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 36807. (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 36753 | . . 3 ⊢ (𝜑 → 𝐾 ∈ HL) |
3 | 2 | adantr 483 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐾 ∈ HL) |
4 | 1 | dalemyeo 36762 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑂) |
5 | 4 | adantr 483 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ 𝑂) |
6 | 1 | dalemzeo 36763 | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑂) |
7 | 6 | adantr 483 | . 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 2821 | . . 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 36802 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑊 ∈ (LVols‘𝐾)) |
17 | 1 | dalemkelat 36754 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) |
18 | 1, 11 | dalemyeb 36779 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
19 | 1, 10 | dalemceb 36768 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾)) |
20 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
21 | 20, 8, 9 | latlej1 17664 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → 𝑌 ≤ (𝑌 ∨ 𝐶)) |
22 | 17, 18, 19, 21 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → 𝑌 ≤ (𝑌 ∨ 𝐶)) |
23 | 22, 15 | breqtrrdi 5100 | . . 3 ⊢ (𝜑 → 𝑌 ≤ 𝑊) |
24 | 23 | adantr 483 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ≤ 𝑊) |
25 | 1, 8, 9, 10, 11, 13, 14, 15 | dalem8 36800 | . . 3 ⊢ (𝜑 → 𝑍 ≤ 𝑊) |
26 | 25 | adantr 483 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑍 ≤ 𝑊) |
27 | simpr 487 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ≠ 𝑍) | |
28 | 8, 9, 11, 12 | 2lplnj 36750 | . 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 1389 | 1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∨ 𝑍) = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 lecple 16566 joincjn 17548 Latclat 17649 Atomscatm 36393 HLchlt 36480 LPlanesclpl 36622 LVolsclvol 36623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-lat 17650 df-clat 17712 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-llines 36628 df-lplanes 36629 df-lvols 36630 |
This theorem is referenced by: dalem14 36807 |
Copyright terms: Public domain | W3C validator |