Theorem pmapglbx 40174

Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 40175, where we read 𝑆 as 𝑆(𝑖). Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)

Hypotheses

Ref	Expression
pmapglb.b	⊢ 𝐵 = (Base‘𝐾)
pmapglb.g	⊢ 𝐺 = (glb‘𝐾)
pmapglb.m	⊢ 𝑀 = (pmap‘𝐾)

Assertion

Ref	Expression
pmapglbx	⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))

Distinct variable groups:   𝑦,𝑖,𝐵   𝑖,𝐼,𝑦   𝑖,𝐾,𝑦   𝑦,𝑆

Allowed substitution hints:   𝑆(𝑖)   𝐺(𝑦,𝑖)   𝑀(𝑦,𝑖)

Proof of Theorem pmapglbx

Dummy variables 𝑝 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hlclat 39763	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
2	1	ad2antrr 727	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ CLat)
3		pmapglb.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
4		eqid 2737	. . . . . . . . 9 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5	3, 4	atbase 39694	. . . . . . . 8 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
6	5	adantl 481	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ 𝐵)
7		r19.29 3101	. . . . . . . . . . 11 ⊢ ((∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆) → ∃𝑖 ∈ 𝐼 (𝑆 ∈ 𝐵 ∧ 𝑦 = 𝑆))
8		eleq1a 2832	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝐵 → (𝑦 = 𝑆 → 𝑦 ∈ 𝐵))
9	8	imp 406	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝐵 ∧ 𝑦 = 𝑆) → 𝑦 ∈ 𝐵)
10	9	rexlimivw 3135	. . . . . . . . . . 11 ⊢ (∃𝑖 ∈ 𝐼 (𝑆 ∈ 𝐵 ∧ 𝑦 = 𝑆) → 𝑦 ∈ 𝐵)
11	7, 10	syl 17	. . . . . . . . . 10 ⊢ ((∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆) → 𝑦 ∈ 𝐵)
12	11	ex 412	. . . . . . . . 9 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → (∃𝑖 ∈ 𝐼 𝑦 = 𝑆 → 𝑦 ∈ 𝐵))
13	12	ad2antlr 728	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (∃𝑖 ∈ 𝐼 𝑦 = 𝑆 → 𝑦 ∈ 𝐵))
14	13	abssdv 4021	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} ⊆ 𝐵)
15		eqid 2737	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
16		pmapglb.g	. . . . . . . 8 ⊢ 𝐺 = (glb‘𝐾)
17	3, 15, 16	clatleglb 18455	. . . . . . 7 ⊢ ((𝐾 ∈ CLat ∧ 𝑝 ∈ 𝐵 ∧ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} ⊆ 𝐵) → (𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}) ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧))
18	2, 6, 14, 17	syl3anc 1374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}) ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧))
19		vex 3446	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
20		eqeq1 2741	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝑆 ↔ 𝑧 = 𝑆))
21	20	rexbidv 3162	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (∃𝑖 ∈ 𝐼 𝑦 = 𝑆 ↔ ∃𝑖 ∈ 𝐼 𝑧 = 𝑆))
22	19, 21	elab 3636	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} ↔ ∃𝑖 ∈ 𝐼 𝑧 = 𝑆)
23	22	imbi1i 349	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ (∃𝑖 ∈ 𝐼 𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
24		r19.23v 3165	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ 𝐼 (𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ (∃𝑖 ∈ 𝐼 𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
25	23, 24	bitr4i 278	. . . . . . . . . 10 ⊢ ((𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑖 ∈ 𝐼 (𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
26	25	albii 1821	. . . . . . . . 9 ⊢ (∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑧∀𝑖 ∈ 𝐼 (𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
27		df-ral 3053	. . . . . . . . 9 ⊢ (∀𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧))
28		ralcom4 3264	. . . . . . . . 9 ⊢ (∀𝑖 ∈ 𝐼 ∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑧∀𝑖 ∈ 𝐼 (𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
29	26, 27, 28	3bitr4i 303	. . . . . . . 8 ⊢ (∀𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑖 ∈ 𝐼 ∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
30		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑝(le‘𝐾)𝑆
31		breq2 5104	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑆 → (𝑝(le‘𝐾)𝑧 ↔ 𝑝(le‘𝐾)𝑆))
32	30, 31	ceqsalg 3478	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝐵 → (∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
33	32	ralimi 3075	. . . . . . . . 9 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → ∀𝑖 ∈ 𝐼 (∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
34		ralbi 3093	. . . . . . . . 9 ⊢ (∀𝑖 ∈ 𝐼 (∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆) → (∀𝑖 ∈ 𝐼 ∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑖 ∈ 𝐼 𝑝(le‘𝐾)𝑆))
35	33, 34	syl 17	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → (∀𝑖 ∈ 𝐼 ∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑖 ∈ 𝐼 𝑝(le‘𝐾)𝑆))
36	29, 35	bitrid 283	. . . . . . 7 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → (∀𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑖 ∈ 𝐼 𝑝(le‘𝐾)𝑆))
37	36	ad2antlr 728	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (∀𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑖 ∈ 𝐼 𝑝(le‘𝐾)𝑆))
38	18, 37	bitrd 279	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}) ↔ ∀𝑖 ∈ 𝐼 𝑝(le‘𝐾)𝑆))
39	38	rabbidva 3407	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖 ∈ 𝐼 𝑝(le‘𝐾)𝑆})
40	39	3adant3 1133	. . 3 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖 ∈ 𝐼 𝑝(le‘𝐾)𝑆})
41		simp1 1137	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → 𝐾 ∈ HL)
42	12	abssdv 4021	. . . . . 6 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} ⊆ 𝐵)
43	3, 16	clatglbcl 18442	. . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} ⊆ 𝐵) → (𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}) ∈ 𝐵)
44	1, 42, 43	syl2an 597	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}) ∈ 𝐵)
45	44	3adant3 1133	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}) ∈ 𝐵)
46		pmapglb.m	. . . . 5 ⊢ 𝑀 = (pmap‘𝐾)
47	3, 15, 4, 46	pmapval 40162	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}) ∈ 𝐵) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})})
48	41, 45, 47	syl2anc 585	. . 3 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})})
49		iinrab 5026	. . . 4 ⊢ (𝐼 ≠ ∅ → ∩ 𝑖 ∈ 𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖 ∈ 𝐼 𝑝(le‘𝐾)𝑆})
50	49	3ad2ant3 1136	. . 3 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → ∩ 𝑖 ∈ 𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖 ∈ 𝐼 𝑝(le‘𝐾)𝑆})
51	40, 48, 50	3eqtr4d 2782	. 2 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = ∩ 𝑖 ∈ 𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
52		nfv 1916	. . . 4 ⊢ Ⅎ𝑖 𝐾 ∈ HL
53		nfra1 3262	. . . 4 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵
54		nfv 1916	. . . 4 ⊢ Ⅎ𝑖 𝐼 ≠ ∅
55	52, 53, 54	nf3an 1903	. . 3 ⊢ Ⅎ𝑖(𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅)
56		simpl1 1193	. . . 4 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) ∧ 𝑖 ∈ 𝐼) → 𝐾 ∈ HL)
57		rspa 3227	. . . . 5 ⊢ ((∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ 𝐵)
58	57	3ad2antl2 1188	. . . 4 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ 𝐵)
59	3, 15, 4, 46	pmapval 40162	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐵) → (𝑀‘𝑆) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
60	56, 58, 59	syl2anc 585	. . 3 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) ∧ 𝑖 ∈ 𝐼) → (𝑀‘𝑆) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
61	55, 60	iineq2d 4972	. 2 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆) = ∩ 𝑖 ∈ 𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
62	51, 61	eqtr4d 2775	1 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3401   ⊆ wss 3903  ∅c0 4287  ∩ ciin 4949   class class class wbr 5100  ‘cfv 6502  Basecbs 17150  lecple 17198  glbcglb 18247  CLatccla 18435  Atomscatm 39668  HLchlt 39755  pmapcpmap 39902

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-poset 18250  df-lub 18281  df-glb 18282  df-join 18283  df-meet 18284  df-lat 18369  df-clat 18436  df-ats 39672  df-hlat 39756  df-pmap 39909

This theorem is referenced by:  pmapglb  40175  pmapglb2xN  40177