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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pcl0bN | Structured version Visualization version GIF version |
Description: The projective subspace closure of the empty subspace. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pcl0b.a | ⊢ 𝐴 = (Atoms‘𝐾) |
pcl0b.c | ⊢ 𝑈 = (PCl‘𝐾) |
Ref | Expression |
---|---|
pcl0bN | ⊢ ((𝐾 ∈ HL ∧ 𝑃 ⊆ 𝐴) → ((𝑈‘𝑃) = ∅ ↔ 𝑃 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pcl0b.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | pcl0b.c | . . . . 5 ⊢ 𝑈 = (PCl‘𝐾) | |
3 | 1, 2 | pclssidN 39260 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ⊆ 𝐴) → 𝑃 ⊆ (𝑈‘𝑃)) |
4 | eqimss 4033 | . . . 4 ⊢ ((𝑈‘𝑃) = ∅ → (𝑈‘𝑃) ⊆ ∅) | |
5 | 3, 4 | sylan9ss 3988 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ⊆ 𝐴) ∧ (𝑈‘𝑃) = ∅) → 𝑃 ⊆ ∅) |
6 | ss0 4391 | . . 3 ⊢ (𝑃 ⊆ ∅ → 𝑃 = ∅) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ⊆ 𝐴) ∧ (𝑈‘𝑃) = ∅) → 𝑃 = ∅) |
8 | fveq2 6882 | . . . 4 ⊢ (𝑃 = ∅ → (𝑈‘𝑃) = (𝑈‘∅)) | |
9 | 2 | pcl0N 39287 | . . . 4 ⊢ (𝐾 ∈ HL → (𝑈‘∅) = ∅) |
10 | 8, 9 | sylan9eqr 2786 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑃 = ∅) → (𝑈‘𝑃) = ∅) |
11 | 10 | adantlr 712 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ⊆ 𝐴) ∧ 𝑃 = ∅) → (𝑈‘𝑃) = ∅) |
12 | 7, 11 | impbida 798 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ⊆ 𝐴) → ((𝑈‘𝑃) = ∅ ↔ 𝑃 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⊆ wss 3941 ∅c0 4315 ‘cfv 6534 Atomscatm 38627 HLchlt 38714 PClcpclN 39252 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-proset 18252 df-poset 18270 df-plt 18287 df-lub 18303 df-glb 18304 df-join 18305 df-meet 18306 df-p0 18382 df-p1 18383 df-lat 18389 df-clat 18456 df-oposet 38540 df-ol 38542 df-oml 38543 df-covers 38630 df-ats 38631 df-atl 38662 df-cvlat 38686 df-hlat 38715 df-psubsp 38868 df-pmap 38869 df-pclN 39253 df-polarityN 39268 |
This theorem is referenced by: pclfinclN 39315 |
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