Theorem pmapsub 40404

Description: The projective map of a Hilbert lattice maps to projective subspaces. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.)

Hypotheses

Ref	Expression
pmapsub.b	⊢ 𝐵 = (Base‘𝐾)
pmapsub.s	⊢ 𝑆 = (PSubSp‘𝐾)
pmapsub.m	⊢ 𝑀 = (pmap‘𝐾)

Assertion

Ref	Expression
pmapsub	⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝑆)

Proof of Theorem pmapsub

Dummy variables 𝑞 𝑝 𝑟 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pmapsub.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		eqid 2765	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2765	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		pmapsub.m	. . 3 ⊢ 𝑀 = (pmap‘𝐾)
5	1, 2, 3, 4	pmapval 40393	. 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})
6		breq1 5108	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑝 → (𝑐(le‘𝐾)𝑋 ↔ 𝑝(le‘𝐾)𝑋))
7	6	elrab 3653	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋))
8	1, 3	atbase 39925	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
9	8	anim1i 626	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) → (𝑝 ∈ 𝐵 ∧ 𝑝(le‘𝐾)𝑋))
10	7, 9	sylbi 220	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} → (𝑝 ∈ 𝐵 ∧ 𝑝(le‘𝐾)𝑋))
11		breq1 5108	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑞 → (𝑐(le‘𝐾)𝑋 ↔ 𝑞(le‘𝐾)𝑋))
12	11	elrab 3653	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋))
13	1, 3	atbase 39925	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ 𝐵)
14	13	anim1i 626	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋) → (𝑞 ∈ 𝐵 ∧ 𝑞(le‘𝐾)𝑋))
15	12, 14	sylbi 220	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} → (𝑞 ∈ 𝐵 ∧ 𝑞(le‘𝐾)𝑋))
16	10, 15	anim12i 624	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}) → ((𝑝 ∈ 𝐵 ∧ 𝑝(le‘𝐾)𝑋) ∧ (𝑞 ∈ 𝐵 ∧ 𝑞(le‘𝐾)𝑋)))
17		an4 668	. . . . . . . . . . 11 ⊢ (((𝑝 ∈ 𝐵 ∧ 𝑝(le‘𝐾)𝑋) ∧ (𝑞 ∈ 𝐵 ∧ 𝑞(le‘𝐾)𝑋)) ↔ ((𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) ∧ (𝑝(le‘𝐾)𝑋 ∧ 𝑞(le‘𝐾)𝑋)))
18	16, 17	sylib 221	. . . . . . . . . 10 ⊢ ((𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}) → ((𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) ∧ (𝑝(le‘𝐾)𝑋 ∧ 𝑞(le‘𝐾)𝑋)))
19	18	anim2i 628	. . . . . . . . 9 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) → ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) ∧ (𝑝(le‘𝐾)𝑋 ∧ 𝑞(le‘𝐾)𝑋))))
20	1, 3	atbase 39925	. . . . . . . . 9 ⊢ (𝑟 ∈ (Atoms‘𝐾) → 𝑟 ∈ 𝐵)
21		eqid 2765	. . . . . . . . . . . . . . . . 17 ⊢ (join‘𝐾) = (join‘𝐾)
22	1, 2, 21	latjle12 18496	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ Lat ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑝(le‘𝐾)𝑋 ∧ 𝑞(le‘𝐾)𝑋) ↔ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋))
23	22	biimpd 232	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑝(le‘𝐾)𝑋 ∧ 𝑞(le‘𝐾)𝑋) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋))
24	23	3exp2 1371	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Lat → (𝑝 ∈ 𝐵 → (𝑞 ∈ 𝐵 → (𝑋 ∈ 𝐵 → ((𝑝(le‘𝐾)𝑋 ∧ 𝑞(le‘𝐾)𝑋) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋)))))
25	24	impd 415	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Lat → ((𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑋 ∈ 𝐵 → ((𝑝(le‘𝐾)𝑋 ∧ 𝑞(le‘𝐾)𝑋) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋))))
26	25	com23 87	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Lat → (𝑋 ∈ 𝐵 → ((𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → ((𝑝(le‘𝐾)𝑋 ∧ 𝑞(le‘𝐾)𝑋) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋))))
27	26	imp43 432	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) ∧ (𝑝(le‘𝐾)𝑋 ∧ 𝑞(le‘𝐾)𝑋))) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋)
28	27	adantr 485	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) ∧ (𝑝(le‘𝐾)𝑋 ∧ 𝑞(le‘𝐾)𝑋))) ∧ 𝑟 ∈ 𝐵) → (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋)
29	1, 21	latjcl 18485	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ 𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵)
30	29	3expib 1138	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Lat → ((𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵))
31	1, 2	lattr 18490	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ (𝑟 ∈ 𝐵 ∧ (𝑝(join‘𝐾)𝑞) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋))
32	31	3exp2 1371	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Lat → (𝑟 ∈ 𝐵 → ((𝑝(join‘𝐾)𝑞) ∈ 𝐵 → (𝑋 ∈ 𝐵 → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋)))))
33	32	com24 96	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Lat → (𝑋 ∈ 𝐵 → ((𝑝(join‘𝐾)𝑞) ∈ 𝐵 → (𝑟 ∈ 𝐵 → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋)))))
34	30, 33	syl5d 74	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Lat → (𝑋 ∈ 𝐵 → ((𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑟 ∈ 𝐵 → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋)))))
35	34	imp41 430	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) ∧ 𝑟 ∈ 𝐵) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋))
36	35	adantlrr 733	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) ∧ (𝑝(le‘𝐾)𝑋 ∧ 𝑞(le‘𝐾)𝑋))) ∧ 𝑟 ∈ 𝐵) → ((𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) ∧ (𝑝(join‘𝐾)𝑞)(le‘𝐾)𝑋) → 𝑟(le‘𝐾)𝑋))
37	28, 36	mpan2d 706	. . . . . . . . 9 ⊢ ((((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) ∧ (𝑝(le‘𝐾)𝑋 ∧ 𝑞(le‘𝐾)𝑋))) ∧ 𝑟 ∈ 𝐵) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟(le‘𝐾)𝑋))
38	19, 20, 37	syl2an 607	. . . . . . . 8 ⊢ ((((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟(le‘𝐾)𝑋))
39		simpr 489	. . . . . . . 8 ⊢ ((((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) ∧ 𝑟 ∈ (Atoms‘𝐾)) → 𝑟 ∈ (Atoms‘𝐾))
40	38, 39	jctild 534	. . . . . . 7 ⊢ ((((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑋)))
41		breq1 5108	. . . . . . . 8 ⊢ (𝑐 = 𝑟 → (𝑐(le‘𝐾)𝑋 ↔ 𝑟(le‘𝐾)𝑋))
42	41	elrab 3653	. . . . . . 7 ⊢ (𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑋))
43	40, 42	imbitrrdi 255	. . . . . 6 ⊢ ((((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) ∧ 𝑟 ∈ (Atoms‘𝐾)) → (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}))
44	43	ralrimiva 3157	. . . . 5 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) ∧ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∧ 𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})) → ∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}))
45	44	ralrimivva 3208	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → ∀𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}))
46		ssrab2 4036	. . . 4 ⊢ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ⊆ (Atoms‘𝐾)
47	45, 46	jctil 528	. . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → ({𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})))
48		pmapsub.s	. . . . 5 ⊢ 𝑆 = (PSubSp‘𝐾)
49	2, 21, 3, 48	ispsubsp 40381	. . . 4 ⊢ (𝐾 ∈ Lat → ({𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∈ 𝑆 ↔ ({𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}))))
50	49	adantr 485	. . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → ({𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∈ 𝑆 ↔ ({𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ⊆ (Atoms‘𝐾) ∧ ∀𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}∀𝑟 ∈ (Atoms‘𝐾)(𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋}))))
51	47, 50	mpbird 260	. 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∈ 𝑆)
52	5, 51	eqeltrd 2865	1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∀wral 3079  {crab 3417   ⊆ wss 3907   class class class wbr 5105  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  lecple 17307  joincjn 18357  Latclat 18477  Atomscatm 39899  PSubSpcpsubsp 40132  pmapcpmap 40133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-poset 18359  df-lub 18390  df-glb 18391  df-join 18392  df-meet 18393  df-lat 18478  df-ats 39903  df-psubsp 40139  df-pmap 40140

This theorem is referenced by:  hlmod1i  40492  polsubN  40543  pl42lem4N  40618