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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem15 | Structured version Visualization version GIF version |
Description: Lemma for dath 36752. 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 2818 | . . . 4 ⊢ (LVols‘𝐾) = (LVols‘𝐾) | |
8 | dalem15.y | . . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
9 | dalem15.z | . . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
10 | eqid 2818 | . . . 4 ⊢ (𝑌 ∨ 𝐶) = (𝑌 ∨ 𝐶) | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | dalem14 36693 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∨ 𝑍) ∈ (LVols‘𝐾)) |
12 | 2 | dalemkehl 36639 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
13 | 2 | dalemyeo 36648 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑂) |
14 | 2 | dalemzeo 36649 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑂) |
15 | dalem15.m | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
16 | dalem15.n | . . . . . 6 ⊢ 𝑁 = (LLines‘𝐾) | |
17 | 4, 15, 16, 6, 7 | 2lplnmj 36638 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) → ((𝑌 ∧ 𝑍) ∈ 𝑁 ↔ (𝑌 ∨ 𝑍) ∈ (LVols‘𝐾))) |
18 | 12, 13, 14, 17 | syl3anc 1363 | . . . 4 ⊢ (𝜑 → ((𝑌 ∧ 𝑍) ∈ 𝑁 ↔ (𝑌 ∨ 𝑍) ∈ (LVols‘𝐾))) |
19 | 18 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → ((𝑌 ∧ 𝑍) ∈ 𝑁 ↔ (𝑌 ∨ 𝑍) ∈ (LVols‘𝐾))) |
20 | 11, 19 | mpbird 258 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∧ 𝑍) ∈ 𝑁) |
21 | 1, 20 | eqeltrid 2914 | 1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑋 ∈ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 lecple 16560 joincjn 17542 meetcmee 17543 Atomscatm 36279 HLchlt 36366 LLinesclln 36507 LPlanesclpl 36508 LVolsclvol 36509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-lat 17644 df-clat 17706 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 df-lvols 36516 |
This theorem is referenced by: dalem16 36695 dalem53 36741 |
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