Theorem pmapglbx 39771

Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 39772, 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 39359	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
2	1	ad2antrr 726	. . . . . . 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 39290	. . . . . . . 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 3114	. . . . . . . . . . 11 ⊢ ((∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆) → ∃𝑖 ∈ 𝐼 (𝑆 ∈ 𝐵 ∧ 𝑦 = 𝑆))
8		eleq1a 2836	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝐵 → (𝑦 = 𝑆 → 𝑦 ∈ 𝐵))
9	8	imp 406	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝐵 ∧ 𝑦 = 𝑆) → 𝑦 ∈ 𝐵)
10	9	rexlimivw 3151	. . . . . . . . . . 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 727	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (∃𝑖 ∈ 𝐼 𝑦 = 𝑆 → 𝑦 ∈ 𝐵))
14	13	abssdv 4068	. . . . . . 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 18563	. . . . . . 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 1373	. . . . . 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 3484	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
20		eqeq1 2741	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝑆 ↔ 𝑧 = 𝑆))
21	20	rexbidv 3179	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (∃𝑖 ∈ 𝐼 𝑦 = 𝑆 ↔ ∃𝑖 ∈ 𝐼 𝑧 = 𝑆))
22	19, 21	elab 3679	. . . . . . . . . . . 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 3183	. . . . . . . . . . 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 1819	. . . . . . . . 9 ⊢ (∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑧∀𝑖 ∈ 𝐼 (𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
27		df-ral 3062	. . . . . . . . 9 ⊢ (∀𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧))
28		ralcom4 3286	. . . . . . . . 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 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑝(le‘𝐾)𝑆
31		breq2 5147	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑆 → (𝑝(le‘𝐾)𝑧 ↔ 𝑝(le‘𝐾)𝑆))
32	30, 31	ceqsalg 3517	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝐵 → (∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
33	32	ralimi 3083	. . . . . . . . 9 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → ∀𝑖 ∈ 𝐼 (∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
34		ralbi 3103	. . . . . . . . 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 727	. . . . . 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 3443	. . . 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 4068	. . . . . 6 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} ⊆ 𝐵)
43	3, 16	clatglbcl 18550	. . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} ⊆ 𝐵) → (𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}) ∈ 𝐵)
44	1, 42, 43	syl2an 596	. . . . 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 39759	. . . 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 584	. . 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 5069	. . . 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 2787	. 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 1914	. . . 4 ⊢ Ⅎ𝑖 𝐾 ∈ HL
53		nfra1 3284	. . . 4 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵
54		nfv 1914	. . . 4 ⊢ Ⅎ𝑖 𝐼 ≠ ∅
55	52, 53, 54	nf3an 1901	. . 3 ⊢ Ⅎ𝑖(𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅)
56		simpl1 1192	. . . 4 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) ∧ 𝑖 ∈ 𝐼) → 𝐾 ∈ HL)
57		rspa 3248	. . . . 5 ⊢ ((∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ 𝐵)
58	57	3ad2antl2 1187	. . . 4 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ 𝐵)
59	3, 15, 4, 46	pmapval 39759	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐵) → (𝑀‘𝑆) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
60	56, 58, 59	syl2anc 584	. . 3 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) ∧ 𝑖 ∈ 𝐼) → (𝑀‘𝑆) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
61	55, 60	iineq2d 5015	. 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 2780	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 1538   = wceq 1540   ∈ wcel 2108  {cab 2714   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3436   ⊆ wss 3951  ∅c0 4333  ∩ ciin 4992   class class class wbr 5143  ‘cfv 6561  Basecbs 17247  lecple 17304  glbcglb 18356  CLatccla 18543  Atomscatm 39264  HLchlt 39351  pmapcpmap 39499

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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-poset 18359  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-lat 18477  df-clat 18544  df-ats 39268  df-hlat 39352  df-pmap 39506

This theorem is referenced by:  pmapglb  39772  pmapglb2xN  39774