Theorem glbval 18521

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 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝐾 ∈ 𝑉)
7	1, 2, 3, 4, 6	glbfval 18515	. . . 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 6879	. . 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 490	. . . . 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 18520	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → ∃!𝑥 ∈ 𝐵 𝜓)
12		raleq 3317	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦))
13		raleq 3317	. . . . . . . . . 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 3186	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
16	12, 15	anbi12d 644	. . . . . . 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	bitr4di 292	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ 𝜓))
18	17	reubidv 3382	. . . . 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 3635	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝑆 ∈ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))})
20	19	fvresd 6897	. . 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 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝑆 ⊆ 𝐵)
23	1	fvexi 6891	. . . . . 6 ⊢ 𝐵 ∈ V
24	23	elpw2 5296	. . . . 5 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵)
25	22, 24	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝑆 ∈ 𝒫 𝐵)
26	17	riotabidv 7371	. . . . 5 ⊢ (𝑠 = 𝑆 → (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) = (℩𝑥 ∈ 𝐵 𝜓))
27		eqid 2761	. . . . 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 7373	. . . . 5 ⊢ (℩𝑥 ∈ 𝐵 𝜓) ∈ V
29	26, 27, 28	fvmpt 6985	. . . 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 18	. . 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 2800	. 2 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → (𝐺‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))
32		ndmfv 6909	. . . 4 ⊢ (¬ 𝑆 ∈ dom 𝐺 → (𝐺‘𝑆) = ∅)
33	32	adantl 487	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝐺) → (𝐺‘𝑆) = ∅)
34	1, 2, 3, 10, 5	glbeldm 18518	. . . . . . 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 708	. . . . 5 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 → 𝑆 ∈ dom 𝐺))
37	36	con3dimp 414	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝐺) → ¬ ∃!𝑥 ∈ 𝐵 𝜓)
38		riotaund 7408	. . . 4 ⊢ (¬ ∃!𝑥 ∈ 𝐵 𝜓 → (℩𝑥 ∈ 𝐵 𝜓) = ∅)
39	37, 38	syl 18	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝐺) → (℩𝑥 ∈ 𝐵 𝜓) = ∅)
40	33, 39	eqtr4d 2799	. 2 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝐺) → (𝐺‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))
41	31, 40	pm2.61dan 825	1 ⊢ (𝜑 → (𝐺‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  ∃!wreu 3364   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557   class class class wbr 5103   ↦ cmpt 5186  dom cdm 5651   ↾ cres 5653  ‘cfv 6531  ℩crio 7368  Basecbs 17367  lecple 17415  glbcglb 18464

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 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-glb 18499

This theorem is used by:  glbcl  18522  glbprop  18523  meetval2  18547  isglbd  18663  tosglb  33518  glb0N  40218  glbconN  40402