Theorem toslub 33055

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 2736	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
6	1, 2, 3, 4, 5	toslublem 33054	. . 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 7334	. 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 2736	. . 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 18277	. 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 18340	. . . . 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 9358	. . 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 2781	1 ⊢ (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060   ⊆ wss 3901   class class class wbr 5098   I cid 5518   Or wor 5531   ↾ cres 5626  ‘cfv 6492  ℩crio 7314  supcsup 9343  Basecbs 17136  lecple 17184  ltcplt 18231  lubclub 18232  Tosetctos 18337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-sup 9345  df-proset 18217  df-poset 18236  df-plt 18251  df-lub 18267  df-toset 18338

This theorem is referenced by:  xrsp1  33095