Theorem glbfval 18371

Description: Value of the greatest lower function of a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 6-Sep-2018.)

Hypotheses

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

Assertion

Ref	Expression
glbfval	⊢ (𝜑 → 𝐺 = ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 𝜓)) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 𝜓}))

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

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

Proof of Theorem glbfval

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		glbfval.k	. 2 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
2		elex 3480	. 2 ⊢ (𝐾 ∈ 𝑉 → 𝐾 ∈ V)
3		fveq2 6875	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4		glbfval.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2788	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
6	5	pweqd 4592	. . . . . 6 ⊢ (𝑝 = 𝐾 → 𝒫 (Base‘𝑝) = 𝒫 𝐵)
7		fveq2 6875	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
8		glbfval.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
9	7, 8	eqtr4di 2788	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ )
10	9	breqd 5130	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → (𝑥(le‘𝑝)𝑦 ↔ 𝑥 ≤ 𝑦))
11	10	ralbidv 3163	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ↔ ∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦))
12	9	breqd 5130	. . . . . . . . . . 11 ⊢ (𝑝 = 𝐾 → (𝑧(le‘𝑝)𝑦 ↔ 𝑧 ≤ 𝑦))
13	12	ralbidv 3163	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 ↔ ∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦))
14	9	breqd 5130	. . . . . . . . . 10 ⊢ (𝑝 = 𝐾 → (𝑧(le‘𝑝)𝑥 ↔ 𝑧 ≤ 𝑥))
15	13, 14	imbi12d 344	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → ((∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥) ↔ (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
16	5, 15	raleqbidv 3325	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥) ↔ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
17	11, 16	anbi12d 632	. . . . . . 7 ⊢ (𝑝 = 𝐾 → ((∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥)) ↔ (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
18	5, 17	riotaeqbidv 7363	. . . . . 6 ⊢ (𝑝 = 𝐾 → (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥))) = (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
19	6, 18	mpteq12dv 5207	. . . . 5 ⊢ (𝑝 = 𝐾 → (𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥)))) = (𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))))
20	17	reubidv 3377	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥)) ↔ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
21		reueq1 3396	. . . . . . . 8 ⊢ ((Base‘𝑝) = 𝐵 → (∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
22	5, 21	syl 17	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
23	20, 22	bitrd 279	. . . . . 6 ⊢ (𝑝 = 𝐾 → (∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥)) ↔ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
24	23	abbidv 2801	. . . . 5 ⊢ (𝑝 = 𝐾 → {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥))} = {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))})
25	19, 24	reseq12d 5967	. . . 4 ⊢ (𝑝 = 𝐾 → ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥))}) = ((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))}))
26		df-glb 18355	. . . 4 ⊢ glb = (𝑝 ∈ V ↦ ((𝑠 ∈ 𝒫 (Base‘𝑝) ↦ (℩𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑥(le‘𝑝)𝑦 ∧ ∀𝑧 ∈ (Base‘𝑝)(∀𝑦 ∈ 𝑠 𝑧(le‘𝑝)𝑦 → 𝑧(le‘𝑝)𝑥))}))
27	4	fvexi 6889	. . . . . . 7 ⊢ 𝐵 ∈ V
28	27	pwex 5350	. . . . . 6 ⊢ 𝒫 𝐵 ∈ V
29	28	mptex 7214	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))) ∈ V
30	29	resex 6016	. . . 4 ⊢ ((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))}) ∈ V
31	25, 26, 30	fvmpt 6985	. . 3 ⊢ (𝐾 ∈ V → (glb‘𝐾) = ((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))}))
32		glbfval.g	. . 3 ⊢ 𝐺 = (glb‘𝐾)
33		glbfval.p	. . . . . . 7 ⊢ (𝜓 ↔ (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
34	33	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ 𝐵 → (𝜓 ↔ (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
35	34	riotabiia 7380	. . . . 5 ⊢ (℩𝑥 ∈ 𝐵 𝜓) = (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
36	35	mpteq2i 5217	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 𝜓)) = (𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
37	33	reubii 3368	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
38	37	abbii 2802	. . . 4 ⊢ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 𝜓} = {𝑠 ∣ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑠 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))}
39	36, 38	reseq12i 5964	. . 3 ⊢ ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 𝜓)) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 𝜓}) = ((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))})
40	31, 32, 39	3eqtr4g 2795	. 2 ⊢ (𝐾 ∈ V → 𝐺 = ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 𝜓)) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 𝜓}))
41	1, 2, 40	3syl 18	1 ⊢ (𝜑 → 𝐺 = ((𝑠 ∈ 𝒫 𝐵 ↦ (℩𝑥 ∈ 𝐵 𝜓)) ↾ {𝑠 ∣ ∃!𝑥 ∈ 𝐵 𝜓}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2713  ∀wral 3051  ∃!wreu 3357  Vcvv 3459  𝒫 cpw 4575   class class class wbr 5119   ↦ cmpt 5201   ↾ cres 5656  ‘cfv 6530  ℩crio 7359  Basecbs 17226  lecple 17276  glbcglb 18320

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-glb 18355

This theorem is referenced by:  glbdm  18372  glbfun  18373  glbval  18377  meet0  18414  odulub  18415  oduglb  18417