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Theorem lmhmclm 23214
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 19354 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2 lmhmlmod2 19353 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
31, 22thd 257 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LMod ↔ 𝑇 ∈ LMod))
4 eqid 2799 . . . . . 6 (Scalar‘𝑆) = (Scalar‘𝑆)
5 eqid 2799 . . . . . 6 (Scalar‘𝑇) = (Scalar‘𝑇)
64, 5lmhmsca 19351 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
76eqcomd 2805 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑆) = (Scalar‘𝑇))
87fveq2d 6415 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑇)))
98oveq2d 6894 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → (ℂflds (Base‘(Scalar‘𝑆))) = (ℂflds (Base‘(Scalar‘𝑇))))
107, 9eqeq12d 2814 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((Scalar‘𝑆) = (ℂflds (Base‘(Scalar‘𝑆))) ↔ (Scalar‘𝑇) = (ℂflds (Base‘(Scalar‘𝑇)))))
118eleq1d 2863 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → ((Base‘(Scalar‘𝑆)) ∈ (SubRing‘ℂfld) ↔ (Base‘(Scalar‘𝑇)) ∈ (SubRing‘ℂfld)))
123, 10, 113anbi123d 1561 . 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 2799 . . 3 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
144, 13isclm 23191 . 2 (𝑆 ∈ ℂMod ↔ (𝑆 ∈ LMod ∧ (Scalar‘𝑆) = (ℂflds (Base‘(Scalar‘𝑆))) ∧ (Base‘(Scalar‘𝑆)) ∈ (SubRing‘ℂfld)))
15 eqid 2799 . . 3 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
165, 15isclm 23191 . 2 (𝑇 ∈ ℂMod ↔ (𝑇 ∈ LMod ∧ (Scalar‘𝑇) = (ℂflds (Base‘(Scalar‘𝑇))) ∧ (Base‘(Scalar‘𝑇)) ∈ (SubRing‘ℂfld)))
1712, 14, 163bitr4g 306 1 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ ℂMod ↔ 𝑇 ∈ ℂMod))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1108   = wceq 1653  wcel 2157  cfv 6101  (class class class)co 6878  Basecbs 16184  s cress 16185  Scalarcsca 16270  SubRingcsubrg 19094  LModclmod 19181   LMHom clmhm 19340  fldccnfld 20068  ℂModcclm 23189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-lmhm 19343  df-clm 23190
This theorem is referenced by: (None)
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