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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem13 | Structured version Visualization version GIF version |
Description: Lemma for dalem14 38072. (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 38018 | . . 3 ⊢ (𝜑 → 𝐾 ∈ HL) |
3 | 2 | adantr 482 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝐾 ∈ HL) |
4 | 1 | dalemyeo 38027 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑂) |
5 | 4 | adantr 482 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ∈ 𝑂) |
6 | 1 | dalemzeo 38028 | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑂) |
7 | 6 | adantr 482 | . 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 2738 | . . 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 38067 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑊 ∈ (LVols‘𝐾)) |
17 | 1 | dalemkelat 38019 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) |
18 | 1, 11 | dalemyeb 38044 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
19 | 1, 10 | dalemceb 38033 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾)) |
20 | eqid 2738 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
21 | 20, 8, 9 | latlej1 18297 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → 𝑌 ≤ (𝑌 ∨ 𝐶)) |
22 | 17, 18, 19, 21 | syl3anc 1372 | . . . 4 ⊢ (𝜑 → 𝑌 ≤ (𝑌 ∨ 𝐶)) |
23 | 22, 15 | breqtrrdi 5146 | . . 3 ⊢ (𝜑 → 𝑌 ≤ 𝑊) |
24 | 23 | adantr 482 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ≤ 𝑊) |
25 | 1, 8, 9, 10, 11, 13, 14, 15 | dalem8 38065 | . . 3 ⊢ (𝜑 → 𝑍 ≤ 𝑊) |
26 | 25 | adantr 482 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑍 ≤ 𝑊) |
27 | simpr 486 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑌 ≠ 𝑍) | |
28 | 8, 9, 11, 12 | 2lplnj 38015 | . 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 1394 | 1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∨ 𝑍) = 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 class class class wbr 5104 ‘cfv 6494 (class class class)co 7352 Basecbs 17043 lecple 17100 joincjn 18160 Latclat 18280 Atomscatm 37657 HLchlt 37744 LPlanesclpl 37887 LVolsclvol 37888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-proset 18144 df-poset 18162 df-plt 18179 df-lub 18195 df-glb 18196 df-join 18197 df-meet 18198 df-p0 18274 df-lat 18281 df-clat 18348 df-oposet 37570 df-ol 37572 df-oml 37573 df-covers 37660 df-ats 37661 df-atl 37692 df-cvlat 37716 df-hlat 37745 df-llines 37893 df-lplanes 37894 df-lvols 37895 |
This theorem is referenced by: dalem14 38072 |
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