Theorem glbprop 18413

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 264	. . . 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 18409	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
8	1, 2, 3, 4, 5, 7	glbval 18411	. . 3 ⊢ (𝜑 → (𝑈‘𝑆) = (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
9	8	eqcomd 2771	. 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) = (𝑈‘𝑆))
10	1, 3, 5, 6	glbcl 18412	. . 3 ⊢ (𝜑 → (𝑈‘𝑆) ∈ 𝐵)
11	1, 2, 3, 4, 5, 6	glbeu 18410	. . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
12		breq1 5107	. . . . . 6 ⊢ (𝑥 = (𝑈‘𝑆) → (𝑥 ≤ 𝑦 ↔ (𝑈‘𝑆) ≤ 𝑦))
13	12	ralbidv 3188	. . . . 5 ⊢ (𝑥 = (𝑈‘𝑆) → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦))
14		breq2 5108	. . . . . . 7 ⊢ (𝑥 = (𝑈‘𝑆) → (𝑧 ≤ 𝑥 ↔ 𝑧 ≤ (𝑈‘𝑆)))
15	14	imbi2d 343	. . . . . 6 ⊢ (𝑥 = (𝑈‘𝑆) → ((∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝑈‘𝑆))))
16	15	ralbidv 3188	. . . . 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 643	. . . 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 7382	. . 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 595	. 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 260	1 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝑈‘𝑆))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃!wreu 3368   class class class wbr 5104  dom cdm 5651  ‘cfv 6525  ℩crio 7356  Basecbs 17257  lecple 17305  glbcglb 18354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-glb 18389

This theorem is referenced by:  glble  18414  clatglb  18560