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Mirrors > Home > MPE Home > Th. List > Mathboxes > atlrelat1 | Structured version Visualization version GIF version |
Description: An atomistic lattice with 0 is relatively atomic. Part of Lemma 7.2 of [MaedaMaeda] p. 30. (chpssati 30124, with ∧ swapped, analog.) (Contributed by NM, 4-Dec-2011.) |
Ref | Expression |
---|---|
atlrelat1.b | ⊢ 𝐵 = (Base‘𝐾) |
atlrelat1.l | ⊢ ≤ = (le‘𝐾) |
atlrelat1.s | ⊢ < = (lt‘𝐾) |
atlrelat1.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atlrelat1 | ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp13 1201 | . . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ AtLat) | |
2 | atlpos 36469 | . . . 4 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Poset) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Poset) |
4 | atlrelat1.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
5 | atlrelat1.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
6 | atlrelat1.s | . . . . 5 ⊢ < = (lt‘𝐾) | |
7 | 4, 5, 6 | pltnle 17559 | . . . 4 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌 ≤ 𝑋) |
8 | 7 | ex 415 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ¬ 𝑌 ≤ 𝑋)) |
9 | 3, 8 | syld3an1 1406 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ¬ 𝑌 ≤ 𝑋)) |
10 | iman 404 | . . . . . . 7 ⊢ ((𝑝 ≤ 𝑌 → 𝑝 ≤ 𝑋) ↔ ¬ (𝑝 ≤ 𝑌 ∧ ¬ 𝑝 ≤ 𝑋)) | |
11 | ancom 463 | . . . . . . 7 ⊢ ((𝑝 ≤ 𝑌 ∧ ¬ 𝑝 ≤ 𝑋) ↔ (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌)) | |
12 | 10, 11 | xchbinx 336 | . . . . . 6 ⊢ ((𝑝 ≤ 𝑌 → 𝑝 ≤ 𝑋) ↔ ¬ (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌)) |
13 | 12 | ralbii 3165 | . . . . 5 ⊢ (∀𝑝 ∈ 𝐴 (𝑝 ≤ 𝑌 → 𝑝 ≤ 𝑋) ↔ ∀𝑝 ∈ 𝐴 ¬ (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌)) |
14 | atlrelat1.a | . . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾) | |
15 | 4, 5, 14 | atlatle 36488 | . . . . . . 7 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌 ≤ 𝑋 ↔ ∀𝑝 ∈ 𝐴 (𝑝 ≤ 𝑌 → 𝑝 ≤ 𝑋))) |
16 | 15 | 3com23 1122 | . . . . . 6 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 ≤ 𝑋 ↔ ∀𝑝 ∈ 𝐴 (𝑝 ≤ 𝑌 → 𝑝 ≤ 𝑋))) |
17 | 16 | biimprd 250 | . . . . 5 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑝 ∈ 𝐴 (𝑝 ≤ 𝑌 → 𝑝 ≤ 𝑋) → 𝑌 ≤ 𝑋)) |
18 | 13, 17 | syl5bir 245 | . . . 4 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑝 ∈ 𝐴 ¬ (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌) → 𝑌 ≤ 𝑋)) |
19 | 18 | con3d 155 | . . 3 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (¬ 𝑌 ≤ 𝑋 → ¬ ∀𝑝 ∈ 𝐴 ¬ (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌))) |
20 | dfrex2 3239 | . . 3 ⊢ (∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌) ↔ ¬ ∀𝑝 ∈ 𝐴 ¬ (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌)) | |
21 | 19, 20 | syl6ibr 254 | . 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (¬ 𝑌 ≤ 𝑋 → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌))) |
22 | 9, 21 | syld 47 | 1 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 class class class wbr 5052 ‘cfv 6341 Basecbs 16466 lecple 16555 Posetcpo 17533 ltcplt 17534 CLatccla 17700 OMLcoml 36343 Atomscatm 36431 AtLatcal 36432 |
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 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
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 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 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-proset 17521 df-poset 17539 df-plt 17551 df-lub 17567 df-glb 17568 df-join 17569 df-meet 17570 df-p0 17632 df-lat 17639 df-clat 17701 df-oposet 36344 df-ol 36346 df-oml 36347 df-covers 36434 df-ats 36435 df-atl 36466 |
This theorem is referenced by: cvlcvr1 36507 hlrelat1 36568 |
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