Theorem lmhmclm 25004

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 20956	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2		lmhmlmod2 20955	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
3	1, 2	2thd 265	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LMod ↔ 𝑇 ∈ LMod))
4		eqid 2729	. . . . . 6 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
5		eqid 2729	. . . . . 6 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
6	4, 5	lmhmsca 20953	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
7	6	eqcomd 2735	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑆) = (Scalar‘𝑇))
8	7	fveq2d 6830	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑇)))
9	8	oveq2d 7369	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (ℂfld ↾s (Base‘(Scalar‘𝑆))) = (ℂfld ↾s (Base‘(Scalar‘𝑇))))
10	7, 9	eqeq12d 2745	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((Scalar‘𝑆) = (ℂfld ↾s (Base‘(Scalar‘𝑆))) ↔ (Scalar‘𝑇) = (ℂfld ↾s (Base‘(Scalar‘𝑇)))))
11	8	eleq1d 2813	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((Base‘(Scalar‘𝑆)) ∈ (SubRing‘ℂfld) ↔ (Base‘(Scalar‘𝑇)) ∈ (SubRing‘ℂfld)))
12	3, 10, 11	3anbi123d 1438	. 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 2729	. . 3 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
14	4, 13	isclm 24981	. 2 ⊢ (𝑆 ∈ ℂMod ↔ (𝑆 ∈ LMod ∧ (Scalar‘𝑆) = (ℂfld ↾s (Base‘(Scalar‘𝑆))) ∧ (Base‘(Scalar‘𝑆)) ∈ (SubRing‘ℂfld)))
15		eqid 2729	. . 3 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
16	5, 15	isclm 24981	. 2 ⊢ (𝑇 ∈ ℂMod ↔ (𝑇 ∈ LMod ∧ (Scalar‘𝑇) = (ℂfld ↾s (Base‘(Scalar‘𝑇))) ∧ (Base‘(Scalar‘𝑇)) ∈ (SubRing‘ℂfld)))
17	12, 14, 16	3bitr4g 314	1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ ℂMod ↔ 𝑇 ∈ ℂMod))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ‘cfv 6486  (class class class)co 7353  Basecbs 17139   ↾_s cress 17160  Scalarcsca 17183  SubRingcsubrg 20473  LModclmod 20782   LMHom clmhm 20942  ℂ_fldccnfld 21280  ℂModcclm 24979

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-lmhm 20945  df-clm 24980

This theorem is referenced by: (None)