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| Mirrors > Home > MPE Home > Th. List > Mathboxes > hlrelat | Structured version Visualization version GIF version | ||
| Description: A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 32383 analog.) (Contributed by NM, 4-Feb-2012.) | 
| Ref | Expression | 
|---|---|
| hlrelat5.b | ⊢ 𝐵 = (Base‘𝐾) | 
| hlrelat5.l | ⊢ ≤ = (le‘𝐾) | 
| hlrelat5.s | ⊢ < = (lt‘𝐾) | 
| hlrelat5.j | ⊢ ∨ = (join‘𝐾) | 
| hlrelat5.a | ⊢ 𝐴 = (Atoms‘𝐾) | 
| Ref | Expression | 
|---|---|
| hlrelat | ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ∃𝑝 ∈ 𝐴 (𝑋 < (𝑋 ∨ 𝑝) ∧ (𝑋 ∨ 𝑝) ≤ 𝑌)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hlrelat5.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | hlrelat5.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
| 3 | hlrelat5.s | . . . 4 ⊢ < = (lt‘𝐾) | |
| 4 | hlrelat5.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 5 | 1, 2, 3, 4 | hlrelat1 39402 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌))) | 
| 6 | 5 | imp 406 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌)) | 
| 7 | simpll1 1213 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ 𝐴) → 𝐾 ∈ HL) | |
| 8 | 7 | hllatd 39365 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ 𝐴) → 𝐾 ∈ Lat) | 
| 9 | simpll2 1214 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
| 10 | 1, 4 | atbase 39290 | . . . . . 6 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵) | 
| 11 | 10 | adantl 481 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐵) | 
| 12 | hlrelat5.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
| 13 | 1, 2, 3, 12 | latnle 18518 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) → (¬ 𝑝 ≤ 𝑋 ↔ 𝑋 < (𝑋 ∨ 𝑝))) | 
| 14 | 8, 9, 11, 13 | syl3anc 1373 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ 𝐴) → (¬ 𝑝 ≤ 𝑋 ↔ 𝑋 < (𝑋 ∨ 𝑝))) | 
| 15 | 2, 3 | pltle 18378 | . . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → 𝑋 ≤ 𝑌)) | 
| 16 | 15 | imp 406 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → 𝑋 ≤ 𝑌) | 
| 17 | 16 | adantr 480 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ 𝐴) → 𝑋 ≤ 𝑌) | 
| 18 | 17 | biantrurd 532 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ 𝐴) → (𝑝 ≤ 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑝 ≤ 𝑌))) | 
| 19 | simpll3 1215 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ 𝐴) → 𝑌 ∈ 𝐵) | |
| 20 | 1, 2, 12 | latjle12 18495 | . . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑝 ≤ 𝑌) ↔ (𝑋 ∨ 𝑝) ≤ 𝑌)) | 
| 21 | 8, 9, 11, 19, 20 | syl13anc 1374 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ 𝐴) → ((𝑋 ≤ 𝑌 ∧ 𝑝 ≤ 𝑌) ↔ (𝑋 ∨ 𝑝) ≤ 𝑌)) | 
| 22 | 18, 21 | bitrd 279 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ 𝐴) → (𝑝 ≤ 𝑌 ↔ (𝑋 ∨ 𝑝) ≤ 𝑌)) | 
| 23 | 14, 22 | anbi12d 632 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) ∧ 𝑝 ∈ 𝐴) → ((¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌) ↔ (𝑋 < (𝑋 ∨ 𝑝) ∧ (𝑋 ∨ 𝑝) ≤ 𝑌))) | 
| 24 | 23 | rexbidva 3177 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → (∃𝑝 ∈ 𝐴 (¬ 𝑝 ≤ 𝑋 ∧ 𝑝 ≤ 𝑌) ↔ ∃𝑝 ∈ 𝐴 (𝑋 < (𝑋 ∨ 𝑝) ∧ (𝑋 ∨ 𝑝) ≤ 𝑌))) | 
| 25 | 6, 24 | mpbid 232 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ∃𝑝 ∈ 𝐴 (𝑋 < (𝑋 ∨ 𝑝) ∧ (𝑋 ∨ 𝑝) ≤ 𝑌)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 lecple 17304 ltcplt 18354 joincjn 18357 Latclat 18476 Atomscatm 39264 HLchlt 39351 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-lat 18477 df-clat 18544 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 | 
| This theorem is referenced by: hlrelat2 39405 atle 39438 2atlt 39441 | 
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