Theorem phlssphl 21872

Description: A subspace of an inner product space (pre-Hilbert space) is an inner product space. (Contributed by AV, 25-Sep-2022.)

Hypotheses

Ref	Expression
phlssphl.x	⊢ 𝑋 = (𝑊 ↾s 𝑈)
phlssphl.s	⊢ 𝑆 = (LSubSp‘𝑊)

Assertion

Ref	Expression
phlssphl	⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ PreHil)

Proof of Theorem phlssphl

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

Step	Hyp	Ref	Expression
1		eqidd 2761	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Base‘𝑋) = (Base‘𝑋))
2		eqidd 2761	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (+g‘𝑋) = (+g‘𝑋))
3		eqidd 2761	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑋))
4		eqidd 2761	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (·𝑖‘𝑋) = (·𝑖‘𝑋))
5		phllmod 21843	. . . 4 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
6		phlssphl.x	. . . . 5 ⊢ 𝑋 = (𝑊 ↾s 𝑈)
7		eqid 2760	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
8		eqid 2760	. . . . 5 ⊢ (0g‘𝑋) = (0g‘𝑋)
9		phlssphl.s	. . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊)
10	6, 7, 8, 9	lss0v 21200	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (0g‘𝑋) = (0g‘𝑊))
11	5, 10	sylan 592	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘𝑋) = (0g‘𝑊))
12	11	eqcomd 2766	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘𝑊) = (0g‘𝑋))
13		eqidd 2761	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) = (Scalar‘𝑋))
14		eqidd 2761	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋)))
15		eqidd 2761	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (+g‘(Scalar‘𝑋)) = (+g‘(Scalar‘𝑋)))
16		eqidd 2761	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (.r‘(Scalar‘𝑋)) = (.r‘(Scalar‘𝑋)))
17		eqidd 2761	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (*𝑟‘(Scalar‘𝑋)) = (*𝑟‘(Scalar‘𝑋)))
18		eqidd 2761	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑋)))
19		phllvec 21842	. . 3 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
20	6, 9	lsslvec 21293	. . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ LVec)
21	19, 20	sylan 592	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ LVec)
22		eqid 2760	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
23	6, 22	resssca 17428	. . . . 5 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑊) = (Scalar‘𝑋))
24	23	eqcomd 2766	. . . 4 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑋) = (Scalar‘𝑊))
25	24	adantl 487	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) = (Scalar‘𝑊))
26	22	phlsrng 21844	. . . 4 ⊢ (𝑊 ∈ PreHil → (Scalar‘𝑊) ∈ *-Ring)
27	26	adantr 486	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑊) ∈ *-Ring)
28	25, 27	eqeltrd 2860	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) ∈ *-Ring)
29		simpl 488	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑊 ∈ PreHil)
30		eqid 2760	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
31	6, 30	ressbasss 17331	. . . . . 6 ⊢ (Base‘𝑋) ⊆ (Base‘𝑊)
32	31	sseli 3927	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝑋) → 𝑥 ∈ (Base‘𝑊))
33	31	sseli 3927	. . . . 5 ⊢ (𝑦 ∈ (Base‘𝑋) → 𝑦 ∈ (Base‘𝑊))
34		eqid 2760	. . . . . 6 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
35		eqid 2760	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
36	22, 34, 30, 35	ipcl 21846	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
37	29, 32, 33, 36	syl3an 1178	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
38	24	fveq2d 6882	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
39	38	eleq2d 2846	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → ((𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
40	39	adantl 487	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ((𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
41	40	3ad2ant1 1151	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
42	37, 41	mpbird 260	. . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)))
43		eqid 2760	. . . . . . . 8 ⊢ (·𝑖‘𝑋) = (·𝑖‘𝑋)
44	6, 34, 43	ssipeq 21869	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (·𝑖‘𝑋) = (·𝑖‘𝑊))
45	44	oveqd 7430	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → (𝑥(·𝑖‘𝑋)𝑦) = (𝑥(·𝑖‘𝑊)𝑦))
46	45	eleq1d 2845	. . . . 5 ⊢ (𝑈 ∈ 𝑆 → ((𝑥(·𝑖‘𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
47	46	adantl 487	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ((𝑥(·𝑖‘𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
48	47	3ad2ant1 1151	. . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
49	42, 48	mpbird 260	. 2 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)))
50	29	3ad2ant1 1151	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑊 ∈ PreHil)
51	5	adantr 486	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑊 ∈ LMod)
52	51	3ad2ant1 1151	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑊 ∈ LMod)
53	25	fveq2d 6882	. . . . . . . . 9 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
54	53	eleq2d 2846	. . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (𝑞 ∈ (Base‘(Scalar‘𝑋)) ↔ 𝑞 ∈ (Base‘(Scalar‘𝑊))))
55	54	biimpa 482	. . . . . . 7 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋))) → 𝑞 ∈ (Base‘(Scalar‘𝑊)))
56	55	3adant3 1150	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑞 ∈ (Base‘(Scalar‘𝑊)))
57	32	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑥 ∈ (Base‘𝑊))
58	57	3ad2ant3 1153	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑥 ∈ (Base‘𝑊))
59		eqid 2760	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
60	30, 22, 59, 35	lmodvscl 21062	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑞( ·𝑠 ‘𝑊)𝑥) ∈ (Base‘𝑊))
61	52, 56, 58, 60	syl3anc 1398	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → (𝑞( ·𝑠 ‘𝑊)𝑥) ∈ (Base‘𝑊))
62	33	3ad2ant2 1152	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑦 ∈ (Base‘𝑊))
63	62	3ad2ant3 1153	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑦 ∈ (Base‘𝑊))
64	31	sseli 3927	. . . . . . 7 ⊢ (𝑧 ∈ (Base‘𝑋) → 𝑧 ∈ (Base‘𝑊))
65	64	3ad2ant3 1153	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑧 ∈ (Base‘𝑊))
66	65	3ad2ant3 1153	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑧 ∈ (Base‘𝑊))
67		eqid 2760	. . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊)
68		eqid 2760	. . . . . 6 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
69	22, 34, 30, 67, 68	ipdir 21852	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ ((𝑞( ·𝑠 ‘𝑊)𝑥) ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧) = (((𝑞( ·𝑠 ‘𝑊)𝑥)(·𝑖‘𝑊)𝑧)(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧)))
70	50, 61, 63, 66, 69	syl13anc 1399	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧) = (((𝑞( ·𝑠 ‘𝑊)𝑥)(·𝑖‘𝑊)𝑧)(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧)))
71		eqid 2760	. . . . . . 7 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
72	22, 34, 30, 35, 59, 71	ipass 21858	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ (𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑞( ·𝑠 ‘𝑊)𝑥)(·𝑖‘𝑊)𝑧) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧)))
73	50, 56, 58, 66, 72	syl13anc 1399	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → ((𝑞( ·𝑠 ‘𝑊)𝑥)(·𝑖‘𝑊)𝑧) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧)))
74	73	oveq1d 7428	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → (((𝑞( ·𝑠 ‘𝑊)𝑥)(·𝑖‘𝑊)𝑧)(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧)) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧)))
75	70, 74	eqtrd 2795	. . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧)))
76	6, 67	ressplusg 17376	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → (+g‘𝑊) = (+g‘𝑋))
77	76	eqcomd 2766	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (+g‘𝑋) = (+g‘𝑊))
78	6, 59	ressvsca 17429	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝑆 → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑋))
79	78	eqcomd 2766	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑊))
80	79	oveqd 7430	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (𝑞( ·𝑠 ‘𝑋)𝑥) = (𝑞( ·𝑠 ‘𝑊)𝑥))
81		eqidd 2761	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑦 = 𝑦)
82	77, 80, 81	oveq123d 7434	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → ((𝑞( ·𝑠 ‘𝑋)𝑥)(+g‘𝑋)𝑦) = ((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦))
83		eqidd 2761	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → 𝑧 = 𝑧)
84	44, 82, 83	oveq123d 7434	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → (((𝑞( ·𝑠 ‘𝑋)𝑥)(+g‘𝑋)𝑦)(·𝑖‘𝑋)𝑧) = (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧))
85	24	fveq2d 6882	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (+g‘(Scalar‘𝑋)) = (+g‘(Scalar‘𝑊)))
86	24	fveq2d 6882	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (.r‘(Scalar‘𝑋)) = (.r‘(Scalar‘𝑊)))
87		eqidd 2761	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑞 = 𝑞)
88	44	oveqd 7430	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (𝑥(·𝑖‘𝑋)𝑧) = (𝑥(·𝑖‘𝑊)𝑧))
89	86, 87, 88	oveq123d 7434	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖‘𝑋)𝑧)) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧)))
90	44	oveqd 7430	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (𝑦(·𝑖‘𝑋)𝑧) = (𝑦(·𝑖‘𝑊)𝑧))
91	85, 89, 90	oveq123d 7434	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → ((𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖‘𝑋)𝑧))(+g‘(Scalar‘𝑋))(𝑦(·𝑖‘𝑋)𝑧)) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧)))
92	84, 91	eqeq12d 2776	. . . . 5 ⊢ (𝑈 ∈ 𝑆 → ((((𝑞( ·𝑠 ‘𝑋)𝑥)(+g‘𝑋)𝑦)(·𝑖‘𝑋)𝑧) = ((𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖‘𝑋)𝑧))(+g‘(Scalar‘𝑋))(𝑦(·𝑖‘𝑋)𝑧)) ↔ (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧))))
93	92	adantl 487	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ((((𝑞( ·𝑠 ‘𝑋)𝑥)(+g‘𝑋)𝑦)(·𝑖‘𝑋)𝑧) = ((𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖‘𝑋)𝑧))(+g‘(Scalar‘𝑋))(𝑦(·𝑖‘𝑋)𝑧)) ↔ (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧))))
94	93	3ad2ant1 1151	. . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → ((((𝑞( ·𝑠 ‘𝑋)𝑥)(+g‘𝑋)𝑦)(·𝑖‘𝑋)𝑧) = ((𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖‘𝑋)𝑧))(+g‘(Scalar‘𝑋))(𝑦(·𝑖‘𝑋)𝑧)) ↔ (((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦)(·𝑖‘𝑊)𝑧) = ((𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧))(+g‘(Scalar‘𝑊))(𝑦(·𝑖‘𝑊)𝑧))))
95	75, 94	mpbird 260	. 2 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → (((𝑞( ·𝑠 ‘𝑋)𝑥)(+g‘𝑋)𝑦)(·𝑖‘𝑋)𝑧) = ((𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖‘𝑋)𝑧))(+g‘(Scalar‘𝑋))(𝑦(·𝑖‘𝑋)𝑧)))
96	44	adantl 487	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (·𝑖‘𝑋) = (·𝑖‘𝑊))
97	96	oveqdr 7441	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑋)𝑥) = (𝑥(·𝑖‘𝑊)𝑥))
98	24	fveq2d 6882	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
99	98	adantl 487	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
100	99	adantr 486	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
101	97, 100	eqeq12d 2776	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑋)𝑥) = (0g‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))))
102		eqid 2760	. . . . . . 7 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
103	22, 34, 30, 102, 7	ipeq0 21851	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊)) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = (0g‘𝑊)))
104	29, 32, 103	syl2an 608	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = (0g‘𝑊)))
105	104	biimpd 232	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g‘𝑊)))
106	101, 105	sylbid 243	. . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑋)𝑥) = (0g‘(Scalar‘𝑋)) → 𝑥 = (0g‘𝑊)))
107	106	3impia 1135	. 2 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ (𝑥(·𝑖‘𝑋)𝑥) = (0g‘(Scalar‘𝑋))) → 𝑥 = (0g‘𝑊))
108	24	fveq2d 6882	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (*𝑟‘(Scalar‘𝑋)) = (*𝑟‘(Scalar‘𝑊)))
109	108	fveq1d 6880	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)))
110	109	adantl 487	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)))
111	110	3ad2ant1 1151	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)))
112		eqid 2760	. . . . . 6 ⊢ (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
113	22, 34, 30, 112	ipcj 21847	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥))
114	29, 32, 33, 113	syl3an 1178	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥))
115	111, 114	eqtrd 2795	. . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥))
116	45	fveq2d 6882	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑋)𝑦)) = ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)))
117	44	oveqd 7430	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → (𝑦(·𝑖‘𝑋)𝑥) = (𝑦(·𝑖‘𝑊)𝑥))
118	116, 117	eqeq12d 2776	. . . . 5 ⊢ (𝑈 ∈ 𝑆 → (((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑋)𝑦)) = (𝑦(·𝑖‘𝑋)𝑥) ↔ ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥)))
119	118	adantl 487	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑋)𝑦)) = (𝑦(·𝑖‘𝑋)𝑥) ↔ ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥)))
120	119	3ad2ant1 1151	. . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑋)𝑦)) = (𝑦(·𝑖‘𝑋)𝑥) ↔ ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥)))
121	115, 120	mpbird 260	. 2 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑋)𝑦)) = (𝑦(·𝑖‘𝑋)𝑥))
122	1, 2, 3, 4, 12, 13, 14, 15, 16, 17, 18, 21, 28, 49, 95, 107, 121	isphld 21867	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ PreHil)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ‘cfv 6533  (class class class)co 7413  Basecbs 17301   ↾_s cress 17322  +_gcplusg 17342  ._rcmulr 17343  *_𝑟cstv 17344  Scalarcsca 17345   ·_𝑠 cvsca 17346  ·_𝑖cip 17347  0_gc0g 17524  *-Ringcsr 21004  LModclmod 21044  LSubSpclss 21115  LVecclvec 21286  PreHilcphl 21837

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-map 8828  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-sets 17256  df-slot 17274  df-ndx 17286  df-base 17302  df-ress 17323  df-plusg 17355  df-mulr 17356  df-sca 17358  df-vsca 17359  df-ip 17360  df-0g 17526  df-mgm 18730  df-sgrp 18821  df-mnd 18837  df-grp 19060  df-minusg 19061  df-sbg 19062  df-subg 19246  df-ghm 19341  df-cmn 19909  df-abl 19910  df-mgp 20274  df-rng 20288  df-ur 20321  df-ring 20374  df-subrg 20732  df-lmod 21046  df-lss 21116  df-lsp 21156  df-lmhm 21206  df-lvec 21287  df-sra 21357  df-rgmod 21358  df-phl 21839

This theorem is used by:  cphsscph  25479