Theorem glbval 17606

Description: Value of the greatest lower bound function of a poset. Out-of-domain arguments (those not satisfying 𝑆 ∈ dom 𝑈) are allowed for convenience, evaluating to the empty set on both sides of the equality. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.)

Hypotheses

Ref	Expression
glbval.b	⊢ 𝐵 = (Base‘𝐾)
glbval.l	⊢ ≤ = (le‘𝐾)
glbval.g	⊢ 𝐺 = (glb‘𝐾)
glbval.p	⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
glbva.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
glbval.ss	⊢ (𝜑 → 𝑆 ⊆ 𝐵)

Assertion

Ref	Expression
glbval	⊢ (𝜑 → (𝐺‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))

Distinct variable groups:   𝑥,𝑧,𝐵   𝑥,𝑦,𝐾,𝑧   𝑥,𝑆,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝐵(𝑦)   𝐺(𝑥,𝑦,𝑧)   ≤ (𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem glbval

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		glbval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		glbval.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		glbval.g	. . . . 5 ⊢ 𝐺 = (glb‘𝐾)
4		biid 263	. . . . 5 ⊢ ((∀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		glbva.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
6	5	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝐾 ∈ 𝑉)
7	1, 2, 3, 4, 6	glbfval 17600	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝐺 = ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))}))
8	7	fveq1d 6671	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → (𝐺‘𝑆) = (((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))})‘𝑆))
9		simpr 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝑆 ∈ dom 𝐺)
10		glbval.p	. . . . . 6 ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
11	1, 2, 3, 10, 6, 9	glbeu 17605	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → ∃!𝑥 ∈ 𝐵 𝜓)
12		raleq 3405	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦))
13		raleq 3405	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦))
14	13	imbi1d 344	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
15	14	ralbidv 3197	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
16	12, 15	anbi12d 632	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
17	16, 10	syl6bbr 291	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ 𝜓))
18	17	reubidv 3389	. . . . 5 ⊢ (𝑠 = 𝑆 → (∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ∃!𝑥 ∈ 𝐵 𝜓))
19	9, 11, 18	elabd 3668	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝑆 ∈ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))})
20	19	fvresd 6689	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → (((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))})‘𝑆) = ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))‘𝑆))
21		glbval.ss	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
22	21	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝑆 ⊆ 𝐵)
23	1	fvexi 6683	. . . . . 6 ⊢ 𝐵 ∈ V
24	23	elpw2 5247	. . . . 5 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵)
25	22, 24	sylibr 236	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝑆 ∈ 𝒫 𝐵)
26	17	riotabidv 7115	. . . . 5 ⊢ (𝑠 = 𝑆 → (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) = (℩𝑥 ∈ 𝐵 𝜓))
27		eqid 2821	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))) = (𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
28		riotaex 7117	. . . . 5 ⊢ (℩𝑥 ∈ 𝐵 𝜓) ∈ V
29	26, 27, 28	fvmpt 6767	. . . 4 ⊢ (𝑆 ∈ 𝒫 𝐵 → ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))
30	25, 29	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))
31	8, 20, 30	3eqtrd 2860	. 2 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → (𝐺‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))
32		ndmfv 6699	. . . 4 ⊢ (¬ 𝑆 ∈ dom 𝐺 → (𝐺‘𝑆) = ∅)
33	32	adantl 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝐺) → (𝐺‘𝑆) = ∅)
34	1, 2, 3, 10, 5	glbeldm 17603	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓)))
35	34	biimprd 250	. . . . . 6 ⊢ (𝜑 → ((𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓) → 𝑆 ∈ dom 𝐺))
36	21, 35	mpand 693	. . . . 5 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 → 𝑆 ∈ dom 𝐺))
37	36	con3dimp 411	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝐺) → ¬ ∃!𝑥 ∈ 𝐵 𝜓)
38		riotaund 7152	. . . 4 ⊢ (¬ ∃!𝑥 ∈ 𝐵 𝜓 → (℩𝑥 ∈ 𝐵 𝜓) = ∅)
39	37, 38	syl 17	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝐺) → (℩𝑥 ∈ 𝐵 𝜓) = ∅)
40	33, 39	eqtr4d 2859	. 2 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝐺) → (𝐺‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))
41	31, 40	pm2.61dan 811	1 ⊢ (𝜑 → (𝐺‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {cab 2799  ∀wral 3138  ∃!wreu 3140   ⊆ wss 3935  ∅c0 4290  𝒫 cpw 4538   class class class wbr 5065   ↦ cmpt 5145  dom cdm 5554   ↾ cres 5556  ‘cfv 6354  ℩crio 7112  Basecbs 16482  lecple 16571  glbcglb 17552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-glb 17584

This theorem is referenced by:  glbcl  17607  glbprop  17608  meetval2  17632  isglbd  17726  tosglb  30657  glb0N  36328  glbconN  36512