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| Mirrors > Home > MPE Home > Th. List > latref | Structured version Visualization version GIF version | ||
| Description: A lattice ordering is reflexive. (ssid 4006 analog.) (Contributed by NM, 8-Oct-2011.) |
| Ref | Expression |
|---|---|
| latref.b | ⊢ 𝐵 = (Base‘𝐾) |
| latref.l | ⊢ ≤ = (le‘𝐾) |
| Ref | Expression |
|---|---|
| latref | ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | latpos 18483 | . 2 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
| 2 | latref.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
| 3 | latref.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
| 4 | 2, 3 | posref 18364 | . 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋) |
| 5 | 1, 4 | sylan 580 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 lecple 17304 Posetcpo 18353 Latclat 18476 |
| 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-ext 2708 ax-nul 5306 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-dm 5695 df-iota 6514 df-fv 6569 df-proset 18340 df-poset 18359 df-lat 18477 |
| This theorem is referenced by: latleeqj1 18496 latjidm 18507 latleeqm1 18512 latmidm 18519 olj01 39226 olm01 39237 cmtidN 39258 ps-1 39479 3at 39492 llnneat 39516 2atnelpln 39546 lplnneat 39547 lplnnelln 39548 3atnelvolN 39588 lvolneatN 39590 lvolnelln 39591 lvolnelpln 39592 4at 39615 lplncvrlvol 39618 lncmp 39785 lhpocnle 40018 ltrnel 40141 ltrncnvel 40144 tendoidcl 40771 cdlemk39u 40970 dia1eldmN 41043 dia1N 41055 dihwN 41291 dihglblem5apreN 41293 dihmeetbclemN 41306 |
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