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Theorem toslub 32408
Description: In a toset, the lowest upper bound lub, defined for partial orders is the supremum, sup(𝐴, 𝐡, < ), 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.)
Hypotheses
Ref Expression
toslub.b 𝐡 = (Baseβ€˜πΎ)
toslub.l < = (ltβ€˜πΎ)
toslub.1 (πœ‘ β†’ 𝐾 ∈ Toset)
toslub.2 (πœ‘ β†’ 𝐴 βŠ† 𝐡)
Assertion
Ref Expression
toslub (πœ‘ β†’ ((lubβ€˜πΎ)β€˜π΄) = sup(𝐴, 𝐡, < ))

Proof of Theorem toslub
Dummy variables π‘Ž 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 toslub.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 toslub.l . . . 4 < = (ltβ€˜πΎ)
3 toslub.1 . . . 4 (πœ‘ β†’ 𝐾 ∈ Toset)
4 toslub.2 . . . 4 (πœ‘ β†’ 𝐴 βŠ† 𝐡)
5 eqid 2730 . . . 4 (leβ€˜πΎ) = (leβ€˜πΎ)
61, 2, 3, 4, 5toslublem 32407 . . 3 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ ((βˆ€π‘ ∈ 𝐴 𝑏(leβ€˜πΎ)π‘Ž ∧ βˆ€π‘ ∈ 𝐡 (βˆ€π‘ ∈ 𝐴 𝑏(leβ€˜πΎ)𝑐 β†’ π‘Ž(leβ€˜πΎ)𝑐)) ↔ (βˆ€π‘ ∈ 𝐴 Β¬ π‘Ž < 𝑏 ∧ βˆ€π‘ ∈ 𝐡 (𝑏 < π‘Ž β†’ βˆƒπ‘‘ ∈ 𝐴 𝑏 < 𝑑))))
76riotabidva 7389 . 2 (πœ‘ β†’ (β„©π‘Ž ∈ 𝐡 (βˆ€π‘ ∈ 𝐴 𝑏(leβ€˜πΎ)π‘Ž ∧ βˆ€π‘ ∈ 𝐡 (βˆ€π‘ ∈ 𝐴 𝑏(leβ€˜πΎ)𝑐 β†’ π‘Ž(leβ€˜πΎ)𝑐))) = (β„©π‘Ž ∈ 𝐡 (βˆ€π‘ ∈ 𝐴 Β¬ π‘Ž < 𝑏 ∧ βˆ€π‘ ∈ 𝐡 (𝑏 < π‘Ž β†’ βˆƒπ‘‘ ∈ 𝐴 𝑏 < 𝑑))))
8 eqid 2730 . . 3 (lubβ€˜πΎ) = (lubβ€˜πΎ)
9 biid 260 . . 3 ((βˆ€π‘ ∈ 𝐴 𝑏(leβ€˜πΎ)π‘Ž ∧ βˆ€π‘ ∈ 𝐡 (βˆ€π‘ ∈ 𝐴 𝑏(leβ€˜πΎ)𝑐 β†’ π‘Ž(leβ€˜πΎ)𝑐)) ↔ (βˆ€π‘ ∈ 𝐴 𝑏(leβ€˜πΎ)π‘Ž ∧ βˆ€π‘ ∈ 𝐡 (βˆ€π‘ ∈ 𝐴 𝑏(leβ€˜πΎ)𝑐 β†’ π‘Ž(leβ€˜πΎ)𝑐)))
101, 5, 8, 9, 3, 4lubval 18315 . 2 (πœ‘ β†’ ((lubβ€˜πΎ)β€˜π΄) = (β„©π‘Ž ∈ 𝐡 (βˆ€π‘ ∈ 𝐴 𝑏(leβ€˜πΎ)π‘Ž ∧ βˆ€π‘ ∈ 𝐡 (βˆ€π‘ ∈ 𝐴 𝑏(leβ€˜πΎ)𝑐 β†’ π‘Ž(leβ€˜πΎ)𝑐))))
111, 5, 2tosso 18378 . . . . 5 (𝐾 ∈ Toset β†’ (𝐾 ∈ Toset ↔ ( < Or 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† (leβ€˜πΎ))))
1211ibi 266 . . . 4 (𝐾 ∈ Toset β†’ ( < Or 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† (leβ€˜πΎ)))
1312simpld 493 . . 3 (𝐾 ∈ Toset β†’ < Or 𝐡)
14 id 22 . . . 4 ( < Or 𝐡 β†’ < Or 𝐡)
1514supval2 9454 . . 3 ( < Or 𝐡 β†’ sup(𝐴, 𝐡, < ) = (β„©π‘Ž ∈ 𝐡 (βˆ€π‘ ∈ 𝐴 Β¬ π‘Ž < 𝑏 ∧ βˆ€π‘ ∈ 𝐡 (𝑏 < π‘Ž β†’ βˆƒπ‘‘ ∈ 𝐴 𝑏 < 𝑑))))
163, 13, 153syl 18 . 2 (πœ‘ β†’ sup(𝐴, 𝐡, < ) = (β„©π‘Ž ∈ 𝐡 (βˆ€π‘ ∈ 𝐴 Β¬ π‘Ž < 𝑏 ∧ βˆ€π‘ ∈ 𝐡 (𝑏 < π‘Ž β†’ βˆƒπ‘‘ ∈ 𝐴 𝑏 < 𝑑))))
177, 10, 163eqtr4d 2780 1 (πœ‘ β†’ ((lubβ€˜πΎ)β€˜π΄) = sup(𝐴, 𝐡, < ))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068   βŠ† wss 3949   class class class wbr 5149   I cid 5574   Or wor 5588   β†Ύ cres 5679  β€˜cfv 6544  β„©crio 7368  supcsup 9439  Basecbs 17150  lecple 17210  ltcplt 18267  lubclub 18268  Tosetctos 18375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7369  df-sup 9441  df-proset 18254  df-poset 18272  df-plt 18289  df-lub 18305  df-toset 18376
This theorem is referenced by:  xrsp1  32448
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