Theorem ocvlss 21557

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 21555	. . 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 21515	. . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
7	6	adantr 480	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → 𝑊 ∈ LMod)
8		eqid 2729	. . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊)
9	1, 8	lmod0vcl 20773	. . . . 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 3943	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑉)
13		eqid 2729	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
14		eqid 2729	. . . . . . 7 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
15		eqid 2729	. . . . . . 7 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
16	13, 14, 1, 15, 8	ip0l 21521	. . . . . 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 3125	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ∀𝑥 ∈ 𝑆 ((0g‘𝑊)(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))
19	1, 14, 13, 15, 2	elocv 21553	. . . 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 4301	. 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 3941	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → 𝑦 ∈ 𝑉)
27		eqid 2729	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
28		eqid 2729	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
29	1, 13, 27, 28	lmodvscl 20760	. . . . . 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 3941	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) ∧ (𝑟 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ ( ⊥ ‘𝑆) ∧ 𝑧 ∈ ( ⊥ ‘𝑆))) → 𝑧 ∈ 𝑉)
33		eqid 2729	. . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊)
34	1, 33	lmodvacl 20757	. . . . 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 2729	. . . . . . . 8 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
41	13, 14, 1, 33, 40	ipdir 21524	. . . . . . 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 2729	. . . . . . . . . 10 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
46	13, 14, 1, 28, 27, 45	ipass 21530	. . . . . . . . 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 21554	. . . . . . . . . 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 7385	. . . . . . . 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 20750	. . . . . . . . . 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 20179	. . . . . . . . 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 2768	. . . . . . 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 21554	. . . . . . . 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 7387	. . . . . 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 20751	. . . . . . 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 2768	. . . . 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 3125	. . . 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 21553	. . . 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 3181	. 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 20816	. 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 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044   ⊆ wss 3911  ∅c0 4292  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  +_gcplusg 17196  ._rcmulr 17197  Scalarcsca 17199   ·_𝑠 cvsca 17200  ·_𝑖cip 17201  0_gc0g 17378  Grpcgrp 18841  Ringcrg 20118  LModclmod 20742  LSubSpclss 20813  PreHilcphl 21509  ocvcocv 21545

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-map 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-plusg 17209  df-sca 17212  df-vsca 17213  df-ip 17214  df-0g 17380  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-grp 18844  df-minusg 18845  df-ghm 19121  df-cmn 19688  df-abl 19689  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-lmod 20744  df-lss 20814  df-lmhm 20905  df-lvec 20986  df-sra 21056  df-rgmod 21057  df-phl 21511  df-ocv 21548

This theorem is referenced by:  ocvin  21559  ocvlsp  21561  csslss  21576  pjdm2  21596  pjff  21597  pjf2  21599  pjfo  21600  ocvpj  21602  pjthlem2  25314  pjth  25315