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Mirrors > Home > MPE Home > Th. List > Mathboxes > lubeldm2 | Structured version Visualization version GIF version |
Description: Member of the domain of the least upper bound function of a poset. (Contributed by Zhi Wang, 26-Sep-2024.) |
Ref | Expression |
---|---|
lubeldm2.b | ⊢ 𝐵 = (Base‘𝐾) |
lubeldm2.l | ⊢ ≤ = (le‘𝐾) |
lubeldm2.u | ⊢ 𝑈 = (lub‘𝐾) |
lubeldm2.p | ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) |
lubeldm2.k | ⊢ (𝜑 → 𝐾 ∈ Poset) |
Ref | Expression |
---|---|
lubeldm2 | ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lubeldm2.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | lubeldm2.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
3 | lubeldm2.u | . . . . 5 ⊢ 𝑈 = (lub‘𝐾) | |
4 | lubeldm2.p | . . . . 5 ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) | |
5 | lubeldm2.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Poset) | |
6 | 1, 2, 3, 4, 5 | lubeldm 18411 | . . . 4 ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓))) |
7 | 6 | biimpa 476 | . . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) |
8 | reurex 3382 | . . . 4 ⊢ (∃!𝑥 ∈ 𝐵 𝜓 → ∃𝑥 ∈ 𝐵 𝜓) | |
9 | 8 | anim2i 617 | . . 3 ⊢ ((𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓) → (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)) |
10 | 7, 9 | syl 17 | . 2 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝑈) → (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)) |
11 | simpl 482 | . . 3 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)) → 𝜑) | |
12 | simprl 771 | . . 3 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)) → 𝑆 ⊆ 𝐵) | |
13 | 2, 1 | poslubmo 18469 | . . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) → ∃*𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) |
14 | 5, 13 | sylan 580 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ⊆ 𝐵) → ∃*𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) |
15 | 4 | rmobii 3386 | . . . . . . 7 ⊢ (∃*𝑥 ∈ 𝐵 𝜓 ↔ ∃*𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) |
16 | 14, 15 | sylibr 234 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑆 ⊆ 𝐵) → ∃*𝑥 ∈ 𝐵 𝜓) |
17 | 16 | anim1ci 616 | . . . . 5 ⊢ (((𝜑 ∧ 𝑆 ⊆ 𝐵) ∧ ∃𝑥 ∈ 𝐵 𝜓) → (∃𝑥 ∈ 𝐵 𝜓 ∧ ∃*𝑥 ∈ 𝐵 𝜓)) |
18 | reu5 3380 | . . . . 5 ⊢ (∃!𝑥 ∈ 𝐵 𝜓 ↔ (∃𝑥 ∈ 𝐵 𝜓 ∧ ∃*𝑥 ∈ 𝐵 𝜓)) | |
19 | 17, 18 | sylibr 234 | . . . 4 ⊢ (((𝜑 ∧ 𝑆 ⊆ 𝐵) ∧ ∃𝑥 ∈ 𝐵 𝜓) → ∃!𝑥 ∈ 𝐵 𝜓) |
20 | 19 | anasss 466 | . . 3 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)) → ∃!𝑥 ∈ 𝐵 𝜓) |
21 | 6 | biimpar 477 | . . 3 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓)) → 𝑆 ∈ dom 𝑈) |
22 | 11, 12, 20, 21 | syl12anc 837 | . 2 ⊢ ((𝜑 ∧ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)) → 𝑆 ∈ dom 𝑈) |
23 | 10, 22 | impbida 801 | 1 ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ∃!wreu 3376 ∃*wrmo 3377 ⊆ wss 3963 class class class wbr 5148 dom cdm 5689 ‘cfv 6563 Basecbs 17245 lecple 17305 Posetcpo 18365 lubclub 18367 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-proset 18352 df-poset 18371 df-lub 18404 |
This theorem is referenced by: lubeldm2d 48755 lubsscl 48757 ipolub00 48782 |
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