Theorem lmhmclm 23685

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

Step	Hyp	Ref	Expression
1		lmhmlmod1 19799	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2		lmhmlmod2 19798	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
3	1, 2	2thd 267	. . 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)
6	4, 5	lmhmsca 19796	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
7	6	eqcomd 2827	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑆) = (Scalar‘𝑇))
8	7	fveq2d 6668	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑇)))
9	8	oveq2d 7166	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (ℂfld ↾s (Base‘(Scalar‘𝑆))) = (ℂfld ↾s (Base‘(Scalar‘𝑇))))
10	7, 9	eqeq12d 2837	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((Scalar‘𝑆) = (ℂfld ↾s (Base‘(Scalar‘𝑆))) ↔ (Scalar‘𝑇) = (ℂfld ↾s (Base‘(Scalar‘𝑇)))))
11	8	eleq1d 2897	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((Base‘(Scalar‘𝑆)) ∈ (SubRing‘ℂfld) ↔ (Base‘(Scalar‘𝑇)) ∈ (SubRing‘ℂfld)))
12	3, 10, 11	3anbi123d 1432	. 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ (Scalar‘𝑆) = (ℂfld ↾s (Base‘(Scalar‘𝑆))) ∧ (Base‘(Scalar‘𝑆)) ∈ (SubRing‘ℂfld)) ↔ (𝑇 ∈ LMod ∧ (Scalar‘𝑇) = (ℂfld ↾s (Base‘(Scalar‘𝑇))) ∧ (Base‘(Scalar‘𝑇)) ∈ (SubRing‘ℂfld))))
13		eqid 2821	. . 3 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
14	4, 13	isclm 23662	. 2 ⊢ (𝑆 ∈ ℂMod ↔ (𝑆 ∈ LMod ∧ (Scalar‘𝑆) = (ℂfld ↾s (Base‘(Scalar‘𝑆))) ∧ (Base‘(Scalar‘𝑆)) ∈ (SubRing‘ℂfld)))
15		eqid 2821	. . 3 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
16	5, 15	isclm 23662	. 2 ⊢ (𝑇 ∈ ℂMod ↔ (𝑇 ∈ LMod ∧ (Scalar‘𝑇) = (ℂfld ↾s (Base‘(Scalar‘𝑇))) ∧ (Base‘(Scalar‘𝑇)) ∈ (SubRing‘ℂfld)))
17	12, 14, 16	3bitr4g 316	1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ ℂMod ↔ 𝑇 ∈ ℂMod))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ‘cfv 6349  (class class class)co 7150  Basecbs 16477   ↾_s cress 16478  Scalarcsca 16562  SubRingcsubrg 19525  LModclmod 19628   LMHom clmhm 19785  ℂ_fldccnfld 20539  ℂModcclm 23660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-lmhm 19788  df-clm 23661

This theorem is referenced by: (None)