Theorem pmapsub 39725

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 2740	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2740	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		pmapsub.m	. . 3 ⊢ 𝑀 = (pmap‘𝐾)
5	1, 2, 3, 4	pmapval 39714	. 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})
6		breq1 5169	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑝 → (𝑐(le‘𝐾)𝑋 ↔ 𝑝(le‘𝐾)𝑋))
7	6	elrab 3708	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋))
8	1, 3	atbase 39245	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ 𝐵)
9	8	anim1i 614	. . . . . . . . . . . . 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 5169	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑞 → (𝑐(le‘𝐾)𝑋 ↔ 𝑞(le‘𝐾)𝑋))
12	11	elrab 3708	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋))
13	1, 3	atbase 39245	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ (Atoms‘𝐾) → 𝑞 ∈ 𝐵)
14	13	anim1i 614	. . . . . . . . . . . . 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 612	. . . . . . . . . . 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 655	. . . . . . . . . . 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 616	. . . . . . . . 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 39245	. . . . . . . . 9 ⊢ (𝑟 ∈ (Atoms‘𝐾) → 𝑟 ∈ 𝐵)
21		eqid 2740	. . . . . . . . . . . . . . . . 17 ⊢ (join‘𝐾) = (join‘𝐾)
22	1, 2, 21	latjle12 18520	. . . . . . . . . . . . . . . 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 1354	. . . . . . . . . . . . . 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 18509	. . . . . . . . . . . . . 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 18514	. . . . . . . . . . . . . . 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 1354	. . . . . . . . . . . . . 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 720	. . . . . . . . . 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 693	. . . . . . . . 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 595	. . . . . . . 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 5169	. . . . . . . 8 ⊢ (𝑐 = 𝑟 → (𝑐(le‘𝐾)𝑋 ↔ 𝑟(le‘𝐾)𝑋))
42	41	elrab 3708	. . . . . . 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 3152	. . . . 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 4103	. . . 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 39702	. . . 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 2844	1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443   ⊆ wss 3976   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  lecple 17318  joincjn 18381  Latclat 18501  Atomscatm 39219  PSubSpcpsubsp 39453  pmapcpmap 39454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-poset 18383  df-lub 18416  df-glb 18417  df-join 18418  df-meet 18419  df-lat 18502  df-ats 39223  df-psubsp 39460  df-pmap 39461

This theorem is referenced by:  hlmod1i  39813  polsubN  39864  pl42lem4N  39939