Theorem glbprop 16999

Description: Properties of greatest lower bound of a poset. (Contributed by NM, 7-Sep-2018.)

Hypotheses

Ref	Expression
glbprop.b	⊢ 𝐵 = (Base‘𝐾)
glbprop.l	⊢ ≤ = (le‘𝐾)
glbprop.u	⊢ 𝑈 = (glb‘𝐾)
glbprop.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
glbprop.s	⊢ (𝜑 → 𝑆 ∈ dom 𝑈)

Assertion

Ref	Expression
glbprop	⊢ (𝜑 → (∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝑈‘𝑆))))

Distinct variable groups:   𝑧,𝐵   𝑦,𝑧,𝐾   𝑦,𝑆,𝑧   𝑦, ≤   𝑦,𝑈,𝑧

Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐵(𝑦)   ≤ (𝑧)   𝑉(𝑦,𝑧)

Proof of Theorem glbprop

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		glbprop.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		glbprop.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		glbprop.u	. . . 4 ⊢ 𝑈 = (glb‘𝐾)
4		biid 251	. . . 4 ⊢ ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
5		glbprop.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
6		glbprop.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ dom 𝑈)
7	1, 2, 3, 5, 6	glbelss 16995	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
8	1, 2, 3, 4, 5, 7	glbval 16997	. . 3 ⊢ (𝜑 → (𝑈‘𝑆) = (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
9	8	eqcomd 2628	. 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) = (𝑈‘𝑆))
10	1, 3, 5, 6	glbcl 16998	. . 3 ⊢ (𝜑 → (𝑈‘𝑆) ∈ 𝐵)
11	1, 2, 3, 4, 5, 6	glbeu 16996	. . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
12		breq1 4656	. . . . . 6 ⊢ (𝑥 = (𝑈‘𝑆) → (𝑥 ≤ 𝑦 ↔ (𝑈‘𝑆) ≤ 𝑦))
13	12	ralbidv 2986	. . . . 5 ⊢ (𝑥 = (𝑈‘𝑆) → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦))
14		breq2 4657	. . . . . . 7 ⊢ (𝑥 = (𝑈‘𝑆) → (𝑧 ≤ 𝑥 ↔ 𝑧 ≤ (𝑈‘𝑆)))
15	14	imbi2d 330	. . . . . 6 ⊢ (𝑥 = (𝑈‘𝑆) → ((∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝑈‘𝑆))))
16	15	ralbidv 2986	. . . . 5 ⊢ (𝑥 = (𝑈‘𝑆) → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝑈‘𝑆))))
17	13, 16	anbi12d 747	. . . 4 ⊢ (𝑥 = (𝑈‘𝑆) → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ (∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝑈‘𝑆)))))
18	17	riota2 6633	. . 3 ⊢ (((𝑈‘𝑆) ∈ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) → ((∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝑈‘𝑆))) ↔ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) = (𝑈‘𝑆)))
19	10, 11, 18	syl2anc 693	. 2 ⊢ (𝜑 → ((∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝑈‘𝑆))) ↔ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) = (𝑈‘𝑆)))
20	9, 19	mpbird 247	1 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝑈‘𝑆))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃!wreu 2914   class class class wbr 4653  dom cdm 5114  ‘cfv 5888  ℩crio 6610  Basecbs 15857  lecple 15948  glbcglb 16943

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-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-glb 16975

This theorem is referenced by:  glble  17000  clatglb  17124