Theorem poslubd 18398

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 18341	. 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 3142	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑇)
10		poslubd.le	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑇 ≤ 𝑦)
11	10	3expia 1119	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑇 ≤ 𝑦))
12	11	ralrimiva 3142	. . . 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 5146	. . . . . . . . 9 ⊢ (𝑧 = 𝑇 → (𝑥 ≤ 𝑧 ↔ 𝑥 ≤ 𝑇))
16	15	ralbidv 3173	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ↔ ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑇))
17		breq1 5145	. . . . . . . . . 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 3173	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦) ↔ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑇 ≤ 𝑦)))
20	16, 19	anbi12d 631	. . . . . . 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 3608	. . . . . 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 583	. . . . 5 ⊢ (𝜑 → ∃𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)))
23	2, 1	poslubmo 18396	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) → ∃*𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)))
24	5, 6, 23	syl2anc 583	. . . . 5 ⊢ (𝜑 → ∃*𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)))
25		reu5 3374	. . . . 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 582	. . . 4 ⊢ (𝜑 → ∃!𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)))
27	20	riota2 7396	. . . 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 583	. . 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 231	. 2 ⊢ (𝜑 → (℩𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦))) = 𝑇)
30	7, 29	eqtrd 2768	1 ⊢ (𝜑 → (𝑈‘𝑆) = 𝑇)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3057  ∃wrex 3066  ∃!wreu 3370  ∃*wrmo 3371   ⊆ wss 3945   class class class wbr 5142  ‘cfv 6542  ℩crio 7369  Basecbs 17173  lecple 17233  Posetcpo 18292  lubclub 18294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-proset 18280  df-poset 18298  df-lub 18331

This theorem is referenced by:  poslubdg  18399  lubsscl  47973