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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem8 | Structured version Visualization version GIF version |
Description: Lemma for dath 37032. 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 36964 | . . . 4 ⊢ (𝜑 → 𝑆 ≤ 𝑊) |
10 | 2, 3, 4, 5, 6, 7, 1, 8 | dalem7 36965 | . . . 4 ⊢ (𝜑 → 𝑇 ≤ 𝑊) |
11 | 2 | dalemkelat 36920 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) |
12 | 2, 5 | dalemseb 36938 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾)) |
13 | 2, 5 | dalemteb 36939 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾)) |
14 | 2, 6 | dalemyeb 36945 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
15 | 2, 5 | dalemceb 36934 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾)) |
16 | eqid 2798 | . . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
17 | 16, 4 | latjcl 17653 | . . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾)) → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾)) |
18 | 11, 14, 15, 17 | syl3anc 1368 | . . . . . 6 ⊢ (𝜑 → (𝑌 ∨ 𝐶) ∈ (Base‘𝐾)) |
19 | 8, 18 | eqeltrid 2894 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾)) |
20 | 16, 3, 4 | latjle12 17664 | . . . . 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 1369 | . . . 4 ⊢ (𝜑 → ((𝑆 ≤ 𝑊 ∧ 𝑇 ≤ 𝑊) ↔ (𝑆 ∨ 𝑇) ≤ 𝑊)) |
22 | 9, 10, 21 | mpbi2and 711 | . . 3 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ 𝑊) |
23 | 2, 3, 4, 5, 6, 7, 8 | dalem5 36963 | . . 3 ⊢ (𝜑 → 𝑈 ≤ 𝑊) |
24 | 2, 4, 5 | dalemsjteb 36942 | . . . 4 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) |
25 | 2, 5 | dalemueb 36940 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾)) |
26 | 16, 3, 4 | latjle12 17664 | . . . 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 1369 | . . 3 ⊢ (𝜑 → (((𝑆 ∨ 𝑇) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊)) |
28 | 22, 23, 27 | mpbi2and 711 | . 2 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ 𝑊) |
29 | 1, 28 | eqbrtrid 5065 | 1 ⊢ (𝜑 → 𝑍 ≤ 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 lecple 16564 joincjn 17546 Latclat 17647 Atomscatm 36559 HLchlt 36646 LPlanesclpl 36788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-lat 17648 df-clat 17710 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 |
This theorem is referenced by: dalem13 36972 |
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