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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlatexch2 | Structured version Visualization version GIF version |
Description: Atom exchange property. (Contributed by NM, 8-Jan-2012.) |
Ref | Expression |
---|---|
hlatexchb.l | ⊢ ≤ = (le‘𝐾) |
hlatexchb.j | ⊢ ∨ = (join‘𝐾) |
hlatexchb.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
hlatexch2 | ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑄 ∨ 𝑅) → 𝑄 ≤ (𝑃 ∨ 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlcvl 38831 | . 2 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CvLat) | |
2 | hlatexchb.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | hlatexchb.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | hlatexchb.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 2, 3, 4 | cvlatexch2 38809 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑄 ∨ 𝑅) → 𝑄 ≤ (𝑃 ∨ 𝑅))) |
6 | 1, 5 | syl3an1 1161 | 1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑄 ∨ 𝑅) → 𝑄 ≤ (𝑃 ∨ 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 lecple 17240 joincjn 18303 Atomscatm 38735 CvLatclc 38737 HLchlt 38822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-proset 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-lat 18424 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 |
This theorem is referenced by: 2llnneN 38882 atexchcvrN 38913 atbtwnex 38921 3dimlem3 38934 3dimlem3OLDN 38935 3dimlem4 38937 3dimlem4OLDN 38938 hlatexch4 38954 3atlem5 38960 dalem27 39172 cdlemblem 39266 paddasslem1 39293 paddasslem6 39298 cdleme3g 39707 cdleme3h 39708 cdleme7d 39719 cdleme11c 39734 cdleme11dN 39735 cdleme36a 39933 cdlemeg46rgv 40001 cdlemk14 40327 dia2dimlem1 40537 dia2dimlem2 40538 dia2dimlem3 40539 |
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