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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 6672 | . 2 ⊢ (𝜑 → (𝑈‘𝑆) = ((lub‘𝐾)‘𝑆)) |
3 | poslubdg.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
4 | eqid 2821 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
5 | eqid 2821 | . . 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 4007 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐾)) |
10 | poslubdg.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝐵) | |
11 | 10, 8 | eleqtrd 2915 | . . 3 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾)) |
12 | poslubdg.ub | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ≤ 𝑇) | |
13 | 8 | eleq2d 2898 | . . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (Base‘𝐾))) |
14 | 13 | biimpar 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑦 ∈ 𝐵) |
15 | 14 | 3adant3 1128 | . . . 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 1407 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘𝐾) ∧ ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑇 ≤ 𝑦) |
18 | 3, 4, 5, 6, 9, 11, 12, 17 | poslubd 17758 | . 2 ⊢ (𝜑 → ((lub‘𝐾)‘𝑆) = 𝑇) |
19 | 2, 18 | eqtrd 2856 | 1 ⊢ (𝜑 → (𝑈‘𝑆) = 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 class class class wbr 5066 ‘cfv 6355 Basecbs 16483 lecple 16572 Posetcpo 17550 lubclub 17552 |
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-rmo 3146 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-proset 17538 df-poset 17556 df-lub 17584 |
This theorem is referenced by: posglbd 17760 mrelatlub 17796 |
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