![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > latref | Structured version Visualization version GIF version |
Description: A lattice ordering is reflexive. (ssid 4018 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 18496 | . 2 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
2 | latref.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
3 | latref.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
4 | 2, 3 | posref 18376 | . 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 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 Basecbs 17245 lecple 17305 Posetcpo 18365 Latclat 18489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-nul 5312 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-dm 5699 df-iota 6516 df-fv 6571 df-proset 18352 df-poset 18371 df-lat 18490 |
This theorem is referenced by: latleeqj1 18509 latjidm 18520 latleeqm1 18525 latmidm 18532 olj01 39207 olm01 39218 cmtidN 39239 ps-1 39460 3at 39473 llnneat 39497 2atnelpln 39527 lplnneat 39528 lplnnelln 39529 3atnelvolN 39569 lvolneatN 39571 lvolnelln 39572 lvolnelpln 39573 4at 39596 lplncvrlvol 39599 lncmp 39766 lhpocnle 39999 ltrnel 40122 ltrncnvel 40125 tendoidcl 40752 cdlemk39u 40951 dia1eldmN 41024 dia1N 41036 dihwN 41272 dihglblem5apreN 41274 dihmeetbclemN 41287 |
Copyright terms: Public domain | W3C validator |