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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemdnee | Structured version Visualization version GIF version |
Description: Lemma for dath 35899. Axis of perspectivity points 𝐷 and 𝐸 are different. (Contributed by NM, 10-Aug-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) |
dalem3.m | ⊢ ∧ = (meet‘𝐾) |
dalem3.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalem3.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
dalem3.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
dalem3.d | ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) |
dalem3.e | ⊢ 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) |
Ref | Expression |
---|---|
dalemdnee | ⊢ (𝜑 → 𝐷 ≠ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 = 𝑄) → 𝐷 = 𝑄) | |
2 | dalema.ph | . . . . . 6 ⊢ (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 | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
4 | dalemc.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
5 | dalemc.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | dalem3.o | . . . . . 6 ⊢ 𝑂 = (LPlanes‘𝐾) | |
7 | dalem3.y | . . . . . 6 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
8 | 2, 3, 4, 5, 6, 7 | dalemqnet 35815 | . . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑇) |
9 | 8 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 = 𝑄) → 𝑄 ≠ 𝑇) |
10 | 1, 9 | eqnetrd 3036 | . . 3 ⊢ ((𝜑 ∧ 𝐷 = 𝑄) → 𝐷 ≠ 𝑇) |
11 | dalem3.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
12 | dalem3.z | . . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
13 | dalem3.d | . . . 4 ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) | |
14 | dalem3.e | . . . 4 ⊢ 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) | |
15 | 2, 3, 4, 5, 11, 6, 7, 12, 13, 14 | dalem4 35828 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ 𝑇) → 𝐷 ≠ 𝐸) |
16 | 10, 15 | syldan 585 | . 2 ⊢ ((𝜑 ∧ 𝐷 = 𝑄) → 𝐷 ≠ 𝐸) |
17 | 2, 3, 4, 5, 11, 6, 7, 12, 13, 14 | dalem3 35827 | . 2 ⊢ ((𝜑 ∧ 𝐷 ≠ 𝑄) → 𝐷 ≠ 𝐸) |
18 | 16, 17 | pm2.61dane 3057 | 1 ⊢ (𝜑 → 𝐷 ≠ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 Basecbs 16266 lecple 16356 joincjn 17341 meetcmee 17342 Atomscatm 35426 HLchlt 35513 LPlanesclpl 35655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-proset 17325 df-poset 17343 df-plt 17355 df-lub 17371 df-glb 17372 df-join 17373 df-meet 17374 df-p0 17436 df-lat 17443 df-clat 17505 df-oposet 35339 df-ol 35341 df-oml 35342 df-covers 35429 df-ats 35430 df-atl 35461 df-cvlat 35485 df-hlat 35514 df-llines 35661 df-lplanes 35662 |
This theorem is referenced by: dalem16 35842 dalem60 35895 |
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