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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem27 | Structured version Visualization version GIF version | ||
| Description: Lemma for dath 39738. Show that the line 𝐺𝑃 intersects the dummy center of perspectivity 𝑐. (Contributed by NM, 8-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)))) | 
| dalem23.m | ⊢ ∧ = (meet‘𝐾) | 
| dalem23.o | ⊢ 𝑂 = (LPlanes‘𝐾) | 
| dalem23.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | 
| dalem23.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | 
| dalem23.g | ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) | 
| Ref | Expression | 
|---|---|
| dalem27 | ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ≤ (𝐺 ∨ 𝑃)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dalem23.g | . . 3 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) | |
| 2 | dalem.ph | . . . . . 6 ⊢ (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 | 2 | dalemkelat 39626 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) | 
| 4 | 3 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat) | 
| 5 | 2 | dalemkehl 39625 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL) | 
| 6 | 5 | 3ad2ant1 1134 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL) | 
| 7 | dalem.ps | . . . . . . 7 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) | |
| 8 | 7 | dalemccea 39685 | . . . . . 6 ⊢ (𝜓 → 𝑐 ∈ 𝐴) | 
| 9 | 8 | 3ad2ant3 1136 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴) | 
| 10 | 2 | dalempea 39628 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴) | 
| 11 | 10 | 3ad2ant1 1134 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ 𝐴) | 
| 12 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 13 | dalem.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
| 14 | dalem.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 15 | 12, 13, 14 | hlatjcl 39368 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾)) | 
| 16 | 6, 9, 11, 15 | syl3anc 1373 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾)) | 
| 17 | 7 | dalemddea 39686 | . . . . . 6 ⊢ (𝜓 → 𝑑 ∈ 𝐴) | 
| 18 | 17 | 3ad2ant3 1136 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑑 ∈ 𝐴) | 
| 19 | 2 | dalemsea 39631 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝐴) | 
| 20 | 19 | 3ad2ant1 1134 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑆 ∈ 𝐴) | 
| 21 | 12, 13, 14 | hlatjcl 39368 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾)) | 
| 22 | 6, 18, 20, 21 | syl3anc 1373 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾)) | 
| 23 | dalem.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
| 24 | dalem23.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
| 25 | 12, 23, 24 | latmle1 18509 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∨ 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ≤ (𝑐 ∨ 𝑃)) | 
| 26 | 4, 16, 22, 25 | syl3anc 1373 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ≤ (𝑐 ∨ 𝑃)) | 
| 27 | 1, 26 | eqbrtrid 5178 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ≤ (𝑐 ∨ 𝑃)) | 
| 28 | dalem23.o | . . . 4 ⊢ 𝑂 = (LPlanes‘𝐾) | |
| 29 | dalem23.y | . . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
| 30 | dalem23.z | . . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
| 31 | 2, 23, 13, 14, 7, 24, 28, 29, 30, 1 | dalem23 39698 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴) | 
| 32 | 2, 23, 13, 14, 28, 29 | dalemply 39656 | . . . . 5 ⊢ (𝜑 → 𝑃 ≤ 𝑌) | 
| 33 | 32 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ≤ 𝑌) | 
| 34 | 2, 23, 13, 14, 7, 24, 28, 29, 30, 1 | dalem24 39699 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝐺 ≤ 𝑌) | 
| 35 | nbrne2 5163 | . . . . 5 ⊢ ((𝑃 ≤ 𝑌 ∧ ¬ 𝐺 ≤ 𝑌) → 𝑃 ≠ 𝐺) | |
| 36 | 35 | necomd 2996 | . . . 4 ⊢ ((𝑃 ≤ 𝑌 ∧ ¬ 𝐺 ≤ 𝑌) → 𝐺 ≠ 𝑃) | 
| 37 | 33, 34, 36 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ≠ 𝑃) | 
| 38 | 23, 13, 14 | hlatexch2 39398 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝐺 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝐺 ≠ 𝑃) → (𝐺 ≤ (𝑐 ∨ 𝑃) → 𝑐 ≤ (𝐺 ∨ 𝑃))) | 
| 39 | 6, 31, 9, 11, 37, 38 | syl131anc 1385 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ≤ (𝑐 ∨ 𝑃) → 𝑐 ≤ (𝐺 ∨ 𝑃))) | 
| 40 | 27, 39 | mpd 15 | 1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ≤ (𝐺 ∨ 𝑃)) | 
| 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 Latclat 18476 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-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 | 
| This theorem is referenced by: dalem28 39702 dalem32 39706 dalem51 39725 dalem52 39726 | 
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