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Mirrors > Home > MPE Home > Th. List > clatglble | Structured version Visualization version GIF version |
Description: The greatest lower bound is the least element. (Contributed by NM, 5-Dec-2011.) |
Ref | Expression |
---|---|
clatglb.b | ⊢ 𝐵 = (Base‘𝐾) |
clatglb.l | ⊢ ≤ = (le‘𝐾) |
clatglb.g | ⊢ 𝐺 = (glb‘𝐾) |
Ref | Expression |
---|---|
clatglble | ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑆) ≤ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clatglb.b | . 2 ⊢ 𝐵 = (Base‘𝐾) | |
2 | clatglb.l | . 2 ⊢ ≤ = (le‘𝐾) | |
3 | clatglb.g | . 2 ⊢ 𝐺 = (glb‘𝐾) | |
4 | simp1 1132 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝐾 ∈ CLat) | |
5 | 1, 3 | clatglbcl2 17725 | . . 3 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺) |
6 | 5 | 3adant3 1128 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑆 ∈ dom 𝐺) |
7 | simp3 1134 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
8 | 1, 2, 3, 4, 6, 7 | glble 17610 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑆) ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 class class class wbr 5066 dom cdm 5555 ‘cfv 6355 Basecbs 16483 lecple 16572 glbcglb 17553 CLatccla 17717 |
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 |
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-glb 17585 df-clat 17718 |
This theorem is referenced by: clatleglb 17736 clatglbss 17737 diaglbN 38206 diaintclN 38209 dibglbN 38317 dibintclN 38318 dihglblem2N 38445 dihglblem4 38448 dihglbcpreN 38451 dochvalr 38508 |
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