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Mirrors > Home > MPE Home > Th. List > lattr | Structured version Visualization version GIF version |
Description: A lattice ordering is transitive. (sstr 3975 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 17660 | . 2 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
2 | latref.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
3 | latref.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
4 | 2, 3 | postr 17563 | . 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)) |
5 | 1, 4 | sylan 582 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 Basecbs 16483 lecple 16572 Posetcpo 17550 Latclat 17655 |
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 2793 ax-nul 5210 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-dm 5565 df-iota 6314 df-fv 6363 df-poset 17556 df-lat 17656 |
This theorem is referenced by: lattrd 17668 latjlej1 17675 latjlej12 17677 latnlej2 17681 latmlem1 17691 latmlem12 17693 clatleglb 17736 lecmtN 36407 hlrelat2 36554 ps-2 36629 dalem3 36815 dalem17 36831 dalem21 36845 dalem25 36849 linepsubN 36903 pmapsub 36919 cdlemblem 36944 pmapjoin 37003 lhpmcvr4N 37177 4atexlemnclw 37221 cdlemd3 37351 cdleme3g 37385 cdleme3h 37386 cdleme7d 37397 cdleme21c 37478 cdleme32b 37593 cdleme35fnpq 37600 cdleme35f 37605 cdleme48bw 37653 cdlemf1 37712 cdlemg2fv2 37751 cdlemg7fvbwN 37758 cdlemg4 37768 cdlemg6c 37771 cdlemg27a 37843 cdlemg33b0 37852 cdlemg33a 37857 cdlemk3 37984 dia2dimlem1 38215 dihord6b 38411 dihord5apre 38413 dihglbcpreN 38451 |
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