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Mirrors > Home > MPE Home > Th. List > lbsextlem1 | Structured version Visualization version GIF version |
Description: Lemma for lbsext 19918. 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 | 2 | fvexi 6670 | . . . . 5 ⊢ 𝑉 ∈ V |
4 | 3 | elpw2 5234 | . . . 4 ⊢ (𝐶 ∈ 𝒫 𝑉 ↔ 𝐶 ⊆ 𝑉) |
5 | 1, 4 | sylibr 236 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝑉) |
6 | lbsext.x | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))) | |
7 | ssid 3977 | . . . 4 ⊢ 𝐶 ⊆ 𝐶 | |
8 | 6, 7 | jctil 522 | . . 3 ⊢ (𝜑 → (𝐶 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))) |
9 | sseq2 3981 | . . . . 5 ⊢ (𝑧 = 𝐶 → (𝐶 ⊆ 𝑧 ↔ 𝐶 ⊆ 𝐶)) | |
10 | difeq1 4080 | . . . . . . . . 9 ⊢ (𝑧 = 𝐶 → (𝑧 ∖ {𝑥}) = (𝐶 ∖ {𝑥})) | |
11 | 10 | fveq2d 6660 | . . . . . . . 8 ⊢ (𝑧 = 𝐶 → (𝑁‘(𝑧 ∖ {𝑥})) = (𝑁‘(𝐶 ∖ {𝑥}))) |
12 | 11 | eleq2d 2898 | . . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))) |
13 | 12 | notbid 320 | . . . . . 6 ⊢ (𝑧 = 𝐶 → (¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))) |
14 | 13 | raleqbi1dv 3403 | . . . . 5 ⊢ (𝑧 = 𝐶 → (∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})) ↔ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥})))) |
15 | 9, 14 | anbi12d 632 | . . . 4 ⊢ (𝑧 = 𝐶 → ((𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥}))) ↔ (𝐶 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))) |
16 | lbsext.s | . . . 4 ⊢ 𝑆 = {𝑧 ∈ 𝒫 𝑉 ∣ (𝐶 ⊆ 𝑧 ∧ ∀𝑥 ∈ 𝑧 ¬ 𝑥 ∈ (𝑁‘(𝑧 ∖ {𝑥})))} | |
17 | 15, 16 | elrab2 3674 | . . 3 ⊢ (𝐶 ∈ 𝑆 ↔ (𝐶 ∈ 𝒫 𝑉 ∧ (𝐶 ⊆ 𝐶 ∧ ∀𝑥 ∈ 𝐶 ¬ 𝑥 ∈ (𝑁‘(𝐶 ∖ {𝑥}))))) |
18 | 5, 8, 17 | sylanbrc 585 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
19 | 18 | ne0d 4287 | 1 ⊢ (𝜑 → 𝑆 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 {crab 3142 ∖ cdif 3921 ⊆ wss 3924 ∅c0 4279 𝒫 cpw 4525 {csn 4553 ‘cfv 6341 Basecbs 16466 LSpanclspn 19726 LBasisclbs 19829 LVecclvec 19857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-iota 6300 df-fv 6349 |
This theorem is referenced by: lbsextlem4 19916 |
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