Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem19 | Structured version Visualization version GIF version |
Description: Lemma for dath 36906. Show that a second dummy atom 𝑑 exists outside of the 𝑌 and 𝑍 planes (when those planes are equal). (Contributed by NM, 15-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) |
dalem19.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalem19.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
dalem19.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
Ref | Expression |
---|---|
dalem19 | ⊢ ((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌) → ∃𝑑 ∈ 𝐴 (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalema.ph | . . . 4 ⊢ (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 | dalemkehl 36793 | . . 3 ⊢ (𝜑 → 𝐾 ∈ HL) |
3 | 2 | ad3antrrr 728 | . 2 ⊢ ((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌) → 𝐾 ∈ HL) |
4 | dalemc.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
5 | dalemc.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
6 | dalemc.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | dalem19.o | . . . 4 ⊢ 𝑂 = (LPlanes‘𝐾) | |
8 | dalem19.y | . . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
9 | 1, 4, 5, 6, 7, 8 | dalemcea 36830 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
10 | 9 | ad3antrrr 728 | . 2 ⊢ ((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌) → 𝐶 ∈ 𝐴) |
11 | simplr 767 | . 2 ⊢ ((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌) → 𝑐 ∈ 𝐴) | |
12 | 1, 7 | dalemyeb 36819 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾)) |
13 | 12 | ad3antrrr 728 | . 2 ⊢ ((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌) → 𝑌 ∈ (Base‘𝐾)) |
14 | dalem19.z | . . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
15 | 1, 4, 5, 6, 7, 8, 14 | dalem17 36850 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝐶 ≤ 𝑌) |
16 | 15 | ad2antrr 724 | . 2 ⊢ ((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌) → 𝐶 ≤ 𝑌) |
17 | simpr 487 | . 2 ⊢ ((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌) → ¬ 𝑐 ≤ 𝑌) | |
18 | eqid 2820 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
19 | 18, 4, 5, 6 | atbtwnex 36618 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑌 ∈ (Base‘𝐾) ∧ 𝐶 ≤ 𝑌 ∧ ¬ 𝑐 ≤ 𝑌)) → ∃𝑑 ∈ 𝐴 (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))) |
20 | 3, 10, 11, 13, 16, 17, 19 | syl33anc 1380 | 1 ⊢ ((((𝜑 ∧ 𝑌 = 𝑍) ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌) → ∃𝑑 ∈ 𝐴 (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 ∃wrex 3138 class class class wbr 5059 ‘cfv 6348 (class class class)co 7149 Basecbs 16476 lecple 16565 joincjn 17547 Atomscatm 36433 HLchlt 36520 LPlanesclpl 36662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-proset 17531 df-poset 17549 df-plt 17561 df-lub 17577 df-glb 17578 df-join 17579 df-meet 17580 df-p0 17642 df-lat 17649 df-clat 17711 df-oposet 36346 df-ol 36348 df-oml 36349 df-covers 36436 df-ats 36437 df-atl 36468 df-cvlat 36492 df-hlat 36521 df-llines 36668 df-lplanes 36669 |
This theorem is referenced by: dalem20 36863 |
Copyright terms: Public domain | W3C validator |