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| Mirrors > Home > MPE Home > Th. List > lattrd | Structured version Visualization version GIF version | ||
| Description: A lattice ordering is transitive. Deduction version of lattr 18489. (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 18489 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)) |
| 10 | 3, 4, 5, 6, 9 | syl13anc 1374 | . 2 ⊢ (𝜑 → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍)) |
| 11 | 1, 2, 10 | mp2and 699 | 1 ⊢ (𝜑 → 𝑋 ≤ 𝑍) |
| 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 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-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-poset 18359 df-lat 18477 |
| This theorem is referenced by: latmlej11 18523 latjass 18528 lubun 18560 cvlcvr1 39340 exatleN 39406 2atjm 39447 2llnmat 39526 llnmlplnN 39541 2llnjaN 39568 2lplnja 39621 dalem5 39669 lncmp 39785 2lnat 39786 2llnma1b 39788 cdlema1N 39793 paddasslem5 39826 paddasslem12 39833 paddasslem13 39834 dalawlem3 39875 dalawlem5 39877 dalawlem6 39878 dalawlem7 39879 dalawlem8 39880 dalawlem11 39883 dalawlem12 39884 pl42lem1N 39981 lhpexle2lem 40011 lhpexle3lem 40013 4atexlemtlw 40069 4atexlemc 40071 cdleme15 40280 cdleme17b 40289 cdleme22e 40346 cdleme22eALTN 40347 cdleme23a 40351 cdleme28a 40372 cdleme30a 40380 cdleme32e 40447 cdleme35b 40452 trlord 40571 cdlemg10 40643 cdlemg11b 40644 cdlemg17a 40663 cdlemg35 40715 tendococl 40774 tendopltp 40782 cdlemi1 40820 cdlemk11 40851 cdlemk5u 40863 cdlemk11u 40873 cdlemk52 40956 dialss 41048 diaglbN 41057 diaintclN 41060 dia2dimlem1 41066 cdlemm10N 41120 djajN 41139 dibglbN 41168 dibintclN 41169 diblss 41172 cdlemn10 41208 dihord1 41220 dihord2pre2 41228 dihopelvalcpre 41250 dihord5apre 41264 dihmeetlem1N 41292 dihglblem2N 41296 dihmeetlem2N 41301 dihglbcpreN 41302 dihmeetlem3N 41307 |
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