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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem43 | Structured version Visualization version GIF version |
Description: Lemma for dath 35811. Planes 𝐺𝐻𝐼 and 𝑌 are different. (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)))) |
dalem38.m | ⊢ ∧ = (meet‘𝐾) |
dalem38.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalem38.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
dalem38.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
dalem38.g | ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) |
dalem38.h | ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇)) |
dalem38.i | ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈)) |
Ref | Expression |
---|---|
dalem43 | ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ≠ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalem.ph | . . . . 5 ⊢ (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 | 1 | dalemkelat 35699 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat) |
3 | 2 | 3ad2ant1 1169 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat) |
4 | 1 | dalemkehl 35698 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
5 | 4 | 3ad2ant1 1169 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL) |
6 | dalem.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
7 | dalem.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
8 | dalem.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | dalem.ps | . . . . 5 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) | |
10 | dalem38.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
11 | dalem38.o | . . . . 5 ⊢ 𝑂 = (LPlanes‘𝐾) | |
12 | dalem38.y | . . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
13 | dalem38.z | . . . . 5 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
14 | dalem38.g | . . . . 5 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) | |
15 | 1, 6, 7, 8, 9, 10, 11, 12, 13, 14 | dalem23 35771 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴) |
16 | dalem38.h | . . . . 5 ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇)) | |
17 | 1, 6, 7, 8, 9, 10, 11, 12, 13, 16 | dalem29 35776 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴) |
18 | eqid 2825 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
19 | 18, 7, 8 | hlatjcl 35442 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾)) |
20 | 5, 15, 17, 19 | syl3anc 1496 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾)) |
21 | dalem38.i | . . . . 5 ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈)) | |
22 | 1, 6, 7, 8, 9, 10, 11, 12, 13, 21 | dalem34 35781 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴) |
23 | 18, 8 | atbase 35364 | . . . 4 ⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾)) |
24 | 22, 23 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾)) |
25 | 18, 6, 7 | latlej2 17414 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → 𝐼 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼)) |
26 | 3, 20, 24, 25 | syl3anc 1496 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼)) |
27 | 1, 6, 7, 8, 9, 10, 11, 12, 13, 21 | dalem35 35782 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝐼 ≤ 𝑌) |
28 | nbrne1 4892 | . 2 ⊢ ((𝐼 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ ¬ 𝐼 ≤ 𝑌) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ≠ 𝑌) | |
29 | 26, 27, 28 | syl2anc 581 | 1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ≠ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 lecple 16312 joincjn 17297 meetcmee 17298 Latclat 17398 Atomscatm 35338 HLchlt 35425 LPlanesclpl 35567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-proset 17281 df-poset 17299 df-plt 17311 df-lub 17327 df-glb 17328 df-join 17329 df-meet 17330 df-p0 17392 df-lat 17399 df-clat 17461 df-oposet 35251 df-ol 35253 df-oml 35254 df-covers 35341 df-ats 35342 df-atl 35373 df-cvlat 35397 df-hlat 35426 df-llines 35573 df-lplanes 35574 |
This theorem is referenced by: dalem44 35791 dalem51 35798 |
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