Proof of Theorem islindf2
Step | Hyp | Ref
| Expression |
1 | | simp1 1116 |
. . 3
⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → 𝑊 ∈ 𝑌) |
2 | | simp3 1118 |
. . . 4
⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → 𝐹:𝐼⟶𝐵) |
3 | | simp2 1117 |
. . . 4
⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → 𝐼 ∈ 𝑋) |
4 | | fex 6813 |
. . . 4
⊢ ((𝐹:𝐼⟶𝐵 ∧ 𝐼 ∈ 𝑋) → 𝐹 ∈ V) |
5 | 2, 3, 4 | syl2anc 576 |
. . 3
⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → 𝐹 ∈ V) |
6 | | islindf.b |
. . . 4
⊢ 𝐵 = (Base‘𝑊) |
7 | | islindf.v |
. . . 4
⊢ · = (
·𝑠 ‘𝑊) |
8 | | islindf.k |
. . . 4
⊢ 𝐾 = (LSpan‘𝑊) |
9 | | islindf.s |
. . . 4
⊢ 𝑆 = (Scalar‘𝑊) |
10 | | islindf.n |
. . . 4
⊢ 𝑁 = (Base‘𝑆) |
11 | | islindf.z |
. . . 4
⊢ 0 =
(0g‘𝑆) |
12 | 6, 7, 8, 9, 10, 11 | islindf 20670 |
. . 3
⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 ∈ V) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))) |
13 | 1, 5, 12 | syl2anc 576 |
. 2
⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))) |
14 | | ffdm 6362 |
. . . . 5
⊢ (𝐹:𝐼⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐼)) |
15 | 14 | simpld 487 |
. . . 4
⊢ (𝐹:𝐼⟶𝐵 → 𝐹:dom 𝐹⟶𝐵) |
16 | 15 | 3ad2ant3 1115 |
. . 3
⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → 𝐹:dom 𝐹⟶𝐵) |
17 | 16 | biantrurd 525 |
. 2
⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (∀𝑥 ∈ dom 𝐹∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))) |
18 | | fdm 6349 |
. . . 4
⊢ (𝐹:𝐼⟶𝐵 → dom 𝐹 = 𝐼) |
19 | 18 | 3ad2ant3 1115 |
. . 3
⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → dom 𝐹 = 𝐼) |
20 | 19 | difeq1d 3982 |
. . . . . . . 8
⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (dom 𝐹 ∖ {𝑥}) = (𝐼 ∖ {𝑥})) |
21 | 20 | imaeq2d 5767 |
. . . . . . 7
⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 “ (dom 𝐹 ∖ {𝑥})) = (𝐹 “ (𝐼 ∖ {𝑥}))) |
22 | 21 | fveq2d 6500 |
. . . . . 6
⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) = (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥})))) |
23 | 22 | eleq2d 2845 |
. . . . 5
⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → ((𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥}))))) |
24 | 23 | notbid 310 |
. . . 4
⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥}))))) |
25 | 24 | ralbidv 3141 |
. . 3
⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥}))))) |
26 | 19, 25 | raleqbidv 3335 |
. 2
⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (∀𝑥 ∈ dom 𝐹∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))) ↔ ∀𝑥 ∈ 𝐼 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥}))))) |
27 | 13, 17, 26 | 3bitr2d 299 |
1
⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥 ∈ 𝐼 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥}))))) |