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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem10 | Structured version Visualization version GIF version |
Description: Lemma for dath 37297. Atom 𝐷 belongs to the axis of perspectivity 𝑋. (Contributed by NM, 19-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) |
dalem10.m | ⊢ ∧ = (meet‘𝐾) |
dalem10.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalem10.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
dalem10.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
dalem10.x | ⊢ 𝑋 = (𝑌 ∧ 𝑍) |
dalem10.d | ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) |
Ref | Expression |
---|---|
dalem10 | ⊢ (𝜑 → 𝐷 ≤ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 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))))) | |
2 | 1 | dalemkelat 37185 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat) |
3 | dalemc.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
4 | dalemc.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 1, 3, 4 | dalempjqeb 37206 | . . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
6 | 1, 4 | dalemreb 37202 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾)) |
7 | eqid 2759 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
8 | dalemc.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
9 | 7, 8, 3 | latlej1 17721 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) |
10 | 2, 5, 6, 9 | syl3anc 1369 | . . 3 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)) |
11 | 1, 3, 4 | dalemsjteb 37207 | . . . 4 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾)) |
12 | 1, 4 | dalemueb 37205 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾)) |
13 | 7, 8, 3 | latlej1 17721 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → (𝑆 ∨ 𝑇) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) |
14 | 2, 11, 12, 13 | syl3anc 1369 | . . 3 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) |
15 | dalem10.y | . . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
16 | dalem10.o | . . . . . 6 ⊢ 𝑂 = (LPlanes‘𝐾) | |
17 | 1, 16 | dalemyeb 37210 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
18 | 15, 17 | eqeltrrid 2856 | . . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (Base‘𝐾)) |
19 | dalem10.z | . . . . 5 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
20 | 1 | dalemzeo 37194 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑂) |
21 | 7, 16 | lplnbase 37095 | . . . . . 6 ⊢ (𝑍 ∈ 𝑂 → 𝑍 ∈ (Base‘𝐾)) |
22 | 20, 21 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ (Base‘𝐾)) |
23 | 19, 22 | eqeltrrid 2856 | . . . 4 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ (Base‘𝐾)) |
24 | dalem10.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
25 | 7, 8, 24 | latmlem12 17744 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (Base‘𝐾)) ∧ ((𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ ((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ (Base‘𝐾))) → (((𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ∧ (𝑆 ∨ 𝑇) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (((𝑃 ∨ 𝑄) ∨ 𝑅) ∧ ((𝑆 ∨ 𝑇) ∨ 𝑈)))) |
26 | 2, 5, 18, 11, 23, 25 | syl122anc 1377 | . . 3 ⊢ (𝜑 → (((𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ∧ (𝑆 ∨ 𝑇) ≤ ((𝑆 ∨ 𝑇) ∨ 𝑈)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (((𝑃 ∨ 𝑄) ∨ 𝑅) ∧ ((𝑆 ∨ 𝑇) ∨ 𝑈)))) |
27 | 10, 14, 26 | mp2and 699 | . 2 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (((𝑃 ∨ 𝑄) ∨ 𝑅) ∧ ((𝑆 ∨ 𝑇) ∨ 𝑈))) |
28 | dalem10.d | . 2 ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) | |
29 | dalem10.x | . . 3 ⊢ 𝑋 = (𝑌 ∧ 𝑍) | |
30 | 15, 19 | oveq12i 7155 | . . 3 ⊢ (𝑌 ∧ 𝑍) = (((𝑃 ∨ 𝑄) ∨ 𝑅) ∧ ((𝑆 ∨ 𝑇) ∨ 𝑈)) |
31 | 29, 30 | eqtri 2782 | . 2 ⊢ 𝑋 = (((𝑃 ∨ 𝑄) ∨ 𝑅) ∧ ((𝑆 ∨ 𝑇) ∨ 𝑈)) |
32 | 27, 28, 31 | 3brtr4g 5059 | 1 ⊢ (𝜑 → 𝐷 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 class class class wbr 5025 ‘cfv 6328 (class class class)co 7143 Basecbs 16526 lecple 16615 joincjn 17605 meetcmee 17606 Latclat 17706 Atomscatm 36824 HLchlt 36911 LPlanesclpl 37053 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-poset 17607 df-lub 17635 df-glb 17636 df-join 17637 df-meet 17638 df-lat 17707 df-ats 36828 df-atl 36859 df-cvlat 36883 df-hlat 36912 df-lplanes 37060 |
This theorem is referenced by: dalem11 37235 dalem16 37240 dalem54 37287 |
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