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Mirrors > Home > MPE Home > Th. List > Mathboxes > dalemqnet | Structured version Visualization version GIF version |
Description: Lemma for dath 36354. Frequently-used utility lemma. (Contributed by NM, 13-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) |
dalempnes.o | ⊢ 𝑂 = (LPlanes‘𝐾) |
dalempnes.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅) |
Ref | Expression |
---|---|
dalemqnet | ⊢ (𝜑 → 𝑄 ≠ 𝑇) |
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 | dalemkelat 36242 | . . 3 ⊢ (𝜑 → 𝐾 ∈ Lat) |
3 | dalemc.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | 1, 3 | dalemceb 36256 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾)) |
5 | 1, 3 | dalemteb 36261 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾)) |
6 | 1, 3 | dalemueb 36262 | . . 3 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾)) |
7 | simp322 1305 | . . . 4 ⊢ ((((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)))) → ¬ 𝐶 ≤ (𝑇 ∨ 𝑈)) | |
8 | 1, 7 | sylbi 209 | . . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑇 ∨ 𝑈)) |
9 | eqid 2771 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
10 | dalemc.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
11 | dalemc.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
12 | 9, 10, 11 | latnlej2l 17552 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈)) → ¬ 𝐶 ≤ 𝑇) |
13 | 2, 4, 5, 6, 8, 12 | syl131anc 1364 | . 2 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝑇) |
14 | 1 | dalemclqjt 36253 | . . . . 5 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇)) |
15 | oveq1 6981 | . . . . . 6 ⊢ (𝑄 = 𝑇 → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑇)) | |
16 | 15 | breq2d 4937 | . . . . 5 ⊢ (𝑄 = 𝑇 → (𝐶 ≤ (𝑄 ∨ 𝑇) ↔ 𝐶 ≤ (𝑇 ∨ 𝑇))) |
17 | 14, 16 | syl5ibcom 237 | . . . 4 ⊢ (𝜑 → (𝑄 = 𝑇 → 𝐶 ≤ (𝑇 ∨ 𝑇))) |
18 | 1 | dalemkehl 36241 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL) |
19 | 1 | dalemtea 36248 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐴) |
20 | 11, 3 | hlatjidm 35987 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴) → (𝑇 ∨ 𝑇) = 𝑇) |
21 | 18, 19, 20 | syl2anc 576 | . . . . 5 ⊢ (𝜑 → (𝑇 ∨ 𝑇) = 𝑇) |
22 | 21 | breq2d 4937 | . . . 4 ⊢ (𝜑 → (𝐶 ≤ (𝑇 ∨ 𝑇) ↔ 𝐶 ≤ 𝑇)) |
23 | 17, 22 | sylibd 231 | . . 3 ⊢ (𝜑 → (𝑄 = 𝑇 → 𝐶 ≤ 𝑇)) |
24 | 23 | necon3bd 2974 | . 2 ⊢ (𝜑 → (¬ 𝐶 ≤ 𝑇 → 𝑄 ≠ 𝑇)) |
25 | 13, 24 | mpd 15 | 1 ⊢ (𝜑 → 𝑄 ≠ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ≠ wne 2960 class class class wbr 4925 ‘cfv 6185 (class class class)co 6974 Basecbs 16337 lecple 16426 joincjn 17424 Latclat 17525 Atomscatm 35881 HLchlt 35968 LPlanesclpl 36110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-proset 17408 df-poset 17426 df-lub 17454 df-glb 17455 df-join 17456 df-meet 17457 df-lat 17526 df-ats 35885 df-atl 35916 df-cvlat 35940 df-hlat 35969 |
This theorem is referenced by: dalemcea 36278 dalem2 36279 dalemdnee 36284 |
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