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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem15 | Structured version Visualization version GIF version |
Description: Lemma for dath 37436. The axis of perspectivity 𝑋 is a line. (Contributed by NM, 21-Jul-2012.) |
Ref | Expression |
---|---|
dalema.ph | ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈))))) |
dalemc.l | ⊢ ≤ = (le‘𝐾) |
dalemc.j | ⊢ ∨ = (join‘𝐾) |
dalemc.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dalem15.m | ⊢ ∧ = (meet‘𝐾) |
dalem15.n | ⊢ 𝑁 = (LLines‘𝐾) |
dalem15.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalem15.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
dalem15.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
dalem15.x | ⊢ 𝑋 = (𝑌 ∧ 𝑍) |
Ref | Expression |
---|---|
dalem15 | ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑋 ∈ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalem15.x | . 2 ⊢ 𝑋 = (𝑌 ∧ 𝑍) | |
2 | dalema.ph | . . . 4 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈))))) | |
3 | dalemc.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
4 | dalemc.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
5 | dalemc.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | dalem15.o | . . . 4 ⊢ 𝑂 = (LPlanes‘𝐾) | |
7 | eqid 2736 | . . . 4 ⊢ (LVols‘𝐾) = (LVols‘𝐾) | |
8 | dalem15.y | . . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
9 | dalem15.z | . . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
10 | eqid 2736 | . . . 4 ⊢ (𝑌 ∨ 𝐶) = (𝑌 ∨ 𝐶) | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | dalem14 37377 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∨ 𝑍) ∈ (LVols‘𝐾)) |
12 | 2 | dalemkehl 37323 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
13 | 2 | dalemyeo 37332 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑂) |
14 | 2 | dalemzeo 37333 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑂) |
15 | dalem15.m | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
16 | dalem15.n | . . . . . 6 ⊢ 𝑁 = (LLines‘𝐾) | |
17 | 4, 15, 16, 6, 7 | 2lplnmj 37322 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) → ((𝑌 ∧ 𝑍) ∈ 𝑁 ↔ (𝑌 ∨ 𝑍) ∈ (LVols‘𝐾))) |
18 | 12, 13, 14, 17 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → ((𝑌 ∧ 𝑍) ∈ 𝑁 ↔ (𝑌 ∨ 𝑍) ∈ (LVols‘𝐾))) |
19 | 18 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → ((𝑌 ∧ 𝑍) ∈ 𝑁 ↔ (𝑌 ∨ 𝑍) ∈ (LVols‘𝐾))) |
20 | 11, 19 | mpbird 260 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∧ 𝑍) ∈ 𝑁) |
21 | 1, 20 | eqeltrid 2835 | 1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑋 ∈ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 class class class wbr 5039 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 lecple 16756 joincjn 17772 meetcmee 17773 Atomscatm 36963 HLchlt 37050 LLinesclln 37191 LPlanesclpl 37192 LVolsclvol 37193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-proset 17756 df-poset 17774 df-plt 17790 df-lub 17806 df-glb 17807 df-join 17808 df-meet 17809 df-p0 17885 df-lat 17892 df-clat 17959 df-oposet 36876 df-ol 36878 df-oml 36879 df-covers 36966 df-ats 36967 df-atl 36998 df-cvlat 37022 df-hlat 37051 df-llines 37198 df-lplanes 37199 df-lvols 37200 |
This theorem is referenced by: dalem16 37379 dalem53 37425 |
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