Theorem toslub 28999

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.)

Hypotheses

Ref	Expression
toslub.b	⊢ 𝐵 = (Base‘𝐾)
toslub.l	⊢ < = (lt‘𝐾)
toslub.1	⊢ (𝜑 → 𝐾 ∈ Toset)
toslub.2	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
toslub	⊢ (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))

Proof of Theorem toslub

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

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 2610	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
6	1, 2, 3, 4, 5	toslublem 28998	. . 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 6527	. 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 2610	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
9		biid 250	. . 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 16807	. 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 16859	. . . . 5 ⊢ (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))))
12	11	ibi 255	. . . 4 ⊢ (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))
13	12	simpld 474	. . 3 ⊢ (𝐾 ∈ Toset → < Or 𝐵)
14		id 22	. . . 4 ⊢ ( < Or 𝐵 → < Or 𝐵)
15	14	supval2 8244	. . 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 2654	1 ⊢ (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540   class class class wbr 4583   I cid 4948   Or wor 4958   ↾ cres 5040  ‘cfv 5804  ℩crio 6510  supcsup 8229  Basecbs 15695  lecple 15775  ltcplt 16764  lubclub 16765  Tosetctos 16856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-sup 8231  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-toset 16857

This theorem is referenced by:  xrsp1  29013