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Mirrors > Home > MPE Home > Th. List > glble | Structured version Visualization version GIF version |
Description: The greatest lower bound is the least element. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.) |
Ref | Expression |
---|---|
glbprop.b | ⊢ 𝐵 = (Base‘𝐾) |
glbprop.l | ⊢ ≤ = (le‘𝐾) |
glbprop.u | ⊢ 𝑈 = (glb‘𝐾) |
glbprop.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
glbprop.s | ⊢ (𝜑 → 𝑆 ∈ dom 𝑈) |
glble.x | ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
Ref | Expression |
---|---|
glble | ⊢ (𝜑 → (𝑈‘𝑆) ≤ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | glbprop.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | glbprop.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | glbprop.u | . . . 4 ⊢ 𝑈 = (glb‘𝐾) | |
4 | glbprop.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
5 | glbprop.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ dom 𝑈) | |
6 | 1, 2, 3, 4, 5 | glbprop 16822 | . . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝑈‘𝑆)))) |
7 | 6 | simpld 474 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦) |
8 | glble.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
9 | breq2 4587 | . . 3 ⊢ (𝑦 = 𝑋 → ((𝑈‘𝑆) ≤ 𝑦 ↔ (𝑈‘𝑆) ≤ 𝑋)) | |
10 | 9 | rspccva 3281 | . 2 ⊢ ((∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦 ∧ 𝑋 ∈ 𝑆) → (𝑈‘𝑆) ≤ 𝑋) |
11 | 7, 8, 10 | syl2anc 691 | 1 ⊢ (𝜑 → (𝑈‘𝑆) ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 Basecbs 15695 lecple 15775 glbcglb 16766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-glb 16798 |
This theorem is referenced by: p0le 16866 clatglble 16948 |
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