![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lmhmlvec | Structured version Visualization version GIF version |
Description: The property for modules to be vector spaces is invariant under module isomorphism. (Contributed by Steven Nguyen, 15-Aug-2023.) |
Ref | Expression |
---|---|
lmhmlvec | ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LVec ↔ 𝑇 ∈ LVec)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmhmlmod1 19798 | . . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod) | |
2 | lmhmlmod2 19797 | . . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod) | |
3 | 1, 2 | 2thd 268 | . . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LMod ↔ 𝑇 ∈ LMod)) |
4 | eqid 2798 | . . . . . 6 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆) | |
5 | eqid 2798 | . . . . . 6 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
6 | 4, 5 | lmhmsca 19795 | . . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆)) |
7 | 6 | eqcomd 2804 | . . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑆) = (Scalar‘𝑇)) |
8 | 7 | eleq1d 2874 | . . 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 19869 | . 2 ⊢ (𝑆 ∈ LVec ↔ (𝑆 ∈ LMod ∧ (Scalar‘𝑆) ∈ DivRing)) |
11 | 5 | islvec 19869 | . 2 ⊢ (𝑇 ∈ LVec ↔ (𝑇 ∈ LMod ∧ (Scalar‘𝑇) ∈ DivRing)) |
12 | 9, 10, 11 | 3bitr4g 317 | 1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ LVec ↔ 𝑇 ∈ LVec)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Scalarcsca 16560 DivRingcdr 19495 LModclmod 19627 LMHom clmhm 19784 LVecclvec 19867 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-lmhm 19787 df-lvec 19868 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |