| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > llni2 | Structured version Visualization version GIF version | ||
| Description: The join of two different atoms is a lattice line. (Contributed by NM, 26-Jun-2012.) |
| Ref | Expression |
|---|---|
| llni2.j | ⊢ ∨ = (join‘𝐾) |
| llni2.a | ⊢ 𝐴 = (Atoms‘𝐾) |
| llni2.n | ⊢ 𝑁 = (LLines‘𝐾) |
| Ref | Expression |
|---|---|
| llni2 | ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑃 ∨ 𝑄) ∈ 𝑁) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpl2 1193 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ 𝐴) | |
| 2 | simpl3 1194 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ 𝐴) | |
| 3 | simpr 484 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝑃 ≠ 𝑄) | |
| 4 | eqidd 2738 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑄)) | |
| 5 | neeq1 3003 | . . . . 5 ⊢ (𝑟 = 𝑃 → (𝑟 ≠ 𝑠 ↔ 𝑃 ≠ 𝑠)) | |
| 6 | oveq1 7438 | . . . . . 6 ⊢ (𝑟 = 𝑃 → (𝑟 ∨ 𝑠) = (𝑃 ∨ 𝑠)) | |
| 7 | 6 | eqeq2d 2748 | . . . . 5 ⊢ (𝑟 = 𝑃 → ((𝑃 ∨ 𝑄) = (𝑟 ∨ 𝑠) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑠))) |
| 8 | 5, 7 | anbi12d 632 | . . . 4 ⊢ (𝑟 = 𝑃 → ((𝑟 ≠ 𝑠 ∧ (𝑃 ∨ 𝑄) = (𝑟 ∨ 𝑠)) ↔ (𝑃 ≠ 𝑠 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑠)))) |
| 9 | neeq2 3004 | . . . . 5 ⊢ (𝑠 = 𝑄 → (𝑃 ≠ 𝑠 ↔ 𝑃 ≠ 𝑄)) | |
| 10 | oveq2 7439 | . . . . . 6 ⊢ (𝑠 = 𝑄 → (𝑃 ∨ 𝑠) = (𝑃 ∨ 𝑄)) | |
| 11 | 10 | eqeq2d 2748 | . . . . 5 ⊢ (𝑠 = 𝑄 → ((𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑠) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑄))) |
| 12 | 9, 11 | anbi12d 632 | . . . 4 ⊢ (𝑠 = 𝑄 → ((𝑃 ≠ 𝑠 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑠)) ↔ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑄)))) |
| 13 | 8, 12 | rspc2ev 3635 | . . 3 ⊢ ((𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑃 ≠ 𝑄 ∧ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑄))) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ (𝑃 ∨ 𝑄) = (𝑟 ∨ 𝑠))) |
| 14 | 1, 2, 3, 4, 13 | syl112anc 1376 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ (𝑃 ∨ 𝑄) = (𝑟 ∨ 𝑠))) |
| 15 | simpl1 1192 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → 𝐾 ∈ HL) | |
| 16 | eqid 2737 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 17 | llni2.j | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
| 18 | llni2.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 19 | 16, 17, 18 | hlatjcl 39368 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
| 20 | 19 | adantr 480 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
| 21 | llni2.n | . . . 4 ⊢ 𝑁 = (LLines‘𝐾) | |
| 22 | 16, 17, 18, 21 | islln3 39512 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∈ 𝑁 ↔ ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ (𝑃 ∨ 𝑄) = (𝑟 ∨ 𝑠)))) |
| 23 | 15, 20, 22 | syl2anc 584 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → ((𝑃 ∨ 𝑄) ∈ 𝑁 ↔ ∃𝑟 ∈ 𝐴 ∃𝑠 ∈ 𝐴 (𝑟 ≠ 𝑠 ∧ (𝑃 ∨ 𝑄) = (𝑟 ∨ 𝑠)))) |
| 24 | 14, 23 | mpbird 257 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑃 ∨ 𝑄) ∈ 𝑁) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 joincjn 18357 Atomscatm 39264 HLchlt 39351 LLinesclln 39493 |
| 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 df-llines 39500 |
| This theorem is referenced by: 2atneat 39517 islln2a 39519 2at0mat0 39527 ps-2c 39530 lplnnle2at 39543 2atmat 39563 lplnexllnN 39566 dalempjsen 39655 dalemcea 39662 dalem2 39663 dalemdea 39664 dalem16 39681 dalemcjden 39694 dalem23 39698 dalem54 39728 dalem60 39734 llnexchb2 39871 arglem1N 40192 cdlemc5 40197 cdleme20l1 40322 cdleme20l2 40323 cdleme20l 40324 cdleme22b 40343 cdlemeg46req 40531 cdlemh 40819 |
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