Theorem glbval 17599

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 264	. . . . 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 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝐾 ∈ 𝑉)
7	1, 2, 3, 4, 6	glbfval 17593	. . . 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 6647	. . 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 488	. . . . 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 17598	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → ∃!𝑥 ∈ 𝐵 𝜓)
12		raleq 3358	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦))
13		raleq 3358	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦))
14	13	imbi1d 345	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
15	14	ralbidv 3162	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
16	12, 15	anbi12d 633	. . . . . . 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 292	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ 𝜓))
18	17	reubidv 3342	. . . . 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 3617	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝑆 ∈ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))})
20	19	fvresd 6665	. . 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 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝑆 ⊆ 𝐵)
23	1	fvexi 6659	. . . . . 6 ⊢ 𝐵 ∈ V
24	23	elpw2 5212	. . . . 5 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵)
25	22, 24	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝑆 ∈ 𝒫 𝐵)
26	17	riotabidv 7095	. . . . 5 ⊢ (𝑠 = 𝑆 → (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) = (℩𝑥 ∈ 𝐵 𝜓))
27		eqid 2798	. . . . 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 7097	. . . . 5 ⊢ (℩𝑥 ∈ 𝐵 𝜓) ∈ V
29	26, 27, 28	fvmpt 6745	. . . 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 2837	. 2 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → (𝐺‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))
32		ndmfv 6675	. . . 4 ⊢ (¬ 𝑆 ∈ dom 𝐺 → (𝐺‘𝑆) = ∅)
33	32	adantl 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝐺) → (𝐺‘𝑆) = ∅)
34	1, 2, 3, 10, 5	glbeldm 17596	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓)))
35	34	biimprd 251	. . . . . 6 ⊢ (𝜑 → ((𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓) → 𝑆 ∈ dom 𝐺))
36	21, 35	mpand 694	. . . . 5 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 → 𝑆 ∈ dom 𝐺))
37	36	con3dimp 412	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝐺) → ¬ ∃!𝑥 ∈ 𝐵 𝜓)
38		riotaund 7132	. . . 4 ⊢ (¬ ∃!𝑥 ∈ 𝐵 𝜓 → (℩𝑥 ∈ 𝐵 𝜓) = ∅)
39	37, 38	syl 17	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝐺) → (℩𝑥 ∈ 𝐵 𝜓) = ∅)
40	33, 39	eqtr4d 2836	. 2 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝐺) → (𝐺‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))
41	31, 40	pm2.61dan 812	1 ⊢ (𝜑 → (𝐺‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃!wreu 3108   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497   class class class wbr 5030   ↦ cmpt 5110  dom cdm 5519   ↾ cres 5521  ‘cfv 6324  ℩crio 7092  Basecbs 16475  lecple 16564  glbcglb 17545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-glb 17577

This theorem is referenced by:  glbcl  17600  glbprop  17601  meetval2  17625  isglbd  17719  tosglb  30683  glb0N  36489  glbconN  36673