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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 36497 | . 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 36475 | . 2 ⊢ ((𝐾 ∈ CvLat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑄 ∨ 𝑅) → 𝑄 ≤ (𝑃 ∨ 𝑅))) |
6 | 1, 5 | syl3an1 1159 | 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 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 lecple 16574 joincjn 17556 Atomscatm 36401 CvLatclc 36403 HLchlt 36488 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-lat 17658 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 |
This theorem is referenced by: 2llnneN 36547 atexchcvrN 36578 atbtwnex 36586 3dimlem3 36599 3dimlem3OLDN 36600 3dimlem4 36602 3dimlem4OLDN 36603 hlatexch4 36619 3atlem5 36625 dalem27 36837 cdlemblem 36931 paddasslem1 36958 paddasslem6 36963 cdleme3g 37372 cdleme3h 37373 cdleme7d 37384 cdleme11c 37399 cdleme11dN 37400 cdleme36a 37598 cdlemeg46rgv 37666 cdlemk14 37992 dia2dimlem1 38202 dia2dimlem2 38203 dia2dimlem3 38204 |
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