Base‘(Scalar‘𝑆))) ↔ (Scalar‘𝑇) = (ℂflds (Base‘(Scalar‘𝑇)))))
118eleq1d 2896 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((Base‘(Scalar‘𝑆)) ∈ (SubRing‘ℂfld) ↔ (Base‘(Scalar‘𝑇)) ∈ (SubRing‘ℂfld)))
123, 10, 113anbi123d 1433 . 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 2821 . . 3 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
144, 13isclm 23647 . 2 (𝑆 ∈ ℂMod ↔ (𝑆 ∈ LMod ∧ (Scalar‘𝑆) = (ℂflds (Base‘(Scalar‘𝑆))) ∧ (Base‘(Scalar‘𝑆)) ∈ (SubRing‘ℂfld)))
15 eqid 2821 . . 3 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
165, 15isclm 23647 . 2 (𝑇 ∈ ℂMod ↔ (𝑇 ∈ LMod ∧ (Scalar‘𝑇) = (ℂflds (Base‘(Scalar‘𝑇))) ∧ (Base‘(Scalar‘𝑇)) ∈ (SubRing‘ℂfld)))
1712, 14, 163bitr4g 317 1 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ ℂMod ↔ 𝑇 ∈ ℂMod))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ‘cfv 6328  (class class class)co 7130  Basecbs 16461   ↾s cress 16462  Scalarcsca 16546  SubRingcsubrg 19506  LModclmod 19609   LMHom clmhm 19766  ℂfldccnfld 20520  ℂModcclm 23645 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-lmhm 19769  df-clm 23646 This theorem is referenced by: (None)
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