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Mirrors > Home > MPE Home > Th. List > dlatjmdi | Structured version Visualization version GIF version |
Description: In a distributive lattice, joins distribute over meets. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
Ref | Expression |
---|---|
dlatjmdi.b | ⊢ 𝐵 = (Base‘𝐾) |
dlatjmdi.j | ⊢ ∨ = (join‘𝐾) |
dlatjmdi.m | ⊢ ∧ = (meet‘𝐾) |
Ref | Expression |
---|---|
dlatjmdi | ⊢ ((𝐾 ∈ DLat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∨ (𝑌 ∧ 𝑍)) = ((𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2819 | . . . 4 ⊢ (ODual‘𝐾) = (ODual‘𝐾) | |
2 | 1 | odudlatb 17798 | . . 3 ⊢ (𝐾 ∈ DLat → (𝐾 ∈ DLat ↔ (ODual‘𝐾) ∈ DLat)) |
3 | 2 | ibi 269 | . 2 ⊢ (𝐾 ∈ DLat → (ODual‘𝐾) ∈ DLat) |
4 | dlatjmdi.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
5 | 1, 4 | odubas 17735 | . . 3 ⊢ 𝐵 = (Base‘(ODual‘𝐾)) |
6 | dlatjmdi.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
7 | 1, 6 | odujoin 17744 | . . 3 ⊢ ∧ = (join‘(ODual‘𝐾)) |
8 | dlatjmdi.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
9 | 1, 8 | odumeet 17742 | . . 3 ⊢ ∨ = (meet‘(ODual‘𝐾)) |
10 | 5, 7, 9 | dlatmjdi 17796 | . 2 ⊢ (((ODual‘𝐾) ∈ DLat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∨ (𝑌 ∧ 𝑍)) = ((𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑍))) |
11 | 3, 10 | sylan 582 | 1 ⊢ ((𝐾 ∈ DLat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∨ (𝑌 ∧ 𝑍)) = ((𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 joincjn 17546 meetcmee 17547 ODualcodu 17730 DLatcdlat 17793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-dec 12091 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ple 16577 df-proset 17530 df-poset 17548 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-lat 17648 df-odu 17731 df-dlat 17794 |
This theorem is referenced by: (None) |
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