Theorem isglbd 17730

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 263	. . 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 17610	. 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 3185	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 𝐻 ≤ 𝑦)
10		isglbd.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝐻)
11	10	3exp 1115	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻)))
12	11	ralrimiv 3184	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻))
13		isglbd.5	. . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐵)
14	1, 3	clatglbcl2 17728	. . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺)
15	5, 6, 14	syl2anc 586	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ dom 𝐺)
16	1, 2, 3, 4, 5, 15	glbeu 17609	. . . 4 ⊢ (𝜑 → ∃!ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ)))
17		breq1 5072	. . . . . . 7 ⊢ (ℎ = 𝐻 → (ℎ ≤ 𝑦 ↔ 𝐻 ≤ 𝑦))
18	17	ralbidv 3200	. . . . . 6 ⊢ (ℎ = 𝐻 → (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝐻 ≤ 𝑦))
19		breq2 5073	. . . . . . . 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 3200	. . . . . 6 ⊢ (ℎ = 𝐻 → (∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ) ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻)))
22	18, 21	anbi12d 632	. . . . 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 7142	. . . 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 586	. . 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 710	. 2 ⊢ (𝜑 → (℩ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ))) = 𝐻)
26	7, 25	eqtrd 2859	1 ⊢ (𝜑 → (𝐺‘𝑆) = 𝐻)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  ∀wral 3141  ∃!wreu 3143   ⊆ wss 3939   class class class wbr 5069  dom cdm 5558  ‘cfv 6358  ℩crio 7116  Basecbs 16486  lecple 16575  glbcglb 17556  CLatccla 17720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-glb 17588  df-clat 17721

This theorem is referenced by:  dihglblem2N  38434