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Mirrors > Home > MPE Home > Th. List > Mathboxes > 4atexlemntlpq | Structured version Visualization version GIF version |
Description: Lemma for 4atexlem7 35361. (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 35353 | . 2 ⊢ (𝜑 → 𝑇 ≤ 𝑊) |
10 | 1 | 4atexlemkc 35344 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ CvLat) |
11 | 1, 2, 3, 4, 5, 6, 7 | 4atexlemu 35350 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ 𝐴) |
12 | 1, 2, 3, 4, 5, 6, 7, 8 | 4atexlemv 35351 | . . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝐴) |
13 | 1 | 4atexlemt 35339 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐴) |
14 | 1, 2, 3, 4, 5, 6, 7, 8 | 4atexlemunv 35352 | . . . . . 6 ⊢ (𝜑 → 𝑈 ≠ 𝑉) |
15 | 1 | 4atexlemutvt 35340 | . . . . . 6 ⊢ (𝜑 → (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇)) |
16 | 5, 3 | cvlsupr5 34633 | . . . . . 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 1344 | . . . . 5 ⊢ (𝜑 → 𝑇 ≠ 𝑈) |
18 | 17 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ≠ 𝑈) |
19 | 1 | 4atexlemk 35333 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ HL) |
20 | 1 | 4atexlemw 35334 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐻) |
21 | 19, 20 | jca 554 | . . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
22 | 21 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
23 | 1 | 4atexlempw 35335 | . . . . . 6 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
24 | 23 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
25 | 1 | 4atexlemq 35337 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
26 | 25 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑄 ∈ 𝐴) |
27 | 13 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ∈ 𝐴) |
28 | 1 | 4atexlempnq 35341 | . . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑄) |
29 | 28 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ≠ 𝑄) |
30 | simpr 477 | . . . . 5 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ≤ (𝑃 ∨ 𝑄)) | |
31 | 2, 3, 4, 5, 6, 7 | lhpat3 35332 | . . . . 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 1342 | . . . 4 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → (¬ 𝑇 ≤ 𝑊 ↔ 𝑇 ≠ 𝑈)) |
33 | 18, 32 | mpbird 247 | . . 3 ⊢ ((𝜑 ∧ 𝑇 ≤ (𝑃 ∨ 𝑄)) → ¬ 𝑇 ≤ 𝑊) |
34 | 33 | ex 450 | . 2 ⊢ (𝜑 → (𝑇 ≤ (𝑃 ∨ 𝑄) → ¬ 𝑇 ≤ 𝑊)) |
35 | 9, 34 | mt2d 131 | 1 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 lecple 15948 joincjn 16944 meetcmee 16945 Atomscatm 34550 CvLatclc 34552 HLchlt 34637 LHypclh 35270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-lhyp 35274 |
This theorem is referenced by: 4atexlemc 35355 4atexlemex2 35357 4atexlemcnd 35358 |
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