Theorem poslubd 17148

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 251	. . 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 16984	. 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 2966	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑇)
10		poslubd.le	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑇 ≤ 𝑦)
11	10	3expia 1267	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑇 ≤ 𝑦))
12	11	ralrimiva 2966	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑇 ≤ 𝑦))
13	9, 12	jca 554	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑇 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑇 ≤ 𝑦)))
14		poslubd.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝐵)
15		breq2 4657	. . . . . . . . 9 ⊢ (𝑧 = 𝑇 → (𝑥 ≤ 𝑧 ↔ 𝑥 ≤ 𝑇))
16	15	ralbidv 2986	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ↔ ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑇))
17		breq1 4656	. . . . . . . . . 10 ⊢ (𝑧 = 𝑇 → (𝑧 ≤ 𝑦 ↔ 𝑇 ≤ 𝑦))
18	17	imbi2d 330	. . . . . . . . 9 ⊢ (𝑧 = 𝑇 → ((∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦) ↔ (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑇 ≤ 𝑦)))
19	18	ralbidv 2986	. . . . . . . 8 ⊢ (𝑧 = 𝑇 → (∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦) ↔ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑇 ≤ 𝑦)))
20	16, 19	anbi12d 747	. . . . . . 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 3309	. . . . . 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 693	. . . . 5 ⊢ (𝜑 → ∃𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)))
23	2, 1	poslubmo 17146	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) → ∃*𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)))
24	5, 6, 23	syl2anc 693	. . . . 5 ⊢ (𝜑 → ∃*𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)))
25		reu5 3159	. . . . 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 698	. . . 4 ⊢ (𝜑 → ∃!𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦)))
27	20	riota2 6633	. . . 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 693	. . 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 222	. 2 ⊢ (𝜑 → (℩𝑧 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑧 ∧ ∀𝑦 ∈ 𝐵 (∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑧 ≤ 𝑦))) = 𝑇)
30	7, 29	eqtrd 2656	1 ⊢ (𝜑 → (𝑈‘𝑆) = 𝑇)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  ∃!wreu 2914  ∃*wrmo 2915   ⊆ wss 3574   class class class wbr 4653  ‘cfv 5888  ℩crio 6610  Basecbs 15857  lecple 15948  Posetcpo 16940  lubclub 16942

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-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-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-preset 16928  df-poset 16946  df-lub 16974

This theorem is referenced by:  poslubdg  17149