Theorem phlssphl 20325

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 2799	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Base‘𝑋) = (Base‘𝑋))
2		eqidd 2799	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (+g‘𝑋) = (+g‘𝑋))
3		eqidd 2799	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑋))
4		eqidd 2799	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (·𝑖‘𝑋) = (·𝑖‘𝑋))
5		phllmod 20296	. . . 4 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
6		phlssphl.x	. . . . 5 ⊢ 𝑋 = (𝑊 ↾s 𝑈)
7		eqid 2798	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
8		eqid 2798	. . . . 5 ⊢ (0g‘𝑋) = (0g‘𝑋)
9		phlssphl.s	. . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊)
10	6, 7, 8, 9	lss0v 19334	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (0g‘𝑋) = (0g‘𝑊))
11	5, 10	sylan 576	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘𝑋) = (0g‘𝑊))
12	11	eqcomd 2804	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘𝑊) = (0g‘𝑋))
13		eqidd 2799	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) = (Scalar‘𝑋))
14		eqidd 2799	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋)))
15		eqidd 2799	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (+g‘(Scalar‘𝑋)) = (+g‘(Scalar‘𝑋)))
16		eqidd 2799	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (.r‘(Scalar‘𝑋)) = (.r‘(Scalar‘𝑋)))
17		eqidd 2799	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (*𝑟‘(Scalar‘𝑋)) = (*𝑟‘(Scalar‘𝑋)))
18		eqidd 2799	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑋)))
19		phllvec 20295	. . 3 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
20	6, 9	lsslvec 19425	. . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ LVec)
21	19, 20	sylan 576	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ LVec)
22		eqid 2798	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
23	6, 22	resssca 16349	. . . . 5 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑊) = (Scalar‘𝑋))
24	23	eqcomd 2804	. . . 4 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑋) = (Scalar‘𝑊))
25	24	adantl 474	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) = (Scalar‘𝑊))
26	22	phlsrng 20297	. . . 4 ⊢ (𝑊 ∈ PreHil → (Scalar‘𝑊) ∈ *-Ring)
27	26	adantr 473	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑊) ∈ *-Ring)
28	25, 27	eqeltrd 2877	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) ∈ *-Ring)
29		simpl 475	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑊 ∈ PreHil)
30		eqid 2798	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
31	6, 30	ressbasss 16254	. . . . . 6 ⊢ (Base‘𝑋) ⊆ (Base‘𝑊)
32	31	sseli 3793	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝑋) → 𝑥 ∈ (Base‘𝑊))
33	31	sseli 3793	. . . . 5 ⊢ (𝑦 ∈ (Base‘𝑋) → 𝑦 ∈ (Base‘𝑊))
34		eqid 2798	. . . . . 6 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
35		eqid 2798	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
36	22, 34, 30, 35	ipcl 20299	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
37	29, 32, 33, 36	syl3an 1200	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
38	24	fveq2d 6414	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
39	38	eleq2d 2863	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → ((𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
40	39	adantl 474	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ((𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊))))
41	40	3ad2ant1 1164	. . . 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 249	. . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋)))
43		eqid 2798	. . . . . . . 8 ⊢ (·𝑖‘𝑋) = (·𝑖‘𝑋)
44	6, 34, 43	ssipeq 20322	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (·𝑖‘𝑋) = (·𝑖‘𝑊))
45	44	oveqd 6894	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → (𝑥(·𝑖‘𝑋)𝑦) = (𝑥(·𝑖‘𝑊)𝑦))
46	45	eleq1d 2862	. . . . 5 ⊢ (𝑈 ∈ 𝑆 → ((𝑥(·𝑖‘𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
47	46	adantl 474	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ((𝑥(·𝑖‘𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑋))))
48	47	3ad2ant1 1164	. . 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 249	. 2 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑋)𝑦) ∈ (Base‘(Scalar‘𝑋)))
50	29	3ad2ant1 1164	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑊 ∈ PreHil)
51	5	adantr 473	. . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑊 ∈ LMod)
52	51	3ad2ant1 1164	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑊 ∈ LMod)
53	25	fveq2d 6414	. . . . . . . . 9 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
54	53	eleq2d 2863	. . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (𝑞 ∈ (Base‘(Scalar‘𝑋)) ↔ 𝑞 ∈ (Base‘(Scalar‘𝑊))))
55	54	biimpa 469	. . . . . . 7 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋))) → 𝑞 ∈ (Base‘(Scalar‘𝑊)))
56	55	3adant3 1163	. . . . . 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 1164	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑥 ∈ (Base‘𝑊))
58	57	3ad2ant3 1166	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑥 ∈ (Base‘𝑊))
59		eqid 2798	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
60	30, 22, 59, 35	lmodvscl 19195	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑞( ·𝑠 ‘𝑊)𝑥) ∈ (Base‘𝑊))
61	52, 56, 58, 60	syl3anc 1491	. . . . 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 1165	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑦 ∈ (Base‘𝑊))
63	62	3ad2ant3 1166	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑦 ∈ (Base‘𝑊))
64	31	sseli 3793	. . . . . . 7 ⊢ (𝑧 ∈ (Base‘𝑋) → 𝑧 ∈ (Base‘𝑊))
65	64	3ad2ant3 1166	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑧 ∈ (Base‘𝑊))
66	65	3ad2ant3 1166	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑧 ∈ (Base‘𝑊))
67		eqid 2798	. . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊)
68		eqid 2798	. . . . . 6 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
69	22, 34, 30, 67, 68	ipdir 20305	. . . . 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 1492	. . . 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 2798	. . . . . . 7 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
72	22, 34, 30, 35, 59, 71	ipass 20311	. . . . . 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 1492	. . . . 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 6892	. . . 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 2832	. . 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 16311	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → (+g‘𝑊) = (+g‘𝑋))
77	76	eqcomd 2804	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (+g‘𝑋) = (+g‘𝑊))
78	6, 59	ressvsca 16350	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝑆 → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑋))
79	78	eqcomd 2804	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑊))
80	79	oveqd 6894	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (𝑞( ·𝑠 ‘𝑋)𝑥) = (𝑞( ·𝑠 ‘𝑊)𝑥))
81		eqidd 2799	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑦 = 𝑦)
82	77, 80, 81	oveq123d 6898	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → ((𝑞( ·𝑠 ‘𝑋)𝑥)(+g‘𝑋)𝑦) = ((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦))
83		eqidd 2799	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → 𝑧 = 𝑧)
84	44, 82, 83	oveq123d 6898	. . . . . 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 6414	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (+g‘(Scalar‘𝑋)) = (+g‘(Scalar‘𝑊)))
86	24	fveq2d 6414	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (.r‘(Scalar‘𝑋)) = (.r‘(Scalar‘𝑊)))
87		eqidd 2799	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑞 = 𝑞)
88	44	oveqd 6894	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (𝑥(·𝑖‘𝑋)𝑧) = (𝑥(·𝑖‘𝑊)𝑧))
89	86, 87, 88	oveq123d 6898	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖‘𝑋)𝑧)) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧)))
90	44	oveqd 6894	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (𝑦(·𝑖‘𝑋)𝑧) = (𝑦(·𝑖‘𝑊)𝑧))
91	85, 89, 90	oveq123d 6898	. . . . . 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 2813	. . . . 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 474	. . . 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 1164	. . 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 249	. 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 474	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (·𝑖‘𝑋) = (·𝑖‘𝑊))
97	96	oveqdr 6905	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑋)𝑥) = (𝑥(·𝑖‘𝑊)𝑥))
98	24	fveq2d 6414	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
99	98	adantl 474	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
100	99	adantr 473	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑊)))
101	97, 100	eqeq12d 2813	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑋)𝑥) = (0g‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))))
102		eqid 2798	. . . . . . 7 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
103	22, 34, 30, 102, 7	ipeq0 20304	. . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊)) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = (0g‘𝑊)))
104	29, 32, 103	syl2an 590	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = (0g‘𝑊)))
105	104	biimpd 221	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g‘𝑊)))
106	101, 105	sylbid 232	. . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑋)𝑥) = (0g‘(Scalar‘𝑋)) → 𝑥 = (0g‘𝑊)))
107	106	3impia 1146	. 2 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ (𝑥(·𝑖‘𝑋)𝑥) = (0g‘(Scalar‘𝑋))) → 𝑥 = (0g‘𝑊))
108	24	fveq2d 6414	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (*𝑟‘(Scalar‘𝑋)) = (*𝑟‘(Scalar‘𝑊)))
109	108	fveq1d 6412	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)))
110	109	adantl 474	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)) = ((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖‘𝑊)𝑦)))
111	110	3ad2ant1 1164	. . . 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 2798	. . . . . 6 ⊢ (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
113	22, 34, 30, 112	ipcj 20300	. . . . 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 1200	. . . 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 2832	. . 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 6414	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑋)𝑦)) = ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)))
117	44	oveqd 6894	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → (𝑦(·𝑖‘𝑋)𝑥) = (𝑦(·𝑖‘𝑊)𝑥))
118	116, 117	eqeq12d 2813	. . . . 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 474	. . . 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 1164	. . 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 249	. 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 20320	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ PreHil)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157  ‘cfv 6100  (class class class)co 6877  Basecbs 16181   ↾_s cress 16182  +_gcplusg 16264  ._rcmulr 16265  *_𝑟cstv 16266  Scalarcsca 16267   ·_𝑠 cvsca 16268  ·_𝑖cip 16269  0_gc0g 16412  *-Ringcsr 19159  LModclmod 19178  LSubSpclss 19247  LVecclvec 19420  PreHilcphl 20290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-rep 4963  ax-sep 4974  ax-nul 4982  ax-pow 5034  ax-pr 5096  ax-un 7182  ax-cnex 10279  ax-resscn 10280  ax-1cn 10281  ax-icn 10282  ax-addcl 10283  ax-addrcl 10284  ax-mulcl 10285  ax-mulrcl 10286  ax-mulcom 10287  ax-addass 10288  ax-mulass 10289  ax-distr 10290  ax-i2m1 10291  ax-1ne0 10292  ax-1rid 10293  ax-rnegex 10294  ax-rrecex 10295  ax-cnre 10296  ax-pre-lttri 10297  ax-pre-lttrn 10298  ax-pre-ltadd 10299  ax-pre-mulgt0 10300

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3386  df-sbc 3633  df-csb 3728  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-pss 3784  df-nul 4115  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4628  df-int 4667  df-iun 4711  df-br 4843  df-opab 4905  df-mpt 4922  df-tr 4945  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5897  df-ord 5943  df-on 5944  df-lim 5945  df-suc 5946  df-iota 6063  df-fun 6102  df-fn 6103  df-f 6104  df-f1 6105  df-fo 6106  df-f1o 6107  df-fv 6108  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7400  df-2nd 7401  df-wrecs 7644  df-recs 7706  df-rdg 7744  df-er 7981  df-en 8195  df-dom 8196  df-sdom 8197  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10557  df-neg 10558  df-nn 11312  df-2 11373  df-3 11374  df-4 11375  df-5 11376  df-6 11377  df-7 11378  df-8 11379  df-ndx 16184  df-slot 16185  df-base 16187  df-sets 16188  df-ress 16189  df-plusg 16277  df-mulr 16278  df-sca 16280  df-vsca 16281  df-ip 16282  df-0g 16414  df-mgm 17554  df-sgrp 17596  df-mnd 17607  df-grp 17738  df-minusg 17739  df-sbg 17740  df-subg 17901  df-ghm 17968  df-mgp 18803  df-ur 18815  df-ring 18862  df-subrg 19093  df-lmod 19180  df-lss 19248  df-lsp 19290  df-lmhm 19340  df-lvec 19421  df-sra 19492  df-rgmod 19493  df-phl 20292

This theorem is referenced by:  cphsscph  23374