Theorem pmapglbx 41017

Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 41018, 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 40606	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
2	1	ad2antrr 739	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ CLat)
3		pmapglb.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
4		eqid 2763	. . . . . . . . 9 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5	3, 4	atbase 40537	. . . . . . . 8 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
6	5	adantl 487	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ 𝐵)
7		r19.29 3129	. . . . . . . . . . 11 ⊢ ((∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆) → ∃𝑖 ∈ 𝐼 (𝑆 ∈ 𝐵 ∧ 𝑦 = 𝑆))
8		eleq1a 2859	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝐵 → (𝑦 = 𝑆 → 𝑦 ∈ 𝐵))
9	8	imp 412	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝐵 ∧ 𝑦 = 𝑆) → 𝑦 ∈ 𝐵)
10	9	rexlimivw 3163	. . . . . . . . . . 11 ⊢ (∃𝑖 ∈ 𝐼 (𝑆 ∈ 𝐵 ∧ 𝑦 = 𝑆) → 𝑦 ∈ 𝐵)
11	7, 10	syl 18	. . . . . . . . . 10 ⊢ ((∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆) → 𝑦 ∈ 𝐵)
12	11	ex 418	. . . . . . . . 9 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → (∃𝑖 ∈ 𝐼 𝑦 = 𝑆 → 𝑦 ∈ 𝐵))
13	12	ad2antlr 740	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (∃𝑖 ∈ 𝐼 𝑦 = 𝑆 → 𝑦 ∈ 𝐵))
14	13	abssdv 3995	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} ⊆ 𝐵)
15		eqid 2763	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
16		pmapglb.g	. . . . . . . 8 ⊢ 𝐺 = (glb‘𝐾)
17	3, 15, 16	clatleglb 18718	. . . . . . 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 1399	. . . . . 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 3458	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
20		eqeq1 2767	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝑆 ↔ 𝑧 = 𝑆))
21	20	rexbidv 3190	. . . . . . . . . . . . 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 352	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ (∃𝑖 ∈ 𝐼 𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
24		r19.23v 3193	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ 𝐼 (𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ (∃𝑖 ∈ 𝐼 𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
25	23, 24	bitr4i 281	. . . . . . . . . 10 ⊢ ((𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑖 ∈ 𝐼 (𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
26	25	albii 1852	. . . . . . . . 9 ⊢ (∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑧∀𝑖 ∈ 𝐼 (𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
27		df-ral 3081	. . . . . . . . 9 ⊢ (∀𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧))
28		ralcom4 3292	. . . . . . . . 9 ⊢ (∀𝑖 ∈ 𝐼 ∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑧∀𝑖 ∈ 𝐼 (𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
29	26, 27, 28	3bitr4i 306	. . . . . . . 8 ⊢ (∀𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑖 ∈ 𝐼 ∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
30		nfv 1949	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑝(le‘𝐾)𝑆
31		breq2 5110	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑆 → (𝑝(le‘𝐾)𝑧 ↔ 𝑝(le‘𝐾)𝑆))
32	30, 31	ceqsalg 3489	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝐵 → (∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
33	32	ralimi 3103	. . . . . . . . 9 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → ∀𝑖 ∈ 𝐼 (∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
34		ralbi 3121	. . . . . . . . 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 18	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → (∀𝑖 ∈ 𝐼 ∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑖 ∈ 𝐼 𝑝(le‘𝐾)𝑆))
36	29, 35	bitrid 286	. . . . . . 7 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → (∀𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑖 ∈ 𝐼 𝑝(le‘𝐾)𝑆))
37	36	ad2antlr 740	. . . . . 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 282	. . . . 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 3422	. . . 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 1150	. . 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 1154	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → 𝐾 ∈ HL)
42	12	abssdv 3995	. . . . . 6 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} ⊆ 𝐵)
43	3, 16	clatglbcl 18705	. . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} ⊆ 𝐵) → (𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}) ∈ 𝐵)
44	1, 42, 43	syl2an 608	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}) ∈ 𝐵)
45	44	3adant3 1150	. . . 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 41005	. . . 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 596	. . 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 5030	. . . 4 ⊢ (𝐼 ≠ ∅ → ∩ 𝑖 ∈ 𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖 ∈ 𝐼 𝑝(le‘𝐾)𝑆})
50	49	3ad2ant3 1153	. . 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 2808	. 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 1949	. . . 4 ⊢ Ⅎ𝑖 𝐾 ∈ HL
53		nfra1 3290	. . . 4 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵
54		nfv 1949	. . . 4 ⊢ Ⅎ𝑖 𝐼 ≠ ∅
55	52, 53, 54	nf3an 1936	. . 3 ⊢ Ⅎ𝑖(𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅)
56		simpl1 1210	. . . 4 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) ∧ 𝑖 ∈ 𝐼) → 𝐾 ∈ HL)
57		rspa 3255	. . . . 5 ⊢ ((∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ 𝐵)
58	57	3ad2antl2 1205	. . . 4 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ 𝐵)
59	3, 15, 4, 46	pmapval 41005	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐵) → (𝑀‘𝑆) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
60	56, 58, 59	syl2anc 596	. . 3 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) ∧ 𝑖 ∈ 𝐼) → (𝑀‘𝑆) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
61	55, 60	iineq2d 4978	. 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 2801	1 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416   ⊆ wss 3900  ∅c0 4281  ∩ ciin 4955   class class class wbr 5106  ‘cfv 6555  Basecbs 17413  lecple 17461  glbcglb 18510  CLatccla 18698  Atomscatm 40511  HLchlt 40598  pmapcpmap 40745

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-poset 18513  df-lub 18544  df-glb 18545  df-join 18546  df-meet 18547  df-lat 18632  df-clat 18699  df-ats 40515  df-hlat 40599  df-pmap 40752

This theorem is used by:  pmapglb  41018  pmapglb2xN  41020