Theorem poslubd 18471

Description: Properties which determine the least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Hypotheses

Ref	Expression
poslubd.l	⊢ ≤ = (le‘𝐾)
poslubd.b	⊢ 𝐵 = (Base‘𝐾)
poslubd.u	⊢ 𝑈 = (lub‘𝐾)
poslubd.k	⊢ (𝜑 → 𝐾 ∈ Poset)
poslubd.s	⊢ (𝜑 → 𝑆 ⊆ 𝐵)
poslubd.t	⊢ (𝜑 → 𝑇 ∈ 𝐵)
poslubd.ub	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ≤ 𝑇)
poslubd.le	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑇 ≤ 𝑦)

Assertion

Ref	Expression
poslubd	⊢ (𝜑 → (𝑈‘𝑆) = 𝑇)

Distinct variable groups:   𝑥, ≤ ,𝑦   𝑥,𝐵,𝑦   𝑥,𝐾,𝑦   𝑥,𝑆,𝑦   𝑥,𝑈,𝑦   𝑥,𝑇,𝑦   𝜑,𝑥,𝑦

Proof of Theorem poslubd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		poslubd.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		poslubd.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		poslubd.u	. . 3 ⊢ 𝑈 = (lub‘𝐾)
4		biid 261	. . 3 ⊢ ((∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)) ↔ (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)))
5		poslubd.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ Poset)
6		poslubd.s	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
7	1, 2, 3, 4, 5, 6	lubval 18414	. 2 ⊢ (𝜑 → (𝑈‘𝑆) = (℩𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦))))
8		poslubd.ub	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ≤ 𝑇)
9	8	ralrimiva 3144	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑇)
10		poslubd.le	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑇 ≤ 𝑦)
11	10	3expia 1120	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑇 ≤ 𝑦))
12	11	ralrimiva 3144	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑇 ≤ 𝑦))
13	9, 12	jca 511	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑇 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑇 ≤ 𝑦)))
14		poslubd.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝐵)
15		breq2 5152	. . . . . . . . 9 ⊢ (𝑧 = 𝑇 → (𝑥 ≤ 𝑧 ↔ 𝑥 ≤ 𝑇))
16	15	ralbidv 3176	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ↔ ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑇))
17		breq1 5151	. . . . . . . . . 10 ⊢ (𝑧 = 𝑇 → (𝑧 ≤ 𝑦 ↔ 𝑇 ≤ 𝑦))
18	17	imbi2d 340	. . . . . . . . 9 ⊢ (𝑧 = 𝑇 → ((∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦) ↔ (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑇 ≤ 𝑦)))
19	18	ralbidv 3176	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦) ↔ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑇 ≤ 𝑦)))
20	16, 19	anbi12d 632	. . . . . . 7 ⊢ (𝑧 = 𝑇 → ((∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)) ↔ (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑇 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑇 ≤ 𝑦))))
21	20	rspcev 3622	. . . . . 6 ⊢ ((𝑇 ∈ 𝐵 ∧ (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑇 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑇 ≤ 𝑦))) → ∃𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)))
22	14, 13, 21	syl2anc 584	. . . . 5 ⊢ (𝜑 → ∃𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)))
23	2, 1	poslubmo 18469	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) → ∃*𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)))
24	5, 6, 23	syl2anc 584	. . . . 5 ⊢ (𝜑 → ∃*𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)))
25		reu5 3380	. . . . 5 ⊢ (∃!𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)) ↔ (∃𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)) ∧ ∃*𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦))))
26	22, 24, 25	sylanbrc 583	. . . 4 ⊢ (𝜑 → ∃!𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)))
27	20	riota2 7413	. . . 4 ⊢ ((𝑇 ∈ 𝐵 ∧ ∃!𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦))) → ((∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑇 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑇 ≤ 𝑦)) ↔ (℩𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦))) = 𝑇))
28	14, 26, 27	syl2anc 584	. . 3 ⊢ (𝜑 → ((∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑇 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑇 ≤ 𝑦)) ↔ (℩𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦))) = 𝑇))
29	13, 28	mpbid 232	. 2 ⊢ (𝜑 → (℩𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦))) = 𝑇)
30	7, 29	eqtrd 2775	1 ⊢ (𝜑 → (𝑈‘𝑆) = 𝑇)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  ∃!wreu 3376  ∃*wrmo 3377   ⊆ wss 3963   class class class wbr 5148  ‘cfv 6563  ℩crio 7387  Basecbs 17245  lecple 17305  Posetcpo 18365  lubclub 18367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-proset 18352  df-poset 18371  df-lub 18404

This theorem is referenced by:  poslubdg  18472  lubsscl  48757