lmhmlvec - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lmhmlvec		Structured version Visualization version GIF version

Theorem lmhmlvec 21032

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 20955	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
2		lmhmlmod2 20954	. . . 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 20952	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
7	6	eqcomd 2735	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑆) = (Scalar‘𝑇))
8	7	eleq1d 2813	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((Scalar‘𝑆) ∈ DivRing ↔ (Scalar‘𝑇) ∈ DivRing))
9	3, 8	anbi12d 632	. 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ (Scalar‘𝑆) ∈ DivRing) ↔ (𝑇 ∈ LMod ∧ (Scalar‘𝑇) ∈ DivRing)))
10	4	islvec 21026	. 2 ⊢ (𝑆 ∈ LVec ↔ (𝑆 ∈ LMod ∧ (Scalar‘𝑆) ∈ DivRing))
11	5	islvec 21026	. 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 2109 ‘cfv 6486 (class class class)co 7353 Scalarcsca 17182 DivRingcdr 20632 LModclmod 20781 LMHom clmhm 20941 LVecclvec 21024

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 20944 df-lvec 21025

This theorem is referenced by: lmimdim 33578

W3C validator