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Mirrors > Home > MPE Home > Th. List > Mathboxes > toslub | Structured version Visualization version GIF version |
Description: In a toset, the lowest upper bound lub, defined for partial orders is the supremum, sup(𝐴, 𝐵, < ), defined for total orders. (these are the set.mm definitions: lowest upper bound and supremum are normally synonymous). Note that those two values are also equal if such a supremum does not exist: in that case, both are equal to the empty set. (Contributed by Thierry Arnoux, 15-Feb-2018.) (Revised by Thierry Arnoux, 24-Sep-2018.) |
Ref | Expression |
---|---|
toslub.b | ⊢ 𝐵 = (Base‘𝐾) |
toslub.l | ⊢ < = (lt‘𝐾) |
toslub.1 | ⊢ (𝜑 → 𝐾 ∈ Toset) |
toslub.2 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
toslub | ⊢ (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toslub.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | toslub.l | . . . 4 ⊢ < = (lt‘𝐾) | |
3 | toslub.1 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Toset) | |
4 | toslub.2 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
5 | eqid 2735 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
6 | 1, 2, 3, 4, 5 | toslublem 32947 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((∀𝑏 ∈ 𝐴 𝑏(le‘𝐾)𝑎 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑏(le‘𝐾)𝑐 → 𝑎(le‘𝐾)𝑐)) ↔ (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏 < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏 < 𝑑)))) |
7 | 6 | riotabidva 7407 | . 2 ⊢ (𝜑 → (℩𝑎 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑏(le‘𝐾)𝑎 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑏(le‘𝐾)𝑐 → 𝑎(le‘𝐾)𝑐))) = (℩𝑎 ∈ 𝐵 (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏 < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏 < 𝑑)))) |
8 | eqid 2735 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
9 | biid 261 | . . 3 ⊢ ((∀𝑏 ∈ 𝐴 𝑏(le‘𝐾)𝑎 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑏(le‘𝐾)𝑐 → 𝑎(le‘𝐾)𝑐)) ↔ (∀𝑏 ∈ 𝐴 𝑏(le‘𝐾)𝑎 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑏(le‘𝐾)𝑐 → 𝑎(le‘𝐾)𝑐))) | |
10 | 1, 5, 8, 9, 3, 4 | lubval 18414 | . 2 ⊢ (𝜑 → ((lub‘𝐾)‘𝐴) = (℩𝑎 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑏(le‘𝐾)𝑎 ∧ ∀𝑐 ∈ 𝐵 (∀𝑏 ∈ 𝐴 𝑏(le‘𝐾)𝑐 → 𝑎(le‘𝐾)𝑐)))) |
11 | 1, 5, 2 | tosso 18477 | . . . . 5 ⊢ (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))) |
12 | 11 | ibi 267 | . . . 4 ⊢ (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))) |
13 | 12 | simpld 494 | . . 3 ⊢ (𝐾 ∈ Toset → < Or 𝐵) |
14 | id 22 | . . . 4 ⊢ ( < Or 𝐵 → < Or 𝐵) | |
15 | 14 | supval2 9493 | . . 3 ⊢ ( < Or 𝐵 → sup(𝐴, 𝐵, < ) = (℩𝑎 ∈ 𝐵 (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏 < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏 < 𝑑)))) |
16 | 3, 13, 15 | 3syl 18 | . 2 ⊢ (𝜑 → sup(𝐴, 𝐵, < ) = (℩𝑎 ∈ 𝐵 (∀𝑏 ∈ 𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ 𝐵 (𝑏 < 𝑎 → ∃𝑑 ∈ 𝐴 𝑏 < 𝑑)))) |
17 | 7, 10, 16 | 3eqtr4d 2785 | 1 ⊢ (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 class class class wbr 5148 I cid 5582 Or wor 5596 ↾ cres 5691 ‘cfv 6563 ℩crio 7387 supcsup 9478 Basecbs 17245 lecple 17305 ltcplt 18366 lubclub 18367 Tosetctos 18474 |
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-3or 1087 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-po 5597 df-so 5598 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-sup 9480 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-toset 18475 |
This theorem is referenced by: xrsp1 32998 |
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