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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalem43 | Structured version Visualization version GIF version |
Description: Lemma for dath 36752. Planes 𝐺𝐻𝐼 and 𝑌 are different. (Contributed by NM, 8-Aug-2012.) |
Ref | Expression |
---|---|
dalem.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))))) |
dalem.l | ⊢ ≤ = (le‘𝐾) |
dalem.j | ⊢ ∨ = (join‘𝐾) |
dalem.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dalem.ps | ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) |
dalem38.m | ⊢ ∧ = (meet‘𝐾) |
dalem38.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalem38.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
dalem38.z | ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) |
dalem38.g | ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) |
dalem38.h | ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇)) |
dalem38.i | ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈)) |
Ref | Expression |
---|---|
dalem43 | ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ≠ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dalem.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 36640 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat) |
3 | 2 | 3ad2ant1 1125 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat) |
4 | 1 | dalemkehl 36639 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
5 | 4 | 3ad2ant1 1125 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL) |
6 | dalem.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
7 | dalem.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
8 | dalem.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | dalem.ps | . . . . 5 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑)))) | |
10 | dalem38.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
11 | dalem38.o | . . . . 5 ⊢ 𝑂 = (LPlanes‘𝐾) | |
12 | dalem38.y | . . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) | |
13 | dalem38.z | . . . . 5 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈) | |
14 | dalem38.g | . . . . 5 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) | |
15 | 1, 6, 7, 8, 9, 10, 11, 12, 13, 14 | dalem23 36712 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴) |
16 | dalem38.h | . . . . 5 ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇)) | |
17 | 1, 6, 7, 8, 9, 10, 11, 12, 13, 16 | dalem29 36717 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴) |
18 | eqid 2818 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
19 | 18, 7, 8 | hlatjcl 36383 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾)) |
20 | 5, 15, 17, 19 | syl3anc 1363 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾)) |
21 | dalem38.i | . . . . 5 ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈)) | |
22 | 1, 6, 7, 8, 9, 10, 11, 12, 13, 21 | dalem34 36722 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴) |
23 | 18, 8 | atbase 36305 | . . . 4 ⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾)) |
24 | 22, 23 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾)) |
25 | 18, 6, 7 | latlej2 17659 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → 𝐼 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼)) |
26 | 3, 20, 24, 25 | syl3anc 1363 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼)) |
27 | 1, 6, 7, 8, 9, 10, 11, 12, 13, 21 | dalem35 36723 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝐼 ≤ 𝑌) |
28 | nbrne1 5076 | . 2 ⊢ ((𝐼 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ ¬ 𝐼 ≤ 𝑌) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ≠ 𝑌) | |
29 | 26, 27, 28 | syl2anc 584 | 1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ≠ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 lecple 16560 joincjn 17542 meetcmee 17543 Latclat 17643 Atomscatm 36279 HLchlt 36366 LPlanesclpl 36508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 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 7103 df-ov 7148 df-oprab 7149 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-lat 17644 df-clat 17706 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 |
This theorem is referenced by: dalem44 36732 dalem51 36739 |
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