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Mirrors > Home > MPE Home > Th. List > latleeqm1 | Structured version Visualization version GIF version |
Description: "Less than or equal to" in terms of meet. (Contributed by NM, 7-Nov-2011.) |
Ref | Expression |
---|---|
latmle.b | ⊢ 𝐵 = (Base‘𝐾) |
latmle.l | ⊢ ≤ = (le‘𝐾) |
latmle.m | ⊢ ∧ = (meet‘𝐾) |
Ref | Expression |
---|---|
latleeqm1 | ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑋 ∧ 𝑌) = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latmle.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐾) | |
2 | latmle.l | . . . . . . 7 ⊢ ≤ = (le‘𝐾) | |
3 | 1, 2 | latref 17663 | . . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋) |
4 | 3 | 3adant3 1128 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ≤ 𝑋) |
5 | 4 | biantrurd 535 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑋 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌))) |
6 | simp1 1132 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Lat) | |
7 | simp2 1133 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
8 | simp3 1134 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵) | |
9 | latmle.m | . . . . . 6 ⊢ ∧ = (meet‘𝐾) | |
10 | 1, 2, 9 | latlem12 17688 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌) ↔ 𝑋 ≤ (𝑋 ∧ 𝑌))) |
11 | 6, 7, 7, 8, 10 | syl13anc 1368 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑋 ∧ 𝑋 ≤ 𝑌) ↔ 𝑋 ≤ (𝑋 ∧ 𝑌))) |
12 | 5, 11 | bitrd 281 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ 𝑋 ≤ (𝑋 ∧ 𝑌))) |
13 | 1, 2, 9 | latmle1 17686 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑋) |
14 | 13 | biantrurd 535 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ (𝑋 ∧ 𝑌) ↔ ((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ 𝑋 ≤ (𝑋 ∧ 𝑌)))) |
15 | 12, 14 | bitrd 281 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ ((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ 𝑋 ≤ (𝑋 ∧ 𝑌)))) |
16 | latpos 17660 | . . . 4 ⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset) | |
17 | 16 | 3ad2ant1 1129 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ Poset) |
18 | 1, 9 | latmcl 17662 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵) |
19 | 1, 2 | posasymb 17562 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∧ 𝑌) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ 𝑋 ≤ (𝑋 ∧ 𝑌)) ↔ (𝑋 ∧ 𝑌) = 𝑋)) |
20 | 17, 18, 7, 19 | syl3anc 1367 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ 𝑋 ≤ (𝑋 ∧ 𝑌)) ↔ (𝑋 ∧ 𝑌) = 𝑋)) |
21 | 15, 20 | bitrd 281 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑋 ∧ 𝑌) = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 lecple 16572 Posetcpo 17550 meetcmee 17555 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-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-proset 17538 df-poset 17556 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-lat 17656 |
This theorem is referenced by: latleeqm2 17690 latnlemlt 17694 latabs2 17698 atnle 36468 2llnmat 36675 llnmlplnN 36690 dalem25 36849 2lnat 36935 lhpm0atN 37180 lhpmatb 37182 cdleme1 37378 cdleme5 37391 cdleme20d 37463 cdleme22e 37495 cdleme22eALTN 37496 cdleme23b 37501 cdleme32e 37596 doca2N 38277 djajN 38288 dihglblem5aN 38443 dihmeetbclemN 38455 |
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