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Mirrors > Home > MPE Home > Th. List > Mathboxes > 4atexlemntlpq | Structured version Visualization version GIF version |
Description: Lemma for 4atexlem7 37205. (Contributed by NM, 24-Nov-2012.) |
Ref | Expression |
---|---|
4thatlem.ph | ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)))) |
4thatlem0.l | ⊢ ≤ = (le‘𝐾) |
4thatlem0.j | ⊢ ∨ = (join‘𝐾) |
4thatlem0.m | ⊢ ∧ = (meet‘𝐾) |
4thatlem0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
4thatlem0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
4thatlem0.u | ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) |
4thatlem0.v | ⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊) |
Ref | Expression |
---|---|
4atexlemntlpq | ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4thatlem.ph | . . 3 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)))) | |
2 | 4thatlem0.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | 4thatlem0.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | 4thatlem0.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
5 | 4thatlem0.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | 4thatlem0.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | 4thatlem0.u | . . 3 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊) | |
8 | 4thatlem0.v | . . 3 ⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | 4atexlemtlw 37197 | . 2 ⊢ (𝜑 → 𝑇 ≤ 𝑊) |
10 | 1 | 4atexlemkc 37188 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ CvLat) |
11 | 1, 2, 3, 4, 5, 6, 7 | 4atexlemu 37194 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
12 | 1, 2, 3, 4, 5, 6, 7, 8 | 4atexlemv 37195 | . . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝐴) |
13 | 1 | 4atexlemt 37183 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐴) |
14 | 1, 2, 3, 4, 5, 6, 7, 8 | 4atexlemunv 37196 | . . . . . 6 ⊢ (𝜑 → 𝑈 ≠ 𝑉) |
15 | 1 | 4atexlemutvt 37184 | . . . . . 6 ⊢ (𝜑 → (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇)) |
16 | 5, 3 | cvlsupr5 36476 | . . . . . 6 ⊢ ((𝐾 ∈ CvLat ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑈 ≠ 𝑉 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) → 𝑇 ≠ 𝑈) |
17 | 10, 11, 12, 13, 14, 15, 16 | syl132anc 1384 | . . . . 5 ⊢ (𝜑 → 𝑇 ≠ 𝑈) |
18 | 17 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ≠ 𝑈) |
19 | 1 | 4atexlemk 37177 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ HL) |
20 | 1 | 4atexlemw 37178 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐻) |
21 | 19, 20 | jca 514 | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
22 | 21 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
23 | 1 | 4atexlempw 37179 | . . . . . 6 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
24 | 23 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
25 | 1 | 4atexlemq 37181 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
26 | 25 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑄 ∈ 𝐴) |
27 | 13 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ∈ 𝐴) |
28 | 1 | 4atexlempnq 37185 | . . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑄) |
29 | 28 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ≠ 𝑄) |
30 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ≤ (𝑃 ∨ 𝑄)) | |
31 | 2, 3, 4, 5, 6, 7 | lhpat3 37176 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄))) → (¬ 𝑇 ≤ 𝑊 ↔ 𝑇 ≠ 𝑈)) |
32 | 22, 24, 26, 27, 29, 30, 31 | syl222anc 1382 | . . . 4 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (¬ 𝑇 ≤ 𝑊 ↔ 𝑇 ≠ 𝑈)) |
33 | 18, 32 | mpbird 259 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑇 ≤ 𝑊) |
34 | 33 | ex 415 | . 2 ⊢ (𝜑 → (𝑇 ≤ (𝑃 ∨ 𝑄) → ¬ 𝑇 ≤ 𝑊)) |
35 | 9, 34 | mt2d 138 | 1 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 lecple 16566 joincjn 17548 meetcmee 17549 Atomscatm 36393 CvLatclc 36395 HLchlt 36480 LHypclh 37114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-lhyp 37118 |
This theorem is referenced by: 4atexlemc 37199 4atexlemex2 37201 4atexlemcnd 37202 |
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