Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6667 | . 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 17752 | . 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 1533 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3936 class class class wbr 5059 ‘cfv 6350 Basecbs 16477 lecple 16566 Posetcpo 17544 lubclub 17546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-proset 17532 df-poset 17550 df-lub 17578 |
This theorem is referenced by: posglbd 17754 mrelatlub 17790 |
Copyright terms: Public domain | W3C validator |