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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem61 | Structured version Visualization version GIF version | ||
| Description: Lemma for dath 39738. Show that atoms 𝐷, 𝐸, and 𝐹 lie on the same line (axis of perspectivity). Eliminate hypotheses containing dummy atoms 𝑐 and 𝑑. (Contributed by NM, 11-Aug-2012.) |
| Ref | Expression |
|---|---|
| dalem.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))))) |
| dalem.l | ⊢ ≤ = (le‘𝐾) |
| dalem.j | ⊢ ∨ = (join‘𝐾) |
| dalem.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| dalem.ps | ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) |
| dalem61.m | ⊢ ∧ = (meet‘𝐾) |
| dalem61.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
| dalem61.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
| dalem61.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
| dalem61.d | ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) |
| dalem61.e | ⊢ 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) |
| dalem61.f | ⊢ 𝐹 = ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆)) |
| Ref | Expression |
|---|---|
| dalem61 | ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐹 ≤ (𝐷 ∨ 𝐸)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dalem.ph | . . 3 ⊢ (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))))) | |
| 2 | dalem.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
| 3 | dalem.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
| 4 | dalem.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 5 | dalem.ps | . . 3 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) | |
| 6 | dalem61.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
| 7 | dalem61.o | . . 3 ⊢ 𝑂 = (LPlanes‘𝐾) | |
| 8 | dalem61.y | . . 3 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
| 9 | dalem61.z | . . 3 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
| 10 | dalem61.f | . . 3 ⊢ 𝐹 = ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆)) | |
| 11 | eqid 2737 | . . 3 ⊢ ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) | |
| 12 | eqid 2737 | . . 3 ⊢ ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇)) = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇)) | |
| 13 | eqid 2737 | . . 3 ⊢ ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈)) = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈)) | |
| 14 | eqid 2737 | . . 3 ⊢ (((((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ∨ ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))) ∨ ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))) ∧ 𝑌) = (((((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ∨ ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))) ∨ ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))) ∧ 𝑌) | |
| 15 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 | dalem59 39733 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐹 ≤ (((((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ∨ ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))) ∨ ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))) ∧ 𝑌)) |
| 16 | dalem61.d | . . 3 ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) | |
| 17 | dalem61.e | . . 3 ⊢ 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) | |
| 18 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17, 11, 12, 13, 14 | dalem60 39734 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐷 ∨ 𝐸) = (((((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ∨ ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))) ∨ ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))) ∧ 𝑌)) |
| 19 | 15, 18 | breqtrrd 5171 | 1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐹 ≤ (𝐷 ∨ 𝐸)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 lecple 17304 joincjn 18357 meetcmee 18358 Atomscatm 39264 HLchlt 39351 LPlanesclpl 39494 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-lat 18477 df-clat 18544 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 |
| This theorem is referenced by: dalem62 39736 |
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