Theorem glbval 18258

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 260	. . . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝐾 ∈ 𝑉)
7	1, 2, 3, 4, 6	glbfval 18252	. . . 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 6844	. . 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 485	. . . . 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 18257	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → ∃!𝑥 ∈ 𝐵 𝜓)
12		raleq 3309	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦))
13		raleq 3309	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦))
14	13	imbi1d 341	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
15	14	ralbidv 3174	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
16	12, 15	anbi12d 631	. . . . . . 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 288	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ 𝜓))
18	17	reubidv 3371	. . . . 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 3633	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝑆 ∈ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))})
20	19	fvresd 6862	. . 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝑆 ⊆ 𝐵)
23	1	fvexi 6856	. . . . . 6 ⊢ 𝐵 ∈ V
24	23	elpw2 5302	. . . . 5 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵)
25	22, 24	sylibr 233	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → 𝑆 ∈ 𝒫 𝐵)
26	17	riotabidv 7315	. . . . 5 ⊢ (𝑠 = 𝑆 → (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) = (℩𝑥 ∈ 𝐵 𝜓))
27		eqid 2736	. . . . 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 7317	. . . . 5 ⊢ (℩𝑥 ∈ 𝐵 𝜓) ∈ V
29	26, 27, 28	fvmpt 6948	. . . 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 2780	. 2 ⊢ ((𝜑 ∧ 𝑆 ∈ dom 𝐺) → (𝐺‘𝑆) = (℩𝑥 ∈ 𝐵 𝜓))
32		ndmfv 6877	. . . 4 ⊢ (¬ 𝑆 ∈ dom 𝐺 → (𝐺‘𝑆) = ∅)
33	32	adantl 482	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝐺) → (𝐺‘𝑆) = ∅)
34	1, 2, 3, 10, 5	glbeldm 18255	. . . . . . 7 ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 𝜓)))
35	34	biimprd 247	. . . . . 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 409	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝐺) → ¬ ∃!𝑥 ∈ 𝐵 𝜓)
38		riotaund 7353	. . . 4 ⊢ (¬ ∃!𝑥 ∈ 𝐵 𝜓 → (℩𝑥 ∈ 𝐵 𝜓) = ∅)
39	37, 38	syl 17	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑆 ∈ dom 𝐺) → (℩𝑥 ∈ 𝐵 𝜓) = ∅)
40	33, 39	eqtr4d 2779	. 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 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2713  ∀wral 3064  ∃!wreu 3351   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560   class class class wbr 5105   ↦ cmpt 5188  dom cdm 5633   ↾ cres 5635  ‘cfv 6496  ℩crio 7312  Basecbs 17083  lecple 17140  glbcglb 18199

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-glb 18236

This theorem is referenced by:  glbcl  18259  glbprop  18260  meetval2  18284  isglbd  18398  tosglb  31835  glb0N  37655  glbconN  37839  glbconNOLD  37840