Theorem phlssphl 21695

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 2736	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Base‘𝑋) = (Base‘𝑋))
2		eqidd 2736	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (+g‘𝑋) = (+g‘𝑋))
3		eqidd 2736	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑋))
4		eqidd 2736	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (·𝑖‘𝑋) = (·𝑖‘𝑋))
5		phllmod 21666	. . . 4 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
6		phlssphl.x	. . . . 5 ⊢ 𝑋 = (𝑊 ↾s 𝑈)
7		eqid 2735	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
8		eqid 2735	. . . . 5 ⊢ (0g‘𝑋) = (0g‘𝑋)
9		phlssphl.s	. . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊)
10	6, 7, 8, 9	lss0v 21033	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (0g‘𝑋) = (0g‘𝑊))
11	5, 10	sylan 580	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘𝑋) = (0g‘𝑊))
12	11	eqcomd 2741	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘𝑊) = (0g‘𝑋))
13		eqidd 2736	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) = (Scalar‘𝑋))
14		eqidd 2736	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋)))
15		eqidd 2736	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (+g‘(Scalar‘𝑋)) = (+g‘(Scalar‘𝑋)))
16		eqidd 2736	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (.r‘(Scalar‘𝑋)) = (.r‘(Scalar‘𝑋)))
17		eqidd 2736	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (*𝑟‘(Scalar‘𝑋)) = (*𝑟‘(Scalar‘𝑋)))
18		eqidd 2736	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑋)))
19		phllvec 21665	. . 3 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
20	6, 9	lsslvec 21126	. . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ LVec)
21	19, 20	sylan 580	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ LVec)
22		eqid 2735	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
23	6, 22	resssca 17389	. . . . 5 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑊) = (Scalar‘𝑋))
24	23	eqcomd 2741	. . . 4 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑋) = (Scalar‘𝑊))
25	24	adantl 481	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) = (Scalar‘𝑊))
26	22	phlsrng 21667	. . . 4 ⊢ (𝑊 ∈ PreHil → (Scalar‘𝑊) ∈ *-Ring)
27	26	adantr 480	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑊) ∈ *-Ring)
28	25, 27	eqeltrd 2839	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) ∈ *-Ring)
29		simpl 482	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑊 ∈ PreHil)
30		eqid 2735	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
31	6, 30	ressbasss 17284	. . . . . 6 ⊢ (Base‘𝑋) ⊆ (Base‘𝑊)
32	31	sseli 3991	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝑋) → 𝑥 ∈ (Base‘𝑊))
33	31	sseli 3991	. . . . 5 ⊢ (𝑦 ∈ (Base‘𝑋) → 𝑦 ∈ (Base‘𝑊))
34		eqid 2735	. . . . . 6 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
35		eqid 2735	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
36	22, 34, 30, 35	ipcl 21669	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
37	29, 32, 33, 36	syl3an 1159	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
38	24	fveq2d 6911	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
39	38	eleq2d 2825	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → ((𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
40	39	adantl 481	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ((𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
41	40	3ad2ant1 1132	. . . 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 257	. . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)))
43		eqid 2735	. . . . . . . 8 ⊢ (·𝑖‘𝑋) = (·𝑖‘𝑋)
44	6, 34, 43	ssipeq 21692	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (·𝑖‘𝑋) = (·𝑖‘𝑊))
45	44	oveqd 7448	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → (𝑥(·𝑖‘𝑋)𝑦) = (𝑥(·𝑖‘𝑊)𝑦))
46	45	eleq1d 2824	. . . . 5 ⊢ (𝑈 ∈ 𝑆 → ((𝑥(·𝑖‘𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
47	46	adantl 481	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ((𝑥(·𝑖‘𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
48	47	3ad2ant1 1132	. . 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 257	. 2 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)))
50	29	3ad2ant1 1132	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑊 ∈ PreHil)
51	5	adantr 480	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑊 ∈ LMod)
52	51	3ad2ant1 1132	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑊 ∈ LMod)
53	25	fveq2d 6911	. . . . . . . . 9 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
54	53	eleq2d 2825	. . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (𝑞 ∈ (Base‘(Scalar‘𝑋)) ↔ 𝑞 ∈ (Base‘(Scalar‘𝑊))))
55	54	biimpa 476	. . . . . . 7 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋))) → 𝑞 ∈ (Base‘(Scalar‘𝑊)))
56	55	3adant3 1131	. . . . . 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 1132	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑥 ∈ (Base‘𝑊))
58	57	3ad2ant3 1134	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑥 ∈ (Base‘𝑊))
59		eqid 2735	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
60	30, 22, 59, 35	lmodvscl 20893	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑞( ·𝑠 ‘𝑊)𝑥) ∈ (Base‘𝑊))
61	52, 56, 58, 60	syl3anc 1370	. . . . 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 1133	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑦 ∈ (Base‘𝑊))
63	62	3ad2ant3 1134	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑦 ∈ (Base‘𝑊))
64	31	sseli 3991	. . . . . . 7 ⊢ (𝑧 ∈ (Base‘𝑋) → 𝑧 ∈ (Base‘𝑊))
65	64	3ad2ant3 1134	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑧 ∈ (Base‘𝑊))
66	65	3ad2ant3 1134	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑧 ∈ (Base‘𝑊))
67		eqid 2735	. . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊)
68		eqid 2735	. . . . . 6 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
69	22, 34, 30, 67, 68	ipdir 21675	. . . . 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 1371	. . . 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 2735	. . . . . . 7 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
72	22, 34, 30, 35, 59, 71	ipass 21681	. . . . . 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 1371	. . . . 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 7446	. . . 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 2775	. . 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 17336	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → (+g‘𝑊) = (+g‘𝑋))
77	76	eqcomd 2741	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (+g‘𝑋) = (+g‘𝑊))
78	6, 59	ressvsca 17390	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝑆 → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑋))
79	78	eqcomd 2741	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑊))
80	79	oveqd 7448	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (𝑞( ·𝑠 ‘𝑋)𝑥) = (𝑞( ·𝑠 ‘𝑊)𝑥))
81		eqidd 2736	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑦 = 𝑦)
82	77, 80, 81	oveq123d 7452	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → ((𝑞( ·𝑠 ‘𝑋)𝑥)(+g‘𝑋)𝑦) = ((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦))
83		eqidd 2736	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → 𝑧 = 𝑧)
84	44, 82, 83	oveq123d 7452	. . . . . 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 6911	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (+g‘(Scalar‘𝑋)) = (+g‘(Scalar‘𝑊)))
86	24	fveq2d 6911	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (.r‘(Scalar‘𝑋)) = (.r‘(Scalar‘𝑊)))
87		eqidd 2736	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑞 = 𝑞)
88	44	oveqd 7448	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (𝑥(·𝑖‘𝑋)𝑧) = (𝑥(·𝑖‘𝑊)𝑧))
89	86, 87, 88	oveq123d 7452	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖‘𝑋)𝑧)) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧)))
90	44	oveqd 7448	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (𝑦(·𝑖‘𝑋)𝑧) = (𝑦(·𝑖‘𝑊)𝑧))
91	85, 89, 90	oveq123d 7452	. . . . . 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 2751	. . . . 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 481	. . . 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 1132	. . 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 257	. 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 481	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (·𝑖‘𝑋) = (·𝑖‘𝑊))
97	96	oveqdr 7459	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑋)𝑥) = (𝑥(·𝑖‘𝑊)𝑥))
98	24	fveq2d 6911	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
99	98	adantl 481	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
100	99	adantr 480	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
101	97, 100	eqeq12d 2751	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑋)𝑥) = (0g‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))))
102		eqid 2735	. . . . . . 7 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
103	22, 34, 30, 102, 7	ipeq0 21674	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊)) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = (0g‘𝑊)))
104	29, 32, 103	syl2an 596	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = (0g‘𝑊)))
105	104	biimpd 229	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g‘𝑊)))
106	101, 105	sylbid 240	. . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑋)𝑥) = (0g‘(Scalar‘𝑋)) → 𝑥 = (0g‘𝑊)))
107	106	3impia 1116	. 2 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ (𝑥(·𝑖‘𝑋)𝑥) = (0g‘(Scalar‘𝑋))) → 𝑥 = (0g‘𝑊))
108	24	fveq2d 6911	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (*𝑟‘(Scalar‘𝑋)) = (*𝑟‘(Scalar‘𝑊)))
109	108	fveq1d 6909	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)))
110	109	adantl 481	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)))
111	110	3ad2ant1 1132	. . . 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 2735	. . . . . 6 ⊢ (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
113	22, 34, 30, 112	ipcj 21670	. . . . 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 1159	. . . 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 2775	. . 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 6911	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑋)𝑦)) = ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)))
117	44	oveqd 7448	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → (𝑦(·𝑖‘𝑋)𝑥) = (𝑦(·𝑖‘𝑊)𝑥))
118	116, 117	eqeq12d 2751	. . . . 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 481	. . . 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 1132	. . 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 257	. 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 21690	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ PreHil)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ‘cfv 6563  (class class class)co 7431  Basecbs 17245   ↾_s cress 17274  +_gcplusg 17298  ._rcmulr 17299  *_𝑟cstv 17300  Scalarcsca 17301   ·_𝑠 cvsca 17302  ·_𝑖cip 17303  0_gc0g 17486  *-Ringcsr 20856  LModclmod 20875  LSubSpclss 20947  LVecclvec 21119  PreHilcphl 21660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-ghm 19244  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-subrg 20587  df-lmod 20877  df-lss 20948  df-lsp 20988  df-lmhm 21039  df-lvec 21120  df-sra 21190  df-rgmod 21191  df-phl 21662

This theorem is referenced by:  cphsscph  25299