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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-isvec | Structured version Visualization version GIF version |
Description: The predicate "is a vector space". (Contributed by BJ, 6-Jan-2024.) |
Ref | Expression |
---|---|
bj-isvec.scal | ⊢ (𝜑 → 𝐾 = (Scalar‘𝑉)) |
Ref | Expression |
---|---|
bj-isvec | ⊢ (𝜑 → (𝑉 ∈ LVec ↔ (𝑉 ∈ LMod ∧ 𝐾 ∈ DivRing))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2820 | . . 3 ⊢ (Scalar‘𝑉) = (Scalar‘𝑉) | |
2 | 1 | islvec 19869 | . 2 ⊢ (𝑉 ∈ LVec ↔ (𝑉 ∈ LMod ∧ (Scalar‘𝑉) ∈ DivRing)) |
3 | bj-isvec.scal | . . . . 5 ⊢ (𝜑 → 𝐾 = (Scalar‘𝑉)) | |
4 | 3 | eqcomd 2826 | . . . 4 ⊢ (𝜑 → (Scalar‘𝑉) = 𝐾) |
5 | 4 | eleq1d 2896 | . . 3 ⊢ (𝜑 → ((Scalar‘𝑉) ∈ DivRing ↔ 𝐾 ∈ DivRing)) |
6 | 5 | anbi2d 630 | . 2 ⊢ (𝜑 → ((𝑉 ∈ LMod ∧ (Scalar‘𝑉) ∈ DivRing) ↔ (𝑉 ∈ LMod ∧ 𝐾 ∈ DivRing))) |
7 | 2, 6 | syl5bb 285 | 1 ⊢ (𝜑 → (𝑉 ∈ LVec ↔ (𝑉 ∈ LMod ∧ 𝐾 ∈ DivRing))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 Scalarcsca 16561 DivRingcdr 19495 LModclmod 19627 LVecclvec 19867 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-iota 6307 df-fv 6356 df-lvec 19868 |
This theorem is referenced by: bj-rvecvec 34602 |
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