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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemdnee | Structured version Visualization version GIF version |
Description: Lemma for dath 35525. 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 35441 | . . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑇) |
9 | 8 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 = 𝑄) → 𝑄 ≠ 𝑇) |
10 | 1, 9 | eqnetrd 2999 | . . 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 35454 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ≠ 𝑇) → 𝐷 ≠ 𝐸) |
16 | 10, 15 | syldan 488 | . 2 ⊢ ((𝜑 ∧ 𝐷 = 𝑄) → 𝐷 ≠ 𝐸) |
17 | 2, 3, 4, 5, 11, 6, 7, 12, 13, 14 | dalem3 35453 | . 2 ⊢ ((𝜑 ∧ 𝐷 ≠ 𝑄) → 𝐷 ≠ 𝐸) |
18 | 16, 17 | pm2.61dane 3019 | 1 ⊢ (𝜑 → 𝐷 ≠ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 class class class wbr 4804 ‘cfv 6049 (class class class)co 6813 Basecbs 16059 lecple 16150 joincjn 17145 meetcmee 17146 Atomscatm 35053 HLchlt 35140 LPlanesclpl 35281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-preset 17129 df-poset 17147 df-plt 17159 df-lub 17175 df-glb 17176 df-join 17177 df-meet 17178 df-p0 17240 df-lat 17247 df-clat 17309 df-oposet 34966 df-ol 34968 df-oml 34969 df-covers 35056 df-ats 35057 df-atl 35088 df-cvlat 35112 df-hlat 35141 df-llines 35287 df-lplanes 35288 |
This theorem is referenced by: dalem16 35468 dalem60 35521 |
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