Theorem lmhmclm 22695

Description: The domain of a linear operator is a complex 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 18854	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2		lmhmlmod2 18853	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
3	1, 2	2thd 254	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LMod ↔ 𝑇 ∈ LMod))
4		eqid 2610	. . . . . 6 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
5		eqid 2610	. . . . . 6 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
6	4, 5	lmhmsca 18851	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
7	6	eqcomd 2616	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑆) = (Scalar‘𝑇))
8	7	fveq2d 6107	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑇)))
9	8	oveq2d 6565	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (ℂfld ↾s (Base‘(Scalar‘𝑆))) = (ℂfld ↾s (Base‘(Scalar‘𝑇))))
10	7, 9	eqeq12d 2625	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((Scalar‘𝑆) = (ℂfld ↾s (Base‘(Scalar‘𝑆))) ↔ (Scalar‘𝑇) = (ℂfld ↾s (Base‘(Scalar‘𝑇)))))
11	8	eleq1d 2672	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((Base‘(Scalar‘𝑆)) ∈ (SubRing‘ℂfld) ↔ (Base‘(Scalar‘𝑇)) ∈ (SubRing‘ℂfld)))
12	3, 10, 11	3anbi123d 1391	. 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 2610	. . 3 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
14	4, 13	isclm 22672	. 2 ⊢ (𝑆 ∈ ℂMod ↔ (𝑆 ∈ LMod ∧ (Scalar‘𝑆) = (ℂfld ↾s (Base‘(Scalar‘𝑆))) ∧ (Base‘(Scalar‘𝑆)) ∈ (SubRing‘ℂfld)))
15		eqid 2610	. . 3 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
16	5, 15	isclm 22672	. 2 ⊢ (𝑇 ∈ ℂMod ↔ (𝑇 ∈ LMod ∧ (Scalar‘𝑇) = (ℂfld ↾s (Base‘(Scalar‘𝑇))) ∧ (Base‘(Scalar‘𝑇)) ∈ (SubRing‘ℂfld)))
17	12, 14, 16	3bitr4g 302	1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ ℂMod ↔ 𝑇 ∈ ℂMod))

Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾_s cress 15696  Scalarcsca 15771  SubRingcsubrg 18599  LModclmod 18686   LMHom clmhm 18840  ℂ_fldccnfld 19567  ℂModcclm 22670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-lmhm 18843  df-clm 22671

This theorem is referenced by: (None)