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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem14 | Structured version Visualization version GIF version |
Description: Lemma for dath 35541. Planes 𝑌 and 𝑍 form a 3-dimensional space (when they are different). (Contributed by NM, 22-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) |
dalem14.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalem14.v | ⊢ 𝑉 = (LVols‘𝐾) |
dalem14.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
dalem14.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
dalem14.w | ⊢ 𝑊 = (𝑌 ∨ 𝐶) |
Ref | Expression |
---|---|
dalem14 | ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∨ 𝑍) ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalema.ph | . . 3 ⊢ (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 | dalemc.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | dalemc.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | dalemc.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | dalem14.o | . . 3 ⊢ 𝑂 = (LPlanes‘𝐾) | |
6 | dalem14.y | . . 3 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
7 | dalem14.z | . . 3 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
8 | dalem14.w | . . 3 ⊢ 𝑊 = (𝑌 ∨ 𝐶) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | dalem13 35481 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∨ 𝑍) = 𝑊) |
10 | dalem14.v | . . 3 ⊢ 𝑉 = (LVols‘𝐾) | |
11 | 1, 2, 3, 4, 5, 10, 6, 7, 8 | dalem9 35477 | . 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → 𝑊 ∈ 𝑉) |
12 | 9, 11 | eqeltrd 2850 | 1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → (𝑌 ∨ 𝑍) ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 class class class wbr 4786 ‘cfv 6029 (class class class)co 6792 Basecbs 16060 lecple 16152 joincjn 17148 Atomscatm 35068 HLchlt 35155 LPlanesclpl 35297 LVolsclvol 35298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6753 df-ov 6795 df-oprab 6796 df-preset 17132 df-poset 17150 df-plt 17162 df-lub 17178 df-glb 17179 df-join 17180 df-meet 17181 df-p0 17243 df-lat 17250 df-clat 17312 df-oposet 34981 df-ol 34983 df-oml 34984 df-covers 35071 df-ats 35072 df-atl 35103 df-cvlat 35127 df-hlat 35156 df-llines 35303 df-lplanes 35304 df-lvols 35305 |
This theorem is referenced by: dalem15 35483 |
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