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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem13 | Structured version Visualization version GIF version |
Description: Lemma for dalem14 36287. (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) |
dalem13.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalem13.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
dalem13.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
dalem13.w | ⊢ 𝑊 = (𝑌 ∨ 𝐶) |
Ref | Expression |
---|---|
dalem13 | ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∨ 𝑍) = 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 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))))) | |
2 | 1 | dalemkehl 36233 | . . 3 ⊢ (𝜑 → 𝐾 ∈ HL) |
3 | 2 | adantr 473 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐾 ∈ HL) |
4 | 1 | dalemyeo 36242 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑂) |
5 | 4 | adantr 473 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ 𝑂) |
6 | 1 | dalemzeo 36243 | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑂) |
7 | 6 | adantr 473 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑍 ∈ 𝑂) |
8 | dalemc.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
9 | dalemc.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
10 | dalemc.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
11 | dalem13.o | . . 3 ⊢ 𝑂 = (LPlanes‘𝐾) | |
12 | eqid 2772 | . . 3 ⊢ (LVols‘𝐾) = (LVols‘𝐾) | |
13 | dalem13.y | . . 3 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
14 | dalem13.z | . . 3 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
15 | dalem13.w | . . 3 ⊢ 𝑊 = (𝑌 ∨ 𝐶) | |
16 | 1, 8, 9, 10, 11, 12, 13, 14, 15 | dalem9 36282 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑊 ∈ (LVols‘𝐾)) |
17 | 1 | dalemkelat 36234 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) |
18 | 1, 11 | dalemyeb 36259 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
19 | 1, 10 | dalemceb 36248 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾)) |
20 | eqid 2772 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
21 | 20, 8, 9 | latlej1 17540 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → 𝑌 ≤ (𝑌 ∨ 𝐶)) |
22 | 17, 18, 19, 21 | syl3anc 1351 | . . . 4 ⊢ (𝜑 → 𝑌 ≤ (𝑌 ∨ 𝐶)) |
23 | 22, 15 | syl6breqr 4967 | . . 3 ⊢ (𝜑 → 𝑌 ≤ 𝑊) |
24 | 23 | adantr 473 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ≤ 𝑊) |
25 | 1, 8, 9, 10, 11, 13, 14, 15 | dalem8 36280 | . . 3 ⊢ (𝜑 → 𝑍 ≤ 𝑊) |
26 | 25 | adantr 473 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑍 ≤ 𝑊) |
27 | simpr 477 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ≠ 𝑍) | |
28 | 8, 9, 11, 12 | 2lplnj 36230 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ∧ 𝑊 ∈ (LVols‘𝐾)) ∧ (𝑌 ≤ 𝑊 ∧ 𝑍 ≤ 𝑊 ∧ 𝑌 ≠ 𝑍)) → (𝑌 ∨ 𝑍) = 𝑊) |
29 | 3, 5, 7, 16, 24, 26, 27, 28 | syl133anc 1373 | 1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∨ 𝑍) = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ≠ wne 2961 class class class wbr 4925 ‘cfv 6185 (class class class)co 6974 Basecbs 16337 lecple 16426 joincjn 17424 Latclat 17525 Atomscatm 35873 HLchlt 35960 LPlanesclpl 36102 LVolsclvol 36103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-proset 17408 df-poset 17426 df-plt 17438 df-lub 17454 df-glb 17455 df-join 17456 df-meet 17457 df-p0 17519 df-lat 17526 df-clat 17588 df-oposet 35786 df-ol 35788 df-oml 35789 df-covers 35876 df-ats 35877 df-atl 35908 df-cvlat 35932 df-hlat 35961 df-llines 36108 df-lplanes 36109 df-lvols 36110 |
This theorem is referenced by: dalem14 36287 |
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