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Theorem toslub 31882
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 2733 . . . 4 (leβ€˜πΎ) = (leβ€˜πΎ)
61, 2, 3, 4, 5toslublem 31881 . . 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 7334 . 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 2733 . . 3 (lubβ€˜πΎ) = (lubβ€˜πΎ)
9 biid 261 . . 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 18250 . 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 18313 . . . . 5 (𝐾 ∈ Toset β†’ (𝐾 ∈ Toset ↔ ( < Or 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† (leβ€˜πΎ))))
1211ibi 267 . . . 4 (𝐾 ∈ Toset β†’ ( < Or 𝐡 ∧ ( I β†Ύ 𝐡) βŠ† (leβ€˜πΎ)))
1312simpld 496 . . 3 (𝐾 ∈ Toset β†’ < Or 𝐡)
14 id 22 . . . 4 ( < Or 𝐡 β†’ < Or 𝐡)
1514supval2 9396 . . 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 2783 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3911   class class class wbr 5106   I cid 5531   Or wor 5545   β†Ύ cres 5636  β€˜cfv 6497  β„©crio 7313  supcsup 9381  Basecbs 17088  lecple 17145  ltcplt 18202  lubclub 18203  Tosetctos 18310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-po 5546  df-so 5547  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-sup 9383  df-proset 18189  df-poset 18207  df-plt 18224  df-lub 18240  df-toset 18311
This theorem is referenced by:  xrsp1  31922
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