Theorem isglbd 18579

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 261	. . 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 18439	. 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 3152	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 𝐻 ≤ 𝑦)
10		isglbd.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝐻)
11	10	3exp 1119	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻)))
12	11	ralrimiv 3151	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻))
13		isglbd.5	. . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐵)
14	1, 3	clatglbcl2 18576	. . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺)
15	5, 6, 14	syl2anc 583	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ dom 𝐺)
16	1, 2, 3, 4, 5, 15	glbeu 18438	. . . 4 ⊢ (𝜑 → ∃!ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ)))
17		breq1 5169	. . . . . . 7 ⊢ (ℎ = 𝐻 → (ℎ ≤ 𝑦 ↔ 𝐻 ≤ 𝑦))
18	17	ralbidv 3184	. . . . . 6 ⊢ (ℎ = 𝐻 → (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝐻 ≤ 𝑦))
19		breq2 5170	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (𝑥 ≤ ℎ ↔ 𝑥 ≤ 𝐻))
20	19	imbi2d 340	. . . . . . 7 ⊢ (ℎ = 𝐻 → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ) ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻)))
21	20	ralbidv 3184	. . . . . 6 ⊢ (ℎ = 𝐻 → (∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ) ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻)))
22	18, 21	anbi12d 631	. . . . 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 7430	. . . 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 583	. . 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 711	. 2 ⊢ (𝜑 → (℩ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ))) = 𝐻)
26	7, 25	eqtrd 2780	1 ⊢ (𝜑 → (𝐺‘𝑆) = 𝐻)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃!wreu 3386   ⊆ wss 3976   class class class wbr 5166  dom cdm 5700  ‘cfv 6573  ℩crio 7403  Basecbs 17258  lecple 17318  glbcglb 18380  CLatccla 18568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-glb 18417  df-clat 18569

This theorem is referenced by:  dihglblem2N  41251