Theorem phlssphl 21602

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 2732	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Base‘𝑋) = (Base‘𝑋))
2		eqidd 2732	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (+g‘𝑋) = (+g‘𝑋))
3		eqidd 2732	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑋))
4		eqidd 2732	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (·𝑖‘𝑋) = (·𝑖‘𝑋))
5		phllmod 21573	. . . 4 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
6		phlssphl.x	. . . . 5 ⊢ 𝑋 = (𝑊 ↾s 𝑈)
7		eqid 2731	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
8		eqid 2731	. . . . 5 ⊢ (0g‘𝑋) = (0g‘𝑋)
9		phlssphl.s	. . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊)
10	6, 7, 8, 9	lss0v 20956	. . . 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 2737	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘𝑊) = (0g‘𝑋))
13		eqidd 2732	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) = (Scalar‘𝑋))
14		eqidd 2732	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑋)))
15		eqidd 2732	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (+g‘(Scalar‘𝑋)) = (+g‘(Scalar‘𝑋)))
16		eqidd 2732	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (.r‘(Scalar‘𝑋)) = (.r‘(Scalar‘𝑋)))
17		eqidd 2732	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (*𝑟‘(Scalar‘𝑋)) = (*𝑟‘(Scalar‘𝑋)))
18		eqidd 2732	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (0g‘(Scalar‘𝑋)) = (0g‘(Scalar‘𝑋)))
19		phllvec 21572	. . 3 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
20	6, 9	lsslvec 21049	. . 3 ⊢ ((𝑊 ∈ LVec ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ LVec)
21	19, 20	sylan 580	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ LVec)
22		eqid 2731	. . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
23	6, 22	resssca 17253	. . . . 5 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑊) = (Scalar‘𝑋))
24	23	eqcomd 2737	. . . 4 ⊢ (𝑈 ∈ 𝑆 → (Scalar‘𝑋) = (Scalar‘𝑊))
25	24	adantl 481	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) = (Scalar‘𝑊))
26	22	phlsrng 21574	. . . 4 ⊢ (𝑊 ∈ PreHil → (Scalar‘𝑊) ∈ *-Ring)
27	26	adantr 480	. . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑊) ∈ *-Ring)
28	25, 27	eqeltrd 2831	. 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Scalar‘𝑋) ∈ *-Ring)
29		simpl 482	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑊 ∈ PreHil)
30		eqid 2731	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
31	6, 30	ressbasss 17156	. . . . . 6 ⊢ (Base‘𝑋) ⊆ (Base‘𝑊)
32	31	sseli 3925	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝑋) → 𝑥 ∈ (Base‘𝑊))
33	31	sseli 3925	. . . . 5 ⊢ (𝑦 ∈ (Base‘𝑋) → 𝑦 ∈ (Base‘𝑊))
34		eqid 2731	. . . . . 6 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
35		eqid 2731	. . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
36	22, 34, 30, 35	ipcl 21576	. . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
37	29, 32, 33, 36	syl3an 1160	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑊)𝑦) ∈ (Base‘(Scalar‘𝑊)))
38	24	fveq2d 6832	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
39	38	eleq2d 2817	. . . . . 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 1133	. . . 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 2731	. . . . . . . 8 ⊢ (·𝑖‘𝑋) = (·𝑖‘𝑋)
44	6, 34, 43	ssipeq 21599	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (·𝑖‘𝑋) = (·𝑖‘𝑊))
45	44	oveqd 7369	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → (𝑥(·𝑖‘𝑋)𝑦) = (𝑥(·𝑖‘𝑊)𝑦))
46	45	eleq1d 2816	. . . . 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 1133	. . 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 1133	. . . . 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 1133	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑊 ∈ LMod)
53	25	fveq2d 6832	. . . . . . . . 9 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → (Base‘(Scalar‘𝑋)) = (Base‘(Scalar‘𝑊)))
54	53	eleq2d 2817	. . . . . . . 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 1132	. . . . . 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 1133	. . . . . . 7 ⊢ ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑥 ∈ (Base‘𝑊))
58	57	3ad2ant3 1135	. . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑥 ∈ (Base‘𝑊))
59		eqid 2731	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
60	30, 22, 59, 35	lmodvscl 20817	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑞 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘𝑊)) → (𝑞( ·𝑠 ‘𝑊)𝑥) ∈ (Base‘𝑊))
61	52, 56, 58, 60	syl3anc 1373	. . . . 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 1134	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑦 ∈ (Base‘𝑊))
63	62	3ad2ant3 1135	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑦 ∈ (Base‘𝑊))
64	31	sseli 3925	. . . . . . 7 ⊢ (𝑧 ∈ (Base‘𝑋) → 𝑧 ∈ (Base‘𝑊))
65	64	3ad2ant3 1135	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋)) → 𝑧 ∈ (Base‘𝑊))
66	65	3ad2ant3 1135	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑞 ∈ (Base‘(Scalar‘𝑋)) ∧ (𝑥 ∈ (Base‘𝑋) ∧ 𝑦 ∈ (Base‘𝑋) ∧ 𝑧 ∈ (Base‘𝑋))) → 𝑧 ∈ (Base‘𝑊))
67		eqid 2731	. . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊)
68		eqid 2731	. . . . . 6 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊))
69	22, 34, 30, 67, 68	ipdir 21582	. . . . 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 1374	. . . 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 2731	. . . . . . 7 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊))
72	22, 34, 30, 35, 59, 71	ipass 21588	. . . . . 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 1374	. . . . 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 7367	. . . 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 2766	. . 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 17201	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → (+g‘𝑊) = (+g‘𝑋))
77	76	eqcomd 2737	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (+g‘𝑋) = (+g‘𝑊))
78	6, 59	ressvsca 17254	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝑆 → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑋))
79	78	eqcomd 2737	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝑆 → ( ·𝑠 ‘𝑋) = ( ·𝑠 ‘𝑊))
80	79	oveqd 7369	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (𝑞( ·𝑠 ‘𝑋)𝑥) = (𝑞( ·𝑠 ‘𝑊)𝑥))
81		eqidd 2732	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑦 = 𝑦)
82	77, 80, 81	oveq123d 7373	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → ((𝑞( ·𝑠 ‘𝑋)𝑥)(+g‘𝑋)𝑦) = ((𝑞( ·𝑠 ‘𝑊)𝑥)(+g‘𝑊)𝑦))
83		eqidd 2732	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → 𝑧 = 𝑧)
84	44, 82, 83	oveq123d 7373	. . . . . 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 6832	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (+g‘(Scalar‘𝑋)) = (+g‘(Scalar‘𝑊)))
86	24	fveq2d 6832	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (.r‘(Scalar‘𝑋)) = (.r‘(Scalar‘𝑊)))
87		eqidd 2732	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑞 = 𝑞)
88	44	oveqd 7369	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → (𝑥(·𝑖‘𝑋)𝑧) = (𝑥(·𝑖‘𝑊)𝑧))
89	86, 87, 88	oveq123d 7373	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (𝑞(.r‘(Scalar‘𝑋))(𝑥(·𝑖‘𝑋)𝑧)) = (𝑞(.r‘(Scalar‘𝑊))(𝑥(·𝑖‘𝑊)𝑧)))
90	44	oveqd 7369	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (𝑦(·𝑖‘𝑋)𝑧) = (𝑦(·𝑖‘𝑊)𝑧))
91	85, 89, 90	oveq123d 7373	. . . . . 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 2747	. . . . 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 1133	. . 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 7380	. . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → (𝑥(·𝑖‘𝑋)𝑥) = (𝑥(·𝑖‘𝑊)𝑥))
98	24	fveq2d 6832	. . . . . . 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 2747	. . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋)) → ((𝑥(·𝑖‘𝑋)𝑥) = (0g‘(Scalar‘𝑋)) ↔ (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))))
102		eqid 2731	. . . . . . 7 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
103	22, 34, 30, 102, 7	ipeq0 21581	. . . . . 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 1117	. 2 ⊢ (((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) ∧ 𝑥 ∈ (Base‘𝑋) ∧ (𝑥(·𝑖‘𝑋)𝑥) = (0g‘(Scalar‘𝑋))) → 𝑥 = (0g‘𝑊))
108	24	fveq2d 6832	. . . . . . 7 ⊢ (𝑈 ∈ 𝑆 → (*𝑟‘(Scalar‘𝑋)) = (*𝑟‘(Scalar‘𝑊)))
109	108	fveq1d 6830	. . . . . 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 1133	. . . 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 2731	. . . . . 6 ⊢ (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
113	22, 34, 30, 112	ipcj 21577	. . . . 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 1160	. . . 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 2766	. . 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 6832	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑋)𝑦)) = ((*𝑟‘(Scalar‘𝑋))‘(𝑥(·𝑖‘𝑊)𝑦)))
117	44	oveqd 7369	. . . . . 6 ⊢ (𝑈 ∈ 𝑆 → (𝑦(·𝑖‘𝑋)𝑥) = (𝑦(·𝑖‘𝑊)𝑥))
118	116, 117	eqeq12d 2747	. . . . 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 1133	. . 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 21597	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ PreHil)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ‘cfv 6487  (class class class)co 7352  Basecbs 17126   ↾_s cress 17147  +_gcplusg 17167  ._rcmulr 17168  *_𝑟cstv 17169  Scalarcsca 17170   ·_𝑠 cvsca 17171  ·_𝑖cip 17172  0_gc0g 17349  *-Ringcsr 20759  LModclmod 20799  LSubSpclss 20870  LVecclvec 21042  PreHilcphl 21567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11068  ax-resscn 11069  ax-1cn 11070  ax-icn 11071  ax-addcl 11072  ax-addrcl 11073  ax-mulcl 11074  ax-mulrcl 11075  ax-mulcom 11076  ax-addass 11077  ax-mulass 11078  ax-distr 11079  ax-i2m1 11080  ax-1ne0 11081  ax-1rid 11082  ax-rnegex 11083  ax-rrecex 11084  ax-cnre 11085  ax-pre-lttri 11086  ax-pre-lttrn 11087  ax-pre-ltadd 11088  ax-pre-mulgt0 11089

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-sdom 8878  df-pnf 11154  df-mnf 11155  df-xr 11156  df-ltxr 11157  df-le 11158  df-sub 11352  df-neg 11353  df-nn 12132  df-2 12194  df-3 12195  df-4 12196  df-5 12197  df-6 12198  df-7 12199  df-8 12200  df-sets 17081  df-slot 17099  df-ndx 17111  df-base 17127  df-ress 17148  df-plusg 17180  df-mulr 17181  df-sca 17183  df-vsca 17184  df-ip 17185  df-0g 17351  df-mgm 18554  df-sgrp 18633  df-mnd 18649  df-grp 18855  df-minusg 18856  df-sbg 18857  df-subg 19042  df-ghm 19131  df-cmn 19700  df-abl 19701  df-mgp 20065  df-rng 20077  df-ur 20106  df-ring 20159  df-subrg 20491  df-lmod 20801  df-lss 20871  df-lsp 20911  df-lmhm 20962  df-lvec 21043  df-sra 21113  df-rgmod 21114  df-phl 21569

This theorem is referenced by:  cphsscph  25184