Theorem pmapsub 39769

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 2730	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2730	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		pmapsub.m	. . 3 ⊢ 𝑀 = (pmap‘𝐾)
5	1, 2, 3, 4	pmapval 39758	. 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})
6		breq1 5113	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑝 → (𝑐(le‘𝐾)𝑋 ↔ 𝑝(le‘𝐾)𝑋))
7	6	elrab 3662	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋))
8	1, 3	atbase 39289	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
9	8	anim1i 615	. . . . . . . . . . . . 13 ⊢ ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋) → (𝑝 ∈ 𝐵 ∧ 𝑝(le‘𝐾)𝑋))
10	7, 9	sylbi 217	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} → (𝑝 ∈ 𝐵 ∧ 𝑝(le‘𝐾)𝑋))
11		breq1 5113	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑞 → (𝑐(le‘𝐾)𝑋 ↔ 𝑞(le‘𝐾)𝑋))
12	11	elrab 3662	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋))
13	1, 3	atbase 39289	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ 𝐵)
14	13	anim1i 615	. . . . . . . . . . . . 13 ⊢ ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋) → (𝑞 ∈ 𝐵 ∧ 𝑞(le‘𝐾)𝑋))
15	12, 14	sylbi 217	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} → (𝑞 ∈ 𝐵 ∧ 𝑞(le‘𝐾)𝑋))
16	10, 15	anim12i 613	. . . . . . . . . . 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 656	. . . . . . . . . . 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 218	. . . . . . . . . 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 617	. . . . . . . . 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 39289	. . . . . . . . 9 ⊢ (𝑟 ∈ (Atoms‘𝐾) → 𝑟 ∈ 𝐵)
21		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ (join‘𝐾) = (join‘𝐾)
22	1, 2, 21	latjle12 18416	. . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . 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 1355	. . . . . . . . . . . . . 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 410	. . . . . . . . . . . . 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 86	. . . . . . . . . . . 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 427	. . . . . . . . . . 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 480	. . . . . . . . . 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 18405	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ 𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵)
30	29	3expib 1122	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Lat → ((𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵))
31	1, 2	lattr 18410	. . . . . . . . . . . . . . 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 1355	. . . . . . . . . . . . . 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 95	. . . . . . . . . . . . 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 73	. . . . . . . . . . . 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 425	. . . . . . . . . . 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 721	. . . . . . . . . 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 694	. . . . . . . . 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 596	. . . . . . . 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 484	. . . . . . . 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 525	. . . . . . 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 5113	. . . . . . . 8 ⊢ (𝑐 = 𝑟 → (𝑐(le‘𝐾)𝑋 ↔ 𝑟(le‘𝐾)𝑋))
42	41	elrab 3662	. . . . . . 7 ⊢ (𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑋))
43	40, 42	imbitrrdi 252	. . . . . 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 3126	. . . . 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 3181	. . . 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 4046	. . . 4 ⊢ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ⊆ (Atoms‘𝐾)
47	45, 46	jctil 519	. . 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 39746	. . . 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 480	. . 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 257	. 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ∈ 𝑆)
52	5, 51	eqeltrd 2829	1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408   ⊆ wss 3917   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  lecple 17234  joincjn 18279  Latclat 18397  Atomscatm 39263  PSubSpcpsubsp 39497  pmapcpmap 39498

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-poset 18281  df-lub 18312  df-glb 18313  df-join 18314  df-meet 18315  df-lat 18398  df-ats 39267  df-psubsp 39504  df-pmap 39505

This theorem is referenced by:  hlmod1i  39857  polsubN  39908  pl42lem4N  39983