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Mirrors > Home > MPE Home > Th. List > lbsextlem1 | Structured version Visualization version GIF version |
Description: Lemma for lbsext 18984. The set 𝑆 is the set of all linearly independent sets containing 𝐶; we show here that it is nonempty. (Contributed by Mario Carneiro, 25-Jun-2014.) |
Ref | Expression |
---|---|
lbsext.v | ⊢ 𝑉 = (Base‘𝑊) |
lbsext.j | ⊢ 𝐽 = (LBasis‘𝑊) |
lbsext.n | ⊢ 𝑁 = (LSpan‘𝑊) |
lbsext.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lbsext.c | ⊢ (𝜑 → 𝐶 ⊆ 𝑉) |
lbsext.x | ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) |
lbsext.s | ⊢ 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))} |
Ref | Expression |
---|---|
lbsextlem1 | ⊢ (𝜑 → 𝑆 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lbsext.c | . . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝑉) | |
2 | lbsext.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
3 | fvex 6113 | . . . . . 6 ⊢ (Base‘𝑊) ∈ V | |
4 | 2, 3 | eqeltri 2684 | . . . . 5 ⊢ 𝑉 ∈ V |
5 | 4 | elpw2 4755 | . . . 4 ⊢ (𝐶 ∈ 𝒫 𝑉 ↔ 𝐶 ⊆ 𝑉) |
6 | 1, 5 | sylibr 223 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝑉) |
7 | lbsext.x | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) | |
8 | ssid 3587 | . . . 4 ⊢ 𝐶 ⊆ 𝐶 | |
9 | 7, 8 | jctil 558 | . . 3 ⊢ (𝜑 → (𝐶 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))) |
10 | sseq2 3590 | . . . . 5 ⊢ (𝑧 = 𝐶 → (𝐶 ⊆ 𝑧 ↔ 𝐶 ⊆ 𝐶)) | |
11 | difeq1 3683 | . . . . . . . . 9 ⊢ (𝑧 = 𝐶 → (𝑧 ∖ {𝑥}) = (𝐶 ∖ {𝑥})) | |
12 | 11 | fveq2d 6107 | . . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝑁‘(𝑧 ∖ {𝑥})) = (𝑁‘(𝐶 ∖ {𝑥}))) |
13 | 12 | eleq2d 2673 | . . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))) |
14 | 13 | notbid 307 | . . . . . 6 ⊢ (𝑧 = 𝐶 → (¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))) |
15 | 14 | raleqbi1dv 3123 | . . . . 5 ⊢ (𝑧 = 𝐶 → (∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))) |
16 | 10, 15 | anbi12d 743 | . . . 4 ⊢ (𝑧 = 𝐶 → ((𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥}))) ↔ (𝐶 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))) |
17 | lbsext.s | . . . 4 ⊢ 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))} | |
18 | 16, 17 | elrab2 3333 | . . 3 ⊢ (𝐶 ∈ 𝑆 ↔ (𝐶 ∈ 𝒫 𝑉 ∧ (𝐶 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))) |
19 | 6, 9, 18 | sylanbrc 695 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
20 | ne0i 3880 | . 2 ⊢ (𝐶 ∈ 𝑆 → 𝑆 ≠ ∅) | |
21 | 19, 20 | syl 17 | 1 ⊢ (𝜑 → 𝑆 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 ‘cfv 5804 Basecbs 15695 LSpanclspn 18792 LBasisclbs 18895 LVecclvec 18923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 |
This theorem is referenced by: lbsextlem4 18982 |
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