Theorem toslub 29668

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 2622	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
6	1, 2, 3, 4, 5	toslublem 29667	. . 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 6627	. 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 2622	. . 3 ⊢ (lub‘𝐾) = (lub‘𝐾)
9		biid 251	. . 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 16984	. 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 17036	. . . . 5 ⊢ (𝐾 ∈ Toset → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾))))
12	11	ibi 256	. . . 4 ⊢ (𝐾 ∈ Toset → ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ (le‘𝐾)))
13	12	simpld 475	. . 3 ⊢ (𝐾 ∈ Toset → < Or 𝐵)
14		id 22	. . . 4 ⊢ ( < Or 𝐵 → < Or 𝐵)
15	14	supval2 8361	. . 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 2666	1 ⊢ (𝜑 → ((lub‘𝐾)‘𝐴) = sup(𝐴, 𝐵, < ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574   class class class wbr 4653   I cid 5023   Or wor 5034   ↾ cres 5116  ‘cfv 5888  ℩crio 6610  supcsup 8346  Basecbs 15857  lecple 15948  ltcplt 16941  lubclub 16942  Tosetctos 17033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-sup 8348  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-toset 17034

This theorem is referenced by:  xrsp1  29682