Theorem pmapsub 39751

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 2735	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2735	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		pmapsub.m	. . 3 ⊢ 𝑀 = (pmap‘𝐾)
5	1, 2, 3, 4	pmapval 39740	. 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})
6		breq1 5151	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑝 → (𝑐(le‘𝐾)𝑋 ↔ 𝑝(le‘𝐾)𝑋))
7	6	elrab 3695	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋))
8	1, 3	atbase 39271	. . . . . . . . . . . . . 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 5151	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑞 → (𝑐(le‘𝐾)𝑋 ↔ 𝑞(le‘𝐾)𝑋))
12	11	elrab 3695	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋))
13	1, 3	atbase 39271	. . . . . . . . . . . . . 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 39271	. . . . . . . . 9 ⊢ (𝑟 ∈ (Atoms‘𝐾) → 𝑟 ∈ 𝐵)
21		eqid 2735	. . . . . . . . . . . . . . . . 17 ⊢ (join‘𝐾) = (join‘𝐾)
22	1, 2, 21	latjle12 18508	. . . . . . . . . . . . . . . 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 1353	. . . . . . . . . . . . . 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 18497	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ 𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵)
30	29	3expib 1121	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Lat → ((𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵))
31	1, 2	lattr 18502	. . . . . . . . . . . . . . 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 1353	. . . . . . . . . . . . . 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 5151	. . . . . . . 8 ⊢ (𝑐 = 𝑟 → (𝑐(le‘𝐾)𝑋 ↔ 𝑟(le‘𝐾)𝑋))
42	41	elrab 3695	. . . . . . 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 3144	. . . . 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 3200	. . . 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 4090	. . . 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 39728	. . . 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 2839	1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433   ⊆ wss 3963   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  Latclat 18489  Atomscatm 39245  PSubSpcpsubsp 39479  pmapcpmap 39480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-poset 18371  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-lat 18490  df-ats 39249  df-psubsp 39486  df-pmap 39487

This theorem is referenced by:  hlmod1i  39839  polsubN  39890  pl42lem4N  39965