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Mirrors > Home > MPE Home > Th. List > poslubdg | Structured version Visualization version GIF version |
Description: Properties which determine the least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
Ref | Expression |
---|---|
poslubdg.l | ⊢ ≤ = (le‘𝐾) |
poslubdg.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
poslubdg.u | ⊢ (𝜑 → 𝑈 = (lub‘𝐾)) |
poslubdg.k | ⊢ (𝜑 → 𝐾 ∈ Poset) |
poslubdg.s | ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
poslubdg.t | ⊢ (𝜑 → 𝑇 ∈ 𝐵) |
poslubdg.ub | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ≤ 𝑇) |
poslubdg.le | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑇 ≤ 𝑦) |
Ref | Expression |
---|---|
poslubdg | ⊢ (𝜑 → (𝑈‘𝑆) = 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | poslubdg.u | . . 3 ⊢ (𝜑 → 𝑈 = (lub‘𝐾)) | |
2 | 1 | fveq1d 6105 | . 2 ⊢ (𝜑 → (𝑈‘𝑆) = ((lub‘𝐾)‘𝑆)) |
3 | poslubdg.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
4 | eqid 2610 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
5 | eqid 2610 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
6 | poslubdg.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ Poset) | |
7 | poslubdg.s | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | |
8 | poslubdg.b | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
9 | 7, 8 | sseqtrd 3604 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐾)) |
10 | poslubdg.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝐵) | |
11 | 10, 8 | eleqtrd 2690 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾)) |
12 | poslubdg.ub | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ≤ 𝑇) | |
13 | 8 | eleq2d 2673 | . . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐾))) |
14 | 13 | biimpar 501 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ 𝐵) |
15 | 14 | 3adant3 1074 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾) ∧ ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑦 ∈ 𝐵) |
16 | poslubdg.le | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑇 ≤ 𝑦) | |
17 | 15, 16 | syld3an2 1365 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾) ∧ ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑇 ≤ 𝑦) |
18 | 3, 4, 5, 6, 9, 11, 12, 17 | poslubd 16971 | . 2 ⊢ (𝜑 → ((lub‘𝐾)‘𝑆) = 𝑇) |
19 | 2, 18 | eqtrd 2644 | 1 ⊢ (𝜑 → (𝑈‘𝑆) = 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 lecple 15775 Posetcpo 16763 lubclub 16765 |
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-rmo 2904 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-preset 16751 df-poset 16769 df-lub 16797 |
This theorem is referenced by: posglbd 16973 mrelatlub 17009 |
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