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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem8 | Structured version Visualization version GIF version |
Description: Lemma for dath 36874. Plane 𝑍 belongs to the 3-dimensional space. (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) |
dalem6.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalem6.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
dalem6.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
dalem6.w | ⊢ 𝑊 = (𝑌 ∨ 𝐶) |
Ref | Expression |
---|---|
dalem8 | ⊢ (𝜑 → 𝑍 ≤ 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalem6.z | . 2 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
2 | dalema.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))))) | |
3 | dalemc.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
4 | dalemc.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
5 | dalemc.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | dalem6.o | . . . . 5 ⊢ 𝑂 = (LPlanes‘𝐾) | |
7 | dalem6.y | . . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
8 | dalem6.w | . . . . 5 ⊢ 𝑊 = (𝑌 ∨ 𝐶) | |
9 | 2, 3, 4, 5, 6, 7, 1, 8 | dalem6 36806 | . . . 4 ⊢ (𝜑 → 𝑆 ≤ 𝑊) |
10 | 2, 3, 4, 5, 6, 7, 1, 8 | dalem7 36807 | . . . 4 ⊢ (𝜑 → 𝑇 ≤ 𝑊) |
11 | 2 | dalemkelat 36762 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) |
12 | 2, 5 | dalemseb 36780 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾)) |
13 | 2, 5 | dalemteb 36781 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾)) |
14 | 2, 6 | dalemyeb 36787 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
15 | 2, 5 | dalemceb 36776 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾)) |
16 | eqid 2823 | . . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
17 | 16, 4 | latjcl 17663 | . . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾)) |
18 | 11, 14, 15, 17 | syl3anc 1367 | . . . . . 6 ⊢ (𝜑 → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾)) |
19 | 8, 18 | eqeltrid 2919 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾)) |
20 | 16, 3, 4 | latjle12 17674 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑆 ≤ 𝑊 ∧ 𝑇 ≤ 𝑊) ↔ (𝑆 ∨ 𝑇) ≤ 𝑊)) |
21 | 11, 12, 13, 19, 20 | syl13anc 1368 | . . . 4 ⊢ (𝜑 → ((𝑆 ≤ 𝑊 ∧ 𝑇 ≤ 𝑊) ↔ (𝑆 ∨ 𝑇) ≤ 𝑊)) |
22 | 9, 10, 21 | mpbi2and 710 | . . 3 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ 𝑊) |
23 | 2, 3, 4, 5, 6, 7, 8 | dalem5 36805 | . . 3 ⊢ (𝜑 → 𝑈 ≤ 𝑊) |
24 | 2, 4, 5 | dalemsjteb 36784 | . . . 4 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) |
25 | 2, 5 | dalemueb 36782 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾)) |
26 | 16, 3, 4 | latjle12 17674 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ ((𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (((𝑆 ∨ 𝑇) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊)) |
27 | 11, 24, 25, 19, 26 | syl13anc 1368 | . . 3 ⊢ (𝜑 → (((𝑆 ∨ 𝑇) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊)) |
28 | 22, 23, 27 | mpbi2and 710 | . 2 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊) |
29 | 1, 28 | eqbrtrid 5103 | 1 ⊢ (𝜑 → 𝑍 ≤ 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 lecple 16574 joincjn 17556 Latclat 17657 Atomscatm 36401 HLchlt 36488 LPlanesclpl 36630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-lat 17658 df-clat 17720 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 df-lplanes 36637 |
This theorem is referenced by: dalem13 36814 |
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