Theorem phlssphl 21691

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 2762	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Base‘𝑋) = (Base‘𝑋))
2		eqidd 2762	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (+g‘𝑋) = (+g‘𝑋))
3		eqidd 2762	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑋))
4		eqidd 2762	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (·𝑖‘𝑋) = (·𝑖‘𝑋))
5		phllmod 21662	. . . 4 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
6		phlssphl.x	. . . . 5 ⊢ 𝑋 = (𝑊 ↾s 𝑈)
7		eqid 2761	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
8		eqid 2761	. . . . 5 ⊢ (0g‘𝑋) = (0g‘𝑋)
9		phlssphl.s	. . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊)
10	6, 7, 8, 9	lss0v 21063	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (0g‘𝑋) = (0g‘𝑊))
11	5, 10	sylan 589	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘𝑋) = (0g‘𝑊))
12	11	eqcomd 2767	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘𝑊) = (0g‘𝑋))
13		eqidd 2762	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) = (Scalar‘𝑋))
14		eqidd 2762	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋)))
15		eqidd 2762	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (+g‘(Scalar‘𝑋)) = (+g‘(Scalar‘𝑋)))
16		eqidd 2762	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (.r‘(Scalar‘𝑋)) = (.r‘(Scalar‘𝑋)))
17		eqidd 2762	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (*𝑟‘(Scalar‘𝑋)) = (*𝑟‘(Scalar‘𝑋)))
18		eqidd 2762	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑋)))
19		phllvec 21661	. . 3 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
20	6, 9	lsslvec 21156	. . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ LVec)
21	19, 20	sylan 589	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ LVec)
22		eqid 2761	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
23	6, 22	resssca 17355	. . . . 5 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑊) = (Scalar‘𝑋))
24	23	eqcomd 2767	. . . 4 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑋) = (Scalar‘𝑊))
25	24	adantl 485	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) = (Scalar‘𝑊))
26	22	phlsrng 21663	. . . 4 ⊢ (𝑊 ∈ PreHil → (Scalar‘𝑊) ∈ *-Ring)
27	26	adantr 484	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑊) ∈ *-Ring)
28	25, 27	eqeltrd 2861	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) ∈ *-Ring)
29		simpl 486	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑊 ∈ PreHil)
30		eqid 2761	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
31	6, 30	ressbasss 17258	. . . . . 6 ⊢ (Base‘𝑋) ⊆ (Base‘𝑊)
32	31	sseli 3932	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝑋) → 𝑥 ∈ (Base‘𝑊))
33	31	sseli 3932	. . . . 5 ⊢ (𝑦 ∈ (Base‘𝑋) → 𝑦 ∈ (Base‘𝑊))
34		eqid 2761	. . . . . 6 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
35		eqid 2761	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
36	22, 34, 30, 35	ipcl 21665	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
37	29, 32, 33, 36	syl3an 1172	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
38	24	fveq2d 6867	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
39	38	eleq2d 2847	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → ((𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
40	39	adantl 485	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ((𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
41	40	3ad2ant1 1145	. . . 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 259	. . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)))
43		eqid 2761	. . . . . . . 8 ⊢ (·𝑖‘𝑋) = (·𝑖‘𝑋)
44	6, 34, 43	ssipeq 21688	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (·𝑖‘𝑋) = (·𝑖‘𝑊))
45	44	oveqd 7409	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → (𝑥(·𝑖‘𝑋)𝑦) = (𝑥(·𝑖‘𝑊)𝑦))
46	45	eleq1d 2846	. . . . 5 ⊢ (𝑈 ∈ 𝑆 → ((𝑥(·𝑖‘𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
47	46	adantl 485	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ((𝑥(·𝑖‘𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
48	47	3ad2ant1 1145	. . 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 259	. 2 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)))
50	29	3ad2ant1 1145	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑊 ∈ PreHil)
51	5	adantr 484	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑊 ∈ LMod)
52	51	3ad2ant1 1145	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑊 ∈ LMod)
53	25	fveq2d 6867	. . . . . . . . 9 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
54	53	eleq2d 2847	. . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (𝑞 ∈ (Base‘(Scalar‘𝑋)) ↔ 𝑞 ∈ (Base‘(Scalar‘𝑊))))
55	54	biimpa 480	. . . . . . 7 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋))) → 𝑞 ∈ (Base‘(Scalar‘𝑊)))
56	55	3adant3 1144	. . . . . 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 1145	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑥 ∈ (Base‘𝑊))
58	57	3ad2ant3 1147	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑥 ∈ (Base‘𝑊))
59		eqid 2761	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
60	30, 22, 59, 35	lmodvscl 20925	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑞( ·𝑠 ‘𝑊)𝑥) ∈ (Base‘𝑊))
61	52, 56, 58, 60	syl3anc 1389	. . . . 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 1146	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑦 ∈ (Base‘𝑊))
63	62	3ad2ant3 1147	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑦 ∈ (Base‘𝑊))
64	31	sseli 3932	. . . . . . 7 ⊢ (𝑧 ∈ (Base‘𝑋) → 𝑧 ∈ (Base‘𝑊))
65	64	3ad2ant3 1147	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑧 ∈ (Base‘𝑊))
66	65	3ad2ant3 1147	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑧 ∈ (Base‘𝑊))
67		eqid 2761	. . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊)
68		eqid 2761	. . . . . 6 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
69	22, 34, 30, 67, 68	ipdir 21671	. . . . 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 1390	. . . 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 2761	. . . . . . 7 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
72	22, 34, 30, 35, 59, 71	ipass 21677	. . . . . 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 1390	. . . . 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 7407	. . . 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 2796	. . 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 17303	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → (+g‘𝑊) = (+g‘𝑋))
77	76	eqcomd 2767	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (+g‘𝑋) = (+g‘𝑊))
78	6, 59	ressvsca 17356	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝑆 → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑋))
79	78	eqcomd 2767	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑊))
80	79	oveqd 7409	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (𝑞( ·𝑠 ‘𝑋)𝑥) = (𝑞( ·𝑠 ‘𝑊)𝑥))
81		eqidd 2762	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑦 = 𝑦)
82	77, 80, 81	oveq123d 7413	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → ((𝑞( ·𝑠 ‘𝑋)𝑥)(+g‘𝑋)𝑦) = ((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦))
83		eqidd 2762	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → 𝑧 = 𝑧)
84	44, 82, 83	oveq123d 7413	. . . . . 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 6867	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (+g‘(Scalar‘𝑋)) = (+g‘(Scalar‘𝑊)))
86	24	fveq2d 6867	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (.r‘(Scalar‘𝑋)) = (.r‘(Scalar‘𝑊)))
87		eqidd 2762	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑞 = 𝑞)
88	44	oveqd 7409	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (𝑥(·𝑖‘𝑋)𝑧) = (𝑥(·𝑖‘𝑊)𝑧))
89	86, 87, 88	oveq123d 7413	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖‘𝑋)𝑧)) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧)))
90	44	oveqd 7409	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (𝑦(·𝑖‘𝑋)𝑧) = (𝑦(·𝑖‘𝑊)𝑧))
91	85, 89, 90	oveq123d 7413	. . . . . 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 2777	. . . . 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 485	. . . 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 1145	. . 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 259	. 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 485	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (·𝑖‘𝑋) = (·𝑖‘𝑊))
97	96	oveqdr 7420	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑋)𝑥) = (𝑥(·𝑖‘𝑊)𝑥))
98	24	fveq2d 6867	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
99	98	adantl 485	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
100	99	adantr 484	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
101	97, 100	eqeq12d 2777	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑋)𝑥) = (0g‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))))
102		eqid 2761	. . . . . . 7 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
103	22, 34, 30, 102, 7	ipeq0 21670	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊)) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = (0g‘𝑊)))
104	29, 32, 103	syl2an 605	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = (0g‘𝑊)))
105	104	biimpd 231	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g‘𝑊)))
106	101, 105	sylbid 242	. . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑋)𝑥) = (0g‘(Scalar‘𝑋)) → 𝑥 = (0g‘𝑊)))
107	106	3impia 1129	. 2 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ (𝑥(·𝑖‘𝑋)𝑥) = (0g‘(Scalar‘𝑋))) → 𝑥 = (0g‘𝑊))
108	24	fveq2d 6867	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (*𝑟‘(Scalar‘𝑋)) = (*𝑟‘(Scalar‘𝑊)))
109	108	fveq1d 6865	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)))
110	109	adantl 485	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)))
111	110	3ad2ant1 1145	. . . 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 2761	. . . . . 6 ⊢ (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
113	22, 34, 30, 112	ipcj 21666	. . . . 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 1172	. . . 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 2796	. . 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 6867	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑋)𝑦)) = ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)))
117	44	oveqd 7409	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → (𝑦(·𝑖‘𝑋)𝑥) = (𝑦(·𝑖‘𝑊)𝑥))
118	116, 117	eqeq12d 2777	. . . . 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 485	. . . 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 1145	. . 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 259	. 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 21686	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ PreHil)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ‘cfv 6517  (class class class)co 7392  Basecbs 17228   ↾_s cress 17249  +_gcplusg 17269  ._rcmulr 17270  *_𝑟cstv 17271  Scalarcsca 17272   ·_𝑠 cvsca 17273  ·_𝑖cip 17274  0_gc0g 17451  *-Ringcsr 20867  LModclmod 20907  LSubSpclss 20978  LVecclvec 21149  PreHilcphl 21656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-er 8673  df-map 8805  df-en 8924  df-dom 8925  df-sdom 8926  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17250  df-plusg 17282  df-mulr 17283  df-sca 17285  df-vsca 17286  df-ip 17287  df-0g 17453  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-grp 18961  df-minusg 18962  df-sbg 18963  df-subg 19148  df-ghm 19237  df-cmn 19805  df-abl 19806  df-mgp 20170  df-rng 20182  df-ur 20211  df-ring 20264  df-subrg 20599  df-lmod 20909  df-lss 20979  df-lsp 21019  df-lmhm 21069  df-lvec 21150  df-sra 21220  df-rgmod 21221  df-phl 21658

This theorem is referenced by:  cphsscph  25293