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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isline4N | Structured version Visualization version GIF version |
Description: Definition of line in terms of original lattice elements. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
isline4.b | ⊢ 𝐵 = (Base‘𝐾) |
isline4.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
isline4.a | ⊢ 𝐴 = (Atoms‘𝐾) |
isline4.n | ⊢ 𝑁 = (Lines‘𝐾) |
isline4.m | ⊢ 𝑀 = (pmap‘𝐾) |
Ref | Expression |
---|---|
isline4N | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) ∈ 𝑁 ↔ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isline4.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2778 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
3 | isline4.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | isline4.n | . . 3 ⊢ 𝑁 = (Lines‘𝐾) | |
5 | isline4.m | . . 3 ⊢ 𝑀 = (pmap‘𝐾) | |
6 | 1, 2, 3, 4, 5 | isline3 35935 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) ∈ 𝑁 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) |
7 | simpll 757 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝐾 ∈ HL) | |
8 | 1, 3 | atbase 35448 | . . . . . 6 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵) |
9 | 8 | adantl 475 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐵) |
10 | simplr 759 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
11 | eqid 2778 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
12 | isline4.c | . . . . . 6 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
13 | 1, 11, 2, 12, 3 | cvrval3 35572 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑝𝐶𝑋 ↔ ∃𝑞 ∈ 𝐴 (¬ 𝑞(le‘𝐾)𝑝 ∧ (𝑝(join‘𝐾)𝑞) = 𝑋))) |
14 | 7, 9, 10, 13 | syl3anc 1439 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (𝑝𝐶𝑋 ↔ ∃𝑞 ∈ 𝐴 (¬ 𝑞(le‘𝐾)𝑝 ∧ (𝑝(join‘𝐾)𝑞) = 𝑋))) |
15 | hlatl 35519 | . . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
16 | 15 | ad3antrrr 720 | . . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → 𝐾 ∈ AtLat) |
17 | simpr 479 | . . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴) | |
18 | simplr 759 | . . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → 𝑝 ∈ 𝐴) | |
19 | 11, 3 | atncmp 35471 | . . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑞 ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) → (¬ 𝑞(le‘𝐾)𝑝 ↔ 𝑞 ≠ 𝑝)) |
20 | 16, 17, 18, 19 | syl3anc 1439 | . . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → (¬ 𝑞(le‘𝐾)𝑝 ↔ 𝑞 ≠ 𝑝)) |
21 | necom 3022 | . . . . . . 7 ⊢ (𝑞 ≠ 𝑝 ↔ 𝑝 ≠ 𝑞) | |
22 | 20, 21 | syl6bb 279 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → (¬ 𝑞(le‘𝐾)𝑝 ↔ 𝑝 ≠ 𝑞)) |
23 | eqcom 2785 | . . . . . . 7 ⊢ ((𝑝(join‘𝐾)𝑞) = 𝑋 ↔ 𝑋 = (𝑝(join‘𝐾)𝑞)) | |
24 | 23 | a1i 11 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → ((𝑝(join‘𝐾)𝑞) = 𝑋 ↔ 𝑋 = (𝑝(join‘𝐾)𝑞))) |
25 | 22, 24 | anbi12d 624 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ 𝐴) → ((¬ 𝑞(le‘𝐾)𝑝 ∧ (𝑝(join‘𝐾)𝑞) = 𝑋) ↔ (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) |
26 | 25 | rexbidva 3234 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (∃𝑞 ∈ 𝐴 (¬ 𝑞(le‘𝐾)𝑝 ∧ (𝑝(join‘𝐾)𝑞) = 𝑋) ↔ ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) |
27 | 14, 26 | bitrd 271 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (𝑝𝐶𝑋 ↔ ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) |
28 | 27 | rexbidva 3234 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (∃𝑝 ∈ 𝐴 𝑝𝐶𝑋 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝(join‘𝐾)𝑞)))) |
29 | 6, 28 | bitr4d 274 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) ∈ 𝑁 ↔ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∃wrex 3091 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 Basecbs 16259 lecple 16349 joincjn 17334 ⋖ ccvr 35421 Atomscatm 35422 AtLatcal 35423 HLchlt 35509 Linesclines 35653 pmapcpmap 35656 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-proset 17318 df-poset 17336 df-plt 17348 df-lub 17364 df-glb 17365 df-join 17366 df-meet 17367 df-p0 17429 df-lat 17436 df-clat 17498 df-oposet 35335 df-ol 35337 df-oml 35338 df-covers 35425 df-ats 35426 df-atl 35457 df-cvlat 35481 df-hlat 35510 df-lines 35660 df-pmap 35663 |
This theorem is referenced by: (None) |
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