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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 | breq2 5073 | . 2 ⊢ (𝑦 = 𝑋 → ((𝑈‘𝑆) ≤ 𝑦 ↔ (𝑈‘𝑆) ≤ 𝑋)) | |
2 | glbprop.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
3 | glbprop.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
4 | glbprop.u | . . . 4 ⊢ 𝑈 = (glb‘𝐾) | |
5 | glbprop.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
6 | glbprop.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ dom 𝑈) | |
7 | 2, 3, 4, 5, 6 | glbprop 17612 | . . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝑈‘𝑆)))) |
8 | 7 | simpld 497 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦) |
9 | glble.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
10 | 1, 8, 9 | rspcdva 3628 | 1 ⊢ (𝜑 → (𝑈‘𝑆) ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∀wral 3141 class class class wbr 5069 dom cdm 5558 ‘cfv 6358 Basecbs 16486 lecple 16575 glbcglb 17556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-glb 17588 |
This theorem is referenced by: p0le 17656 clatglble 17738 |
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