Theorem pmapsub 37709

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 2738	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2738	. . 3 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
4		pmapsub.m	. . 3 ⊢ 𝑀 = (pmap‘𝐾)
5	1, 2, 3, 4	pmapval 37698	. 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋})
6		breq1 5073	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑝 → (𝑐(le‘𝐾)𝑋 ↔ 𝑝(le‘𝐾)𝑋))
7	6	elrab 3617	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝(le‘𝐾)𝑋))
8	1, 3	atbase 37230	. . . . . . . . . . . . . 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 216	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} → (𝑝 ∈ 𝐵 ∧ 𝑝(le‘𝐾)𝑋))
11		breq1 5073	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑞 → (𝑐(le‘𝐾)𝑋 ↔ 𝑞(le‘𝐾)𝑋))
12	11	elrab 3617	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑞(le‘𝐾)𝑋))
13	1, 3	atbase 37230	. . . . . . . . . . . . . 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 216	. . . . . . . . . . . 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 652	. . . . . . . . . . 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 217	. . . . . . . . . 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 37230	. . . . . . . . 9 ⊢ (𝑟 ∈ (Atoms‘𝐾) → 𝑟 ∈ 𝐵)
21		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (join‘𝐾) = (join‘𝐾)
22	1, 2, 21	latjle12 18083	. . . . . . . . . . . . . . . 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 228	. . . . . . . . . . . . . . 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 1352	. . . . . . . . . . . . . 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 18072	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ 𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵)
30	29	3expib 1120	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Lat → ((𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑝(join‘𝐾)𝑞) ∈ 𝐵))
31	1, 2	lattr 18077	. . . . . . . . . . . . . . 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 1352	. . . . . . . . . . . . . 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 717	. . . . . . . . . 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 690	. . . . . . . . 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 5073	. . . . . . . 8 ⊢ (𝑐 = 𝑟 → (𝑐(le‘𝐾)𝑋 ↔ 𝑟(le‘𝐾)𝑋))
42	41	elrab 3617	. . . . . . 7 ⊢ (𝑟 ∈ {𝑐 ∈ (Atoms‘𝐾) ∣ 𝑐(le‘𝐾)𝑋} ↔ (𝑟 ∈ (Atoms‘𝐾) ∧ 𝑟(le‘𝐾)𝑋))
43	40, 42	syl6ibr 251	. . . . . 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 3107	. . . . 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 3114	. . . 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 4009	. . . 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 37686	. . . 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 256	. 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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067   ⊆ wss 3883   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  lecple 16895  joincjn 17944  Latclat 18064  Atomscatm 37204  PSubSpcpsubsp 37437  pmapcpmap 37438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-poset 17946  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-lat 18065  df-ats 37208  df-psubsp 37444  df-pmap 37445

This theorem is referenced by:  hlmod1i  37797  polsubN  37848  pl42lem4N  37923