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Mirrors > Home > MPE Home > Th. List > Mathboxes > 4atexlemntlpq | Structured version Visualization version GIF version |
Description: Lemma for 4atexlem7 37213. (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 37205 | . 2 ⊢ (𝜑 → 𝑇 ≤ 𝑊) |
10 | 1 | 4atexlemkc 37196 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ CvLat) |
11 | 1, 2, 3, 4, 5, 6, 7 | 4atexlemu 37202 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
12 | 1, 2, 3, 4, 5, 6, 7, 8 | 4atexlemv 37203 | . . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝐴) |
13 | 1 | 4atexlemt 37191 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐴) |
14 | 1, 2, 3, 4, 5, 6, 7, 8 | 4atexlemunv 37204 | . . . . . 6 ⊢ (𝜑 → 𝑈 ≠ 𝑉) |
15 | 1 | 4atexlemutvt 37192 | . . . . . 6 ⊢ (𝜑 → (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇)) |
16 | 5, 3 | cvlsupr5 36484 | . . . . . 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 37185 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ HL) |
20 | 1 | 4atexlemw 37186 | . . . . . . 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 37187 | . . . . . 6 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
24 | 23 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
25 | 1 | 4atexlemq 37189 | . . . . . 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 37193 | . . . . . 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 37184 | . . . . 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 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 lecple 16574 joincjn 17556 meetcmee 17557 Atomscatm 36401 CvLatclc 36403 HLchlt 36488 LHypclh 37122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-lhyp 37126 |
This theorem is referenced by: 4atexlemc 37207 4atexlemex2 37209 4atexlemcnd 37210 |
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