Theorem isglbd 18676

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 264	. . 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 18534	. 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 3155	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 𝐻 ≤ 𝑦)
10		isglbd.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝐻)
11	10	3exp 1137	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻)))
12	11	ralrimiv 3154	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻))
13		isglbd.5	. . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐵)
14	1, 3	clatglbcl2 18673	. . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺)
15	5, 6, 14	syl2anc 596	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ dom 𝐺)
16	1, 2, 3, 4, 5, 15	glbeu 18533	. . . 4 ⊢ (𝜑 → ∃!ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ)))
17		breq1 5106	. . . . . . 7 ⊢ (ℎ = 𝐻 → (ℎ ≤ 𝑦 ↔ 𝐻 ≤ 𝑦))
18	17	ralbidv 3186	. . . . . 6 ⊢ (ℎ = 𝐻 → (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝐻 ≤ 𝑦))
19		breq2 5107	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (𝑥 ≤ ℎ ↔ 𝑥 ≤ 𝐻))
20	19	imbi2d 343	. . . . . . 7 ⊢ (ℎ = 𝐻 → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ) ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻)))
21	20	ralbidv 3186	. . . . . 6 ⊢ (ℎ = 𝐻 → (∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ) ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻)))
22	18, 21	anbi12d 644	. . . . 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 7400	. . . 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 596	. . 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 725	. 2 ⊢ (𝜑 → (℩ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ))) = 𝐻)
26	7, 25	eqtrd 2796	1 ⊢ (𝜑 → (𝐺‘𝑆) = 𝐻)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃!wreu 3364   ⊆ wss 3899   class class class wbr 5103  dom cdm 5651  ‘cfv 6537  ℩crio 7374  Basecbs 17380  lecple 17428  glbcglb 18477  CLatccla 18665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  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 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-glb 18512  df-clat 18666

This theorem is used by:  dihglblem2N  42331