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Mirrors > Home > MPE Home > Th. List > latmlej21 | Structured version Visualization version GIF version |
Description: Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.) |
Ref | Expression |
---|---|
latledi.b | ⊢ 𝐵 = (Base‘𝐾) |
latledi.l | ⊢ ≤ = (le‘𝐾) |
latledi.j | ⊢ ∨ = (join‘𝐾) |
latledi.m | ⊢ ∧ = (meet‘𝐾) |
Ref | Expression |
---|---|
latmlej21 | ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌 ∧ 𝑋) ≤ (𝑋 ∨ 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | latledi.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | latledi.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
3 | 1, 2 | latmcom 17679 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) |
4 | 3 | 3adant3r3 1180 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) |
5 | latledi.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
6 | latledi.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
7 | 1, 5, 6, 2 | latmlej11 17694 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∧ 𝑌) ≤ (𝑋 ∨ 𝑍)) |
8 | 4, 7 | eqbrtrrd 5083 | 1 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌 ∧ 𝑋) ≤ (𝑋 ∨ 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 lecple 16566 joincjn 17548 meetcmee 17549 Latclat 17649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-poset 17550 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-lat 17650 |
This theorem is referenced by: dalawlem3 37003 dalawlem6 37006 |
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