Theorem ocvlss 21604

Description: The orthocomplement of a subset is a linear subspace of the pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)

Hypotheses

Ref	Expression
ocvss.v	⊢ 𝑉 = (Base‘𝑊)
ocvss.o	⊢ ⊥ = (ocv‘𝑊)
ocvlss.l	⊢ 𝐿 = (LSubSp‘𝑊)

Assertion

Ref	Expression
ocvlss	⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘𝑆) ∈ 𝐿)

Proof of Theorem ocvlss

Dummy variables 𝑥 𝑟 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ocvss.v	. . . 4 ⊢ 𝑉 = (Base‘𝑊)
2		ocvss.o	. . . 4 ⊢ ⊥ = (ocv‘𝑊)
3	1, 2	ocvss 21602	. . 3 ⊢ ( ⊥ ‘𝑆) ⊆ 𝑉
4	3	a1i 11	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘𝑆) ⊆ 𝑉)
5		simpr 484	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ 𝑉)
6		phllmod 21562	. . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
7	6	adantr 480	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → 𝑊 ∈ LMod)
8		eqid 2731	. . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊)
9	1, 8	lmod0vcl 20819	. . . . 5 ⊢ (𝑊 ∈ LMod → (0g‘𝑊) ∈ 𝑉)
10	7, 9	syl 17	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (0g‘𝑊) ∈ 𝑉)
11		simpll 766	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ 𝑆) → 𝑊 ∈ PreHil)
12	5	sselda 3929	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑉)
13		eqid 2731	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
14		eqid 2731	. . . . . . 7 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
15		eqid 2731	. . . . . . 7 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
16	13, 14, 1, 15, 8	ip0l 21568	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ 𝑉) → ((0g‘𝑊)(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))
17	11, 12, 16	syl2anc 584	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ 𝑆) → ((0g‘𝑊)(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))
18	17	ralrimiva 3124	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ∀𝑥 ∈ 𝑆 ((0g‘𝑊)(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))
19	1, 14, 13, 15, 2	elocv 21600	. . . 4 ⊢ ((0g‘𝑊) ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ (0g‘𝑊) ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 ((0g‘𝑊)(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))))
20	5, 10, 18, 19	syl3anbrc 1344	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (0g‘𝑊) ∈ ( ⊥ ‘𝑆))
21	20	ne0d 4287	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘𝑆) ≠ ∅)
22	5	adantr 480	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → 𝑆 ⊆ 𝑉)
23	7	adantr 480	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → 𝑊 ∈ LMod)
24		simpr1 1195	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → 𝑟 ∈ (Base‘(Scalar‘𝑊)))
25		simpr2 1196	. . . . . . 7 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → 𝑦 ∈ ( ⊥ ‘𝑆))
26	3, 25	sselid 3927	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → 𝑦 ∈ 𝑉)
27		eqid 2731	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
28		eqid 2731	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
29	1, 13, 27, 28	lmodvscl 20806	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑉) → (𝑟( ·𝑠 ‘𝑊)𝑦) ∈ 𝑉)
30	23, 24, 26, 29	syl3anc 1373	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → (𝑟( ·𝑠 ‘𝑊)𝑦) ∈ 𝑉)
31		simpr3 1197	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → 𝑧 ∈ ( ⊥ ‘𝑆))
32	3, 31	sselid 3927	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → 𝑧 ∈ 𝑉)
33		eqid 2731	. . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊)
34	1, 33	lmodvacl 20803	. . . . 5 ⊢ ((𝑊 ∈ LMod ∧ (𝑟( ·𝑠 ‘𝑊)𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝑟( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) ∈ 𝑉)
35	23, 30, 32, 34	syl3anc 1373	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → ((𝑟( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) ∈ 𝑉)
36	11	adantlr 715	. . . . . . 7 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑊 ∈ PreHil)
37	30	adantr 480	. . . . . . 7 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (𝑟( ·𝑠 ‘𝑊)𝑦) ∈ 𝑉)
38	32	adantr 480	. . . . . . 7 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑧 ∈ 𝑉)
39	12	adantlr 715	. . . . . . 7 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑉)
40		eqid 2731	. . . . . . . 8 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
41	13, 14, 1, 33, 40	ipdir 21571	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ ((𝑟( ·𝑠 ‘𝑊)𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉)) → (((𝑟( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)(·𝑖‘𝑊)𝑥) = (((𝑟( ·𝑠 ‘𝑊)𝑦)(·𝑖‘𝑊)𝑥)(+g‘(Scalar‘𝑊))(𝑧(·𝑖‘𝑊)𝑥)))
42	36, 37, 38, 39, 41	syl13anc 1374	. . . . . 6 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (((𝑟( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)(·𝑖‘𝑊)𝑥) = (((𝑟( ·𝑠 ‘𝑊)𝑦)(·𝑖‘𝑊)𝑥)(+g‘(Scalar‘𝑊))(𝑧(·𝑖‘𝑊)𝑥)))
43	24	adantr 480	. . . . . . . . 9 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑟 ∈ (Base‘(Scalar‘𝑊)))
44	26	adantr 480	. . . . . . . . 9 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑦 ∈ 𝑉)
45		eqid 2731	. . . . . . . . . 10 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
46	13, 14, 1, 28, 27, 45	ipass 21577	. . . . . . . . 9 ⊢ ((𝑊 ∈ PreHil ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉)) → ((𝑟( ·𝑠 ‘𝑊)𝑦)(·𝑖‘𝑊)𝑥) = (𝑟(.r‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑥)))
47	36, 43, 44, 39, 46	syl13anc 1374	. . . . . . . 8 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → ((𝑟( ·𝑠 ‘𝑊)𝑦)(·𝑖‘𝑊)𝑥) = (𝑟(.r‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑥)))
48	1, 14, 13, 15, 2	ocvi 21601	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑦(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))
49	25, 48	sylan 580	. . . . . . . . 9 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (𝑦(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))
50	49	oveq2d 7357	. . . . . . . 8 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (𝑟(.r‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑥)) = (𝑟(.r‘(Scalar‘𝑊))(0g‘(Scalar‘𝑊))))
51	23	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → 𝑊 ∈ LMod)
52	13	lmodring 20796	. . . . . . . . . 10 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Ring)
53	51, 52	syl 17	. . . . . . . . 9 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (Scalar‘𝑊) ∈ Ring)
54	28, 45, 15	ringrz 20207	. . . . . . . . 9 ⊢ (((Scalar‘𝑊) ∈ Ring ∧ 𝑟 ∈ (Base‘(Scalar‘𝑊))) → (𝑟(.r‘(Scalar‘𝑊))(0g‘(Scalar‘𝑊))) = (0g‘(Scalar‘𝑊)))
55	53, 43, 54	syl2anc 584	. . . . . . . 8 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (𝑟(.r‘(Scalar‘𝑊))(0g‘(Scalar‘𝑊))) = (0g‘(Scalar‘𝑊)))
56	47, 50, 55	3eqtrd 2770	. . . . . . 7 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → ((𝑟( ·𝑠 ‘𝑊)𝑦)(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))
57	1, 14, 13, 15, 2	ocvi 21601	. . . . . . . 8 ⊢ ((𝑧 ∈ ( ⊥ ‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑧(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))
58	31, 57	sylan 580	. . . . . . 7 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (𝑧(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))
59	56, 58	oveq12d 7359	. . . . . 6 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (((𝑟( ·𝑠 ‘𝑊)𝑦)(·𝑖‘𝑊)𝑥)(+g‘(Scalar‘𝑊))(𝑧(·𝑖‘𝑊)𝑥)) = ((0g‘(Scalar‘𝑊))(+g‘(Scalar‘𝑊))(0g‘(Scalar‘𝑊))))
60	13	lmodfgrp 20797	. . . . . . 7 ⊢ (𝑊 ∈ LMod → (Scalar‘𝑊) ∈ Grp)
61	28, 15	grpidcl 18873	. . . . . . . 8 ⊢ ((Scalar‘𝑊) ∈ Grp → (0g‘(Scalar‘𝑊)) ∈ (Base‘(Scalar‘𝑊)))
62	28, 40, 15	grplid 18875	. . . . . . . 8 ⊢ (((Scalar‘𝑊) ∈ Grp ∧ (0g‘(Scalar‘𝑊)) ∈ (Base‘(Scalar‘𝑊))) → ((0g‘(Scalar‘𝑊))(+g‘(Scalar‘𝑊))(0g‘(Scalar‘𝑊))) = (0g‘(Scalar‘𝑊)))
63	61, 62	mpdan 687	. . . . . . 7 ⊢ ((Scalar‘𝑊) ∈ Grp → ((0g‘(Scalar‘𝑊))(+g‘(Scalar‘𝑊))(0g‘(Scalar‘𝑊))) = (0g‘(Scalar‘𝑊)))
64	51, 60, 63	3syl 18	. . . . . 6 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → ((0g‘(Scalar‘𝑊))(+g‘(Scalar‘𝑊))(0g‘(Scalar‘𝑊))) = (0g‘(Scalar‘𝑊)))
65	42, 59, 64	3eqtrd 2770	. . . . 5 ⊢ ((((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) ∧ 𝑥 ∈ 𝑆) → (((𝑟( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))
66	65	ralrimiva 3124	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → ∀𝑥 ∈ 𝑆 (((𝑟( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))
67	1, 14, 13, 15, 2	elocv 21600	. . . 4 ⊢ (((𝑟( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ ((𝑟( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (((𝑟( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧)(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))))
68	22, 35, 66, 67	syl3anbrc 1344	. . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → ((𝑟( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) ∈ ( ⊥ ‘𝑆))
69	68	ralrimivvva 3178	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ ( ⊥ ‘𝑆)∀𝑧 ∈ ( ⊥ ‘𝑆)((𝑟( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) ∈ ( ⊥ ‘𝑆))
70		ocvlss.l	. . 3 ⊢ 𝐿 = (LSubSp‘𝑊)
71	13, 28, 1, 33, 27, 70	islss 20862	. 2 ⊢ (( ⊥ ‘𝑆) ∈ 𝐿 ↔ (( ⊥ ‘𝑆) ⊆ 𝑉 ∧ ( ⊥ ‘𝑆) ≠ ∅ ∧ ∀𝑟 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ ( ⊥ ‘𝑆)∀𝑧 ∈ ( ⊥ ‘𝑆)((𝑟( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)𝑧) ∈ ( ⊥ ‘𝑆)))
72	4, 21, 69, 71	syl3anbrc 1344	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘𝑆) ∈ 𝐿)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047   ⊆ wss 3897  ∅c0 4278  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  +_gcplusg 17156  ._rcmulr 17157  Scalarcsca 17159   ·_𝑠 cvsca 17160  ·_𝑖cip 17161  0_gc0g 17338  Grpcgrp 18841  Ringcrg 20146  LModclmod 20788  LSubSpclss 20859  PreHilcphl 21556  ocvcocv 21592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-er 8617  df-map 8747  df-en 8865  df-dom 8866  df-sdom 8867  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-2 12183  df-3 12184  df-4 12185  df-5 12186  df-6 12187  df-7 12188  df-8 12189  df-sets 17070  df-slot 17088  df-ndx 17100  df-base 17116  df-plusg 17169  df-sca 17172  df-vsca 17173  df-ip 17174  df-0g 17340  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-grp 18844  df-minusg 18845  df-ghm 19120  df-cmn 19689  df-abl 19690  df-mgp 20054  df-rng 20066  df-ur 20095  df-ring 20148  df-lmod 20790  df-lss 20860  df-lmhm 20951  df-lvec 21032  df-sra 21102  df-rgmod 21103  df-phl 21558  df-ocv 21595

This theorem is referenced by:  ocvin  21606  ocvlsp  21608  csslss  21623  pjdm2  21643  pjff  21644  pjf2  21646  pjfo  21647  ocvpj  21649  pjthlem2  25360  pjth  25361