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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 2738 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
6 | 1, 2, 3, 4, 5 | toslublem 30827 | . . 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 7147 | . 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 2738 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
9 | biid 264 | . . 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 17710 | . 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 17762 | . . . . 5 ⊢ (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))) |
12 | 11 | ibi 270 | . . . 4 ⊢ (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))) |
13 | 12 | simpld 498 | . . 3 ⊢ (𝐾 ∈ Toset → < Or 𝐵) |
14 | id 22 | . . . 4 ⊢ ( < Or 𝐵 → < Or 𝐵) | |
15 | 14 | supval2 8992 | . . 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 2783 | 1 ⊢ (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∀wral 3053 ∃wrex 3054 ⊆ wss 3843 class class class wbr 5030 I cid 5428 Or wor 5441 ↾ cres 5527 ‘cfv 6339 ℩crio 7126 supcsup 8977 Basecbs 16586 lecple 16675 ltcplt 17667 lubclub 17668 Tosetctos 17759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-po 5442 df-so 5443 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-sup 8979 df-proset 17654 df-poset 17672 df-plt 17684 df-lub 17700 df-toset 17760 |
This theorem is referenced by: xrsp1 30868 |
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