Theorem pmapglbx 35055

Description: The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 35056, 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 34645	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ CLat)
2	1	ad2antrr 762	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝐾 ∈ CLat)
3		pmapglb.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
4		eqid 2622	. . . . . . . . 9 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5	3, 4	atbase 34576	. . . . . . . 8 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
6	5	adantl 482	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → 𝑝 ∈ 𝐵)
7		r19.29 3072	. . . . . . . . . . 11 ⊢ ((∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆) → ∃𝑖 ∈ 𝐼 (𝑆 ∈ 𝐵 ∧ 𝑦 = 𝑆))
8		eleq1a 2696	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ 𝐵 → (𝑦 = 𝑆 → 𝑦 ∈ 𝐵))
9	8	imp 445	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ 𝐵 ∧ 𝑦 = 𝑆) → 𝑦 ∈ 𝐵)
10	9	rexlimivw 3029	. . . . . . . . . . 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 450	. . . . . . . . 9 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → (∃𝑖 ∈ 𝐼 𝑦 = 𝑆 → 𝑦 ∈ 𝐵))
13	12	ad2antlr 763	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → (∃𝑖 ∈ 𝐼 𝑦 = 𝑆 → 𝑦 ∈ 𝐵))
14	13	abssdv 3676	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) ∧ 𝑝 ∈ (Atoms‘𝐾)) → {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} ⊆ 𝐵)
15		eqid 2622	. . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾)
16		pmapglb.g	. . . . . . . 8 ⊢ 𝐺 = (glb‘𝐾)
17	3, 15, 16	clatleglb 17126	. . . . . . 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 1326	. . . . . 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 3203	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
20		eqeq1 2626	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝑆 ↔ 𝑧 = 𝑆))
21	20	rexbidv 3052	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (∃𝑖 ∈ 𝐼 𝑦 = 𝑆 ↔ ∃𝑖 ∈ 𝐼 𝑧 = 𝑆))
22	19, 21	elab 3350	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} ↔ ∃𝑖 ∈ 𝐼 𝑧 = 𝑆)
23	22	imbi1i 339	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ (∃𝑖 ∈ 𝐼 𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
24		r19.23v 3023	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ 𝐼 (𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ (∃𝑖 ∈ 𝐼 𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
25	23, 24	bitr4i 267	. . . . . . . . . 10 ⊢ ((𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑖 ∈ 𝐼 (𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
26	25	albii 1747	. . . . . . . . 9 ⊢ (∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑧∀𝑖 ∈ 𝐼 (𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
27		df-ral 2917	. . . . . . . . 9 ⊢ (∀𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑧(𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} → 𝑝(le‘𝐾)𝑧))
28		ralcom4 3224	. . . . . . . . 9 ⊢ (∀𝑖 ∈ 𝐼 ∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ ∀𝑧∀𝑖 ∈ 𝐼 (𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
29	26, 27, 28	3bitr4i 292	. . . . . . . 8 ⊢ (∀𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑖 ∈ 𝐼 ∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧))
30		nfv 1843	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑝(le‘𝐾)𝑆
31		breq2 4657	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑆 → (𝑝(le‘𝐾)𝑧 ↔ 𝑝(le‘𝐾)𝑆))
32	30, 31	ceqsalg 3230	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝐵 → (∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
33	32	ralimi 2952	. . . . . . . . 9 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → ∀𝑖 ∈ 𝐼 (∀𝑧(𝑧 = 𝑆 → 𝑝(le‘𝐾)𝑧) ↔ 𝑝(le‘𝐾)𝑆))
34		ralbi 3068	. . . . . . . . 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	syl5bb 272	. . . . . . 7 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → (∀𝑧 ∈ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}𝑝(le‘𝐾)𝑧 ↔ ∀𝑖 ∈ 𝐼 𝑝(le‘𝐾)𝑆))
37	36	ad2antlr 763	. . . . . 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 268	. . . . 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 3188	. . . 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 1081	. . 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 1061	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → 𝐾 ∈ HL)
42	12	abssdv 3676	. . . . . 6 ⊢ (∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 → {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} ⊆ 𝐵)
43	3, 16	clatglbcl 17114	. . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ {𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆} ⊆ 𝐵) → (𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}) ∈ 𝐵)
44	1, 42, 43	syl2an 494	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆}) ∈ 𝐵)
45	44	3adant3 1081	. . . 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 35043	. . . 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 693	. . 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 4582	. . . 4 ⊢ (𝐼 ≠ ∅ → ∩ 𝑖 ∈ 𝐼 {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆} = {𝑝 ∈ (Atoms‘𝐾) ∣ ∀𝑖 ∈ 𝐼 𝑝(le‘𝐾)𝑆})
50	49	3ad2ant3 1084	. . 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 2666	. 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 1843	. . . 4 ⊢ Ⅎ𝑖 𝐾 ∈ HL
53		nfra1 2941	. . . 4 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵
54		nfv 1843	. . . 4 ⊢ Ⅎ𝑖 𝐼 ≠ ∅
55	52, 53, 54	nf3an 1831	. . 3 ⊢ Ⅎ𝑖(𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅)
56		simpl1 1064	. . . 4 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) ∧ 𝑖 ∈ 𝐼) → 𝐾 ∈ HL)
57		rspa 2930	. . . . 5 ⊢ ((∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ 𝐵)
58	57	3ad2antl2 1224	. . . 4 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) ∧ 𝑖 ∈ 𝐼) → 𝑆 ∈ 𝐵)
59	3, 15, 4, 46	pmapval 35043	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐵) → (𝑀‘𝑆) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
60	56, 58, 59	syl2anc 693	. . 3 ⊢ (((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) ∧ 𝑖 ∈ 𝐼) → (𝑀‘𝑆) = {𝑝 ∈ (Atoms‘𝐾) ∣ 𝑝(le‘𝐾)𝑆})
61	55, 60	iineq2d 4541	. 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 2659	1 ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916   ⊆ wss 3574  ∅c0 3915  ∩ ciin 4521   class class class wbr 4653  ‘cfv 5888  Basecbs 15857  lecple 15948  glbcglb 16943  CLatccla 17107  Atomscatm 34550  HLchlt 34637  pmapcpmap 34783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-poset 16946  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-lat 17046  df-clat 17108  df-ats 34554  df-hlat 34638  df-pmap 34790

This theorem is referenced by:  pmapglb  35056  pmapglb2xN  35058