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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 18411 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋)) |
4 | 3 | 3adant3r3 1185 | . 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 18426 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∧ 𝑌) ≤ (𝑋 ∨ 𝑍)) |
8 | 4, 7 | eqbrtrrd 5170 | 1 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌 ∧ 𝑋) ≤ (𝑋 ∨ 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 class class class wbr 5146 ‘cfv 6539 (class class class)co 7403 Basecbs 17139 lecple 17199 joincjn 18259 meetcmee 18260 Latclat 18379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-poset 18261 df-lub 18294 df-glb 18295 df-join 18296 df-meet 18297 df-lat 18380 |
This theorem is referenced by: dalawlem3 38681 dalawlem6 38684 |
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