![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem13 | Structured version Visualization version GIF version |
Description: Lemma for dalem14 35485. (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 35431 | . . 3 ⊢ (𝜑 → 𝐾 ∈ HL) |
3 | 2 | adantr 466 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐾 ∈ HL) |
4 | 1 | dalemyeo 35440 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑂) |
5 | 4 | adantr 466 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ 𝑂) |
6 | 1 | dalemzeo 35441 | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑂) |
7 | 6 | adantr 466 | . 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 2771 | . . 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 35480 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑊 ∈ (LVols‘𝐾)) |
17 | 1 | dalemkelat 35432 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) |
18 | 1, 11 | dalemyeb 35457 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
19 | 1, 10 | dalemceb 35446 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾)) |
20 | eqid 2771 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
21 | 20, 8, 9 | latlej1 17268 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → 𝑌 ≤ (𝑌 ∨ 𝐶)) |
22 | 17, 18, 19, 21 | syl3anc 1476 | . . . 4 ⊢ (𝜑 → 𝑌 ≤ (𝑌 ∨ 𝐶)) |
23 | 22, 15 | syl6breqr 4828 | . . 3 ⊢ (𝜑 → 𝑌 ≤ 𝑊) |
24 | 23 | adantr 466 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ≤ 𝑊) |
25 | 1, 8, 9, 10, 11, 13, 14, 15 | dalem8 35478 | . . 3 ⊢ (𝜑 → 𝑍 ≤ 𝑊) |
26 | 25 | adantr 466 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑍 ≤ 𝑊) |
27 | simpr 471 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ≠ 𝑍) | |
28 | 8, 9, 11, 12 | 2lplnj 35428 | . 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 1499 | 1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∨ 𝑍) = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 class class class wbr 4786 ‘cfv 6031 (class class class)co 6793 Basecbs 16064 lecple 16156 joincjn 17152 Latclat 17253 Atomscatm 35072 HLchlt 35159 LPlanesclpl 35300 LVolsclvol 35301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-preset 17136 df-poset 17154 df-plt 17166 df-lub 17182 df-glb 17183 df-join 17184 df-meet 17185 df-p0 17247 df-lat 17254 df-clat 17316 df-oposet 34985 df-ol 34987 df-oml 34988 df-covers 35075 df-ats 35076 df-atl 35107 df-cvlat 35131 df-hlat 35160 df-llines 35306 df-lplanes 35307 df-lvols 35308 |
This theorem is referenced by: dalem14 35485 |
Copyright terms: Public domain | W3C validator |