Theorem isglbd 17503

Description: Properties that determine the greatest lower bound of a complete lattice. (Contributed by Mario Carneiro, 19-Mar-2014.)

Hypotheses

Ref	Expression
isglbd.b	⊢ 𝐵 = (Base‘𝐾)
isglbd.l	⊢ ≤ = (le‘𝐾)
isglbd.g	⊢ 𝐺 = (glb‘𝐾)
isglbd.1	⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝐻 ≤ 𝑦)
isglbd.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝐻)
isglbd.3	⊢ (𝜑 → 𝐾 ∈ CLat)
isglbd.4	⊢ (𝜑 → 𝑆 ⊆ 𝐵)
isglbd.5	⊢ (𝜑 → 𝐻 ∈ 𝐵)

Assertion

Ref	Expression
isglbd	⊢ (𝜑 → (𝐺‘𝑆) = 𝐻)

Distinct variable groups:   𝑥,𝐵   𝑥,𝑦,𝐻   𝑥,𝐾,𝑦   𝜑,𝑥,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐵(𝑦)   𝐺(𝑥,𝑦)   ≤ (𝑥,𝑦)

Proof of Theorem isglbd

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isglbd.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		isglbd.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		isglbd.g	. . 3 ⊢ 𝐺 = (glb‘𝐾)
4		biid 253	. . 3 ⊢ ((∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ)) ↔ (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ)))
5		isglbd.3	. . 3 ⊢ (𝜑 → 𝐾 ∈ CLat)
6		isglbd.4	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
7	1, 2, 3, 4, 5, 6	glbval 17383	. 2 ⊢ (𝜑 → (𝐺‘𝑆) = (℩ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ))))
8		isglbd.1	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝐻 ≤ 𝑦)
9	8	ralrimiva 3147	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 𝐻 ≤ 𝑦)
10		isglbd.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝐻)
11	10	3exp 1109	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻)))
12	11	ralrimiv 3146	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻))
13		isglbd.5	. . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐵)
14	1, 3	clatglbcl2 17501	. . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺)
15	5, 6, 14	syl2anc 579	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ dom 𝐺)
16	1, 2, 3, 4, 5, 15	glbeu 17382	. . . 4 ⊢ (𝜑 → ∃!ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ)))
17		breq1 4889	. . . . . . 7 ⊢ (ℎ = 𝐻 → (ℎ ≤ 𝑦 ↔ 𝐻 ≤ 𝑦))
18	17	ralbidv 3167	. . . . . 6 ⊢ (ℎ = 𝐻 → (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝐻 ≤ 𝑦))
19		breq2 4890	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (𝑥 ≤ ℎ ↔ 𝑥 ≤ 𝐻))
20	19	imbi2d 332	. . . . . . 7 ⊢ (ℎ = 𝐻 → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ) ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻)))
21	20	ralbidv 3167	. . . . . 6 ⊢ (ℎ = 𝐻 → (∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ) ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻)))
22	18, 21	anbi12d 624	. . . . 5 ⊢ (ℎ = 𝐻 → ((∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ)) ↔ (∀𝑦 ∈ 𝑆 𝐻 ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻))))
23	22	riota2 6905	. . . 4 ⊢ ((𝐻 ∈ 𝐵 ∧ ∃!ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ))) → ((∀𝑦 ∈ 𝑆 𝐻 ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻)) ↔ (℩ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ))) = 𝐻))
24	13, 16, 23	syl2anc 579	. . 3 ⊢ (𝜑 → ((∀𝑦 ∈ 𝑆 𝐻 ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻)) ↔ (℩ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ))) = 𝐻))
25	9, 12, 24	mpbi2and 702	. 2 ⊢ (𝜑 → (℩ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ))) = 𝐻)
26	7, 25	eqtrd 2813	1 ⊢ (𝜑 → (𝐺‘𝑆) = 𝐻)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106  ∀wral 3089  ∃!wreu 3091   ⊆ wss 3791   class class class wbr 4886  dom cdm 5355  ‘cfv 6135  ℩crio 6882  Basecbs 16255  lecple 16345  glbcglb 17329  CLatccla 17493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-glb 17361  df-clat 17494

This theorem is referenced by:  dihglblem2N  37432