Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lattrd | Structured version Visualization version GIF version |
Description: A lattice ordering is transitive. Deduction version of lattr 16879. (Contributed by NM, 3-Sep-2012.) |
Ref | Expression |
---|---|
lattrd.b | ⊢ 𝐵 = (Base‘𝐾) |
lattrd.l | ⊢ ≤ = (le‘𝐾) |
lattrd.1 | ⊢ (𝜑 → 𝐾 ∈ Lat) |
lattrd.2 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
lattrd.3 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
lattrd.4 | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
lattrd.5 | ⊢ (𝜑 → 𝑋 ≤ 𝑌) |
lattrd.6 | ⊢ (𝜑 → 𝑌 ≤ 𝑍) |
Ref | Expression |
---|---|
lattrd | ⊢ (𝜑 → 𝑋 ≤ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lattrd.5 | . 2 ⊢ (𝜑 → 𝑋 ≤ 𝑌) | |
2 | lattrd.6 | . 2 ⊢ (𝜑 → 𝑌 ≤ 𝑍) | |
3 | lattrd.1 | . . 3 ⊢ (𝜑 → 𝐾 ∈ Lat) | |
4 | lattrd.2 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | lattrd.3 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | lattrd.4 | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
7 | lattrd.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
8 | lattrd.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
9 | 7, 8 | lattr 16879 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)) |
10 | 3, 4, 5, 6, 9 | syl13anc 1320 | . 2 ⊢ (𝜑 → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)) |
11 | 1, 2, 10 | mp2and 711 | 1 ⊢ (𝜑 → 𝑋 ≤ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 lecple 15775 Latclat 16868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-dm 5048 df-iota 5768 df-fv 5812 df-poset 16769 df-lat 16869 |
This theorem is referenced by: latmlej11 16913 latjass 16918 lubun 16946 cvlcvr1 33644 exatleN 33708 2atjm 33749 2llnmat 33828 llnmlplnN 33843 2llnjaN 33870 2lplnja 33923 dalem5 33971 lncmp 34087 2lnat 34088 2llnma1b 34090 cdlema1N 34095 paddasslem5 34128 paddasslem12 34135 paddasslem13 34136 dalawlem3 34177 dalawlem5 34179 dalawlem6 34180 dalawlem7 34181 dalawlem8 34182 dalawlem11 34185 dalawlem12 34186 pl42lem1N 34283 lhpexle2lem 34313 lhpexle3lem 34315 4atexlemtlw 34371 4atexlemc 34373 cdleme15 34583 cdleme17b 34592 cdleme22e 34650 cdleme22eALTN 34651 cdleme23a 34655 cdleme28a 34676 cdleme30a 34684 cdleme32e 34751 cdleme35b 34756 trlord 34875 cdlemg10 34947 cdlemg11b 34948 cdlemg17a 34967 cdlemg35 35019 tendococl 35078 tendopltp 35086 cdlemi1 35124 cdlemk11 35155 cdlemk5u 35167 cdlemk11u 35177 cdlemk52 35260 dialss 35353 diaglbN 35362 diaintclN 35365 dia2dimlem1 35371 cdlemm10N 35425 djajN 35444 dibglbN 35473 dibintclN 35474 diblss 35477 cdlemn10 35513 dihord1 35525 dihord2pre2 35533 dihopelvalcpre 35555 dihord5apre 35569 dihmeetlem1N 35597 dihglblem2N 35601 dihmeetlem2N 35606 dihglbcpreN 35607 dihmeetlem3N 35612 |
Copyright terms: Public domain | W3C validator |