lmhmlvec - Metamath Proof Explorer

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Theorem lmhmlvec 21105

Description: The property for modules to be vector spaces is invariant under module isomorphism. (Contributed by Steven Nguyen, 15-Aug-2023.)

**Assertion**
Ref	Expression
lmhmlvec	⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LVec ↔ 𝑇 ∈ LVec))

**Proof of Theorem lmhmlvec**
Step	Hyp	Ref	Expression
1		lmhmlmod1 21028	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2		lmhmlmod2 21027	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
3	1, 2	2thd 265	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LMod ↔ 𝑇 ∈ LMod))
4		eqid 2736	. . . . . 6 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
5		eqid 2736	. . . . . 6 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
6	4, 5	lmhmsca 21025	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
7	6	eqcomd 2742	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑆) = (Scalar‘𝑇))
8	7	eleq1d 2821	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((Scalar‘𝑆) ∈ DivRing ↔ (Scalar‘𝑇) ∈ DivRing))
9	3, 8	anbi12d 633	. 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ (Scalar‘𝑆) ∈ DivRing) ↔ (𝑇 ∈ LMod ∧ (Scalar‘𝑇) ∈ DivRing)))
10	4	islvec 21099	. 2 ⊢ (𝑆 ∈ LVec ↔ (𝑆 ∈ LMod ∧ (Scalar‘𝑆) ∈ DivRing))
11	5	islvec 21099	. 2 ⊢ (𝑇 ∈ LVec ↔ (𝑇 ∈ LMod ∧ (Scalar‘𝑇) ∈ DivRing))
12	9, 10, 11	3bitr4g 314	1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LVec ↔ 𝑇 ∈ LVec))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2114 ‘cfv 6498 (class class class)co 7367 Scalarcsca 17223 DivRingcdr 20706 LModclmod 20855 LMHom clmhm 21014 LVecclvec 21097

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 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pr 5375

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-sbc 3729 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6454 df-fun 6500 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-lmhm 21017 df-lvec 21098

This theorem is referenced by: lmimdim 33748

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