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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 1136 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝐾 ∈ CLat) | |
5 | 1, 3 | clatglbcl2 18441 | . . 3 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺) |
6 | 5 | 3adant3 1132 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑆 ∈ dom 𝐺) |
7 | simp3 1138 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
8 | 1, 2, 3, 4, 6, 7 | glble 18307 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑆) ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ⊆ wss 3944 class class class wbr 5141 dom cdm 5669 ‘cfv 6532 Basecbs 17126 lecple 17186 glbcglb 18245 CLatccla 18433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-glb 18282 df-clat 18434 |
This theorem is referenced by: clatleglb 18453 clatglbss 18454 diaglbN 39729 diaintclN 39732 dibglbN 39840 dibintclN 39841 dihglblem2N 39968 dihglblem4 39971 dihglbcpreN 39974 dochvalr 40031 |
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