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Mirrors > Home > MPE Home > Th. List > lattr | Structured version Visualization version GIF version |
Description: A lattice ordering is transitive. (sstr 3972 analog.) (Contributed by NM, 17-Nov-2011.) |
Ref | Expression |
---|---|
latref.b | ⊢ 𝐵 = (Base‘𝐾) |
latref.l | ⊢ ≤ = (le‘𝐾) |
Ref | Expression |
---|---|
lattr | ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latpos 17648 | . 2 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
2 | latref.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
3 | latref.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
4 | 2, 3 | postr 17551 | . 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)) |
5 | 1, 4 | sylan 580 | 1 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 Basecbs 16471 lecple 16560 Posetcpo 17538 Latclat 17643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-dm 5558 df-iota 6307 df-fv 6356 df-poset 17544 df-lat 17644 |
This theorem is referenced by: lattrd 17656 latjlej1 17663 latjlej12 17665 latnlej2 17669 latmlem1 17679 latmlem12 17681 clatleglb 17724 lecmtN 36272 hlrelat2 36419 ps-2 36494 dalem3 36680 dalem17 36696 dalem21 36710 dalem25 36714 linepsubN 36768 pmapsub 36784 cdlemblem 36809 pmapjoin 36868 lhpmcvr4N 37042 4atexlemnclw 37086 cdlemd3 37216 cdleme3g 37250 cdleme3h 37251 cdleme7d 37262 cdleme21c 37343 cdleme32b 37458 cdleme35fnpq 37465 cdleme35f 37470 cdleme48bw 37518 cdlemf1 37577 cdlemg2fv2 37616 cdlemg7fvbwN 37623 cdlemg4 37633 cdlemg6c 37636 cdlemg27a 37708 cdlemg33b0 37717 cdlemg33a 37722 cdlemk3 37849 dia2dimlem1 38080 dihord6b 38276 dihord5apre 38278 dihglbcpreN 38316 |
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