Theorem phlpropd 20344

Description: If two structures have the same components (properties), one is a pre-Hilbert space iff the other one is. (Contributed by Mario Carneiro, 8-Oct-2015.)

Hypotheses

Ref	Expression
phlpropd.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
phlpropd.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
phlpropd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
phlpropd.4	⊢ (𝜑 → 𝐹 = (Scalar‘𝐾))
phlpropd.5	⊢ (𝜑 → 𝐹 = (Scalar‘𝐿))
phlpropd.6	⊢ 𝑃 = (Base‘𝐹)
phlpropd.7	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))
phlpropd.8	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(·𝑖‘𝐾)𝑦) = (𝑥(·𝑖‘𝐿)𝑦))

Assertion

Ref	Expression
phlpropd	⊢ (𝜑 → (𝐾 ∈ PreHil ↔ 𝐿 ∈ PreHil))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑥,𝑃,𝑦   𝜑,𝑥,𝑦

Proof of Theorem phlpropd

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		phlpropd.1	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
2		phlpropd.2	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
3		phlpropd.3	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
4		phlpropd.4	. . . 4 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐾))
5		phlpropd.5	. . . 4 ⊢ (𝜑 → 𝐹 = (Scalar‘𝐿))
6		phlpropd.6	. . . 4 ⊢ 𝑃 = (Base‘𝐹)
7		phlpropd.7	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))
8	1, 2, 3, 4, 5, 6, 7	lvecpropd 19932	. . 3 ⊢ (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec))
9	4, 5	eqtr3d 2835	. . . 4 ⊢ (𝜑 → (Scalar‘𝐾) = (Scalar‘𝐿))
10	9	eleq1d 2874	. . 3 ⊢ (𝜑 → ((Scalar‘𝐾) ∈ *-Ring ↔ (Scalar‘𝐿) ∈ *-Ring))
11		phlpropd.8	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(·𝑖‘𝐾)𝑦) = (𝑥(·𝑖‘𝐿)𝑦))
12	11	oveqrspc2v 7162	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵)) → (𝑏(·𝑖‘𝐾)𝑎) = (𝑏(·𝑖‘𝐿)𝑎))
13	12	anass1rs 654	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑏(·𝑖‘𝐾)𝑎) = (𝑏(·𝑖‘𝐿)𝑎))
14	13	mpteq2dva 5125	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ 𝐵 ↦ (𝑏(·𝑖‘𝐾)𝑎)) = (𝑏 ∈ 𝐵 ↦ (𝑏(·𝑖‘𝐿)𝑎)))
15	1	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 = (Base‘𝐾))
16	15	mpteq1d 5119	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ 𝐵 ↦ (𝑏(·𝑖‘𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖‘𝐾)𝑎)))
17	2	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 = (Base‘𝐿))
18	17	mpteq1d 5119	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ 𝐵 ↦ (𝑏(·𝑖‘𝐿)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)))
19	14, 16, 18	3eqtr3d 2841	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖‘𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)))
20		rlmbas 19960	. . . . . . . . . . . 12 ⊢ (Base‘𝐹) = (Base‘(ringLMod‘𝐹))
21	6, 20	eqtri 2821	. . . . . . . . . . 11 ⊢ 𝑃 = (Base‘(ringLMod‘𝐹))
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 = (Base‘(ringLMod‘𝐹)))
23		fvex 6658	. . . . . . . . . . . 12 ⊢ (Scalar‘𝐾) ∈ V
24	4, 23	eqeltrdi 2898	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ V)
25		rlmsca 19965	. . . . . . . . . . 11 ⊢ (𝐹 ∈ V → 𝐹 = (Scalar‘(ringLMod‘𝐹)))
26	24, 25	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (Scalar‘(ringLMod‘𝐹)))
27		eqidd 2799	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘(ringLMod‘𝐹))𝑦) = (𝑥(+g‘(ringLMod‘𝐹))𝑦))
28		eqidd 2799	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥( ·𝑠 ‘(ringLMod‘𝐹))𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐹))𝑦))
29	1, 22, 2, 22, 4, 26, 5, 26, 6, 6, 3, 27, 7, 28	lmhmpropd 19838	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘𝐹)))
30	4	fveq2d 6649	. . . . . . . . . 10 ⊢ (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐾)))
31	30	oveq2d 7151	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))))
32	5	fveq2d 6649	. . . . . . . . . 10 ⊢ (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐿)))
33	32	oveq2d 7151	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
34	29, 31, 33	3eqtr3d 2841	. . . . . . . 8 ⊢ (𝜑 → (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
35	34	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
36	19, 35	eleq12d 2884	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖‘𝐾)𝑎)) ∈ (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) ↔ (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)) ∈ (𝐿 LMHom (ringLMod‘(Scalar‘𝐿)))))
37	11	oveqrspc2v 7162	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵)) → (𝑎(·𝑖‘𝐾)𝑎) = (𝑎(·𝑖‘𝐿)𝑎))
38	37	anabsan2 673	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎(·𝑖‘𝐾)𝑎) = (𝑎(·𝑖‘𝐿)𝑎))
39	9	fveq2d 6649	. . . . . . . . 9 ⊢ (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
40	39	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
41	38, 40	eqeq12d 2814	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑎(·𝑖‘𝐾)𝑎) = (0g‘(Scalar‘𝐾)) ↔ (𝑎(·𝑖‘𝐿)𝑎) = (0g‘(Scalar‘𝐿))))
42	1, 2, 3	grpidpropd 17864	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
43	42	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (0g‘𝐾) = (0g‘𝐿))
44	43	eqeq2d 2809	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 = (0g‘𝐾) ↔ 𝑎 = (0g‘𝐿)))
45	41, 44	imbi12d 348	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (((𝑎(·𝑖‘𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g‘𝐾)) ↔ ((𝑎(·𝑖‘𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g‘𝐿))))
46	9	fveq2d 6649	. . . . . . . . . . . 12 ⊢ (𝜑 → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
47	46	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
48	11	oveqrspc2v 7162	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(·𝑖‘𝐾)𝑏) = (𝑎(·𝑖‘𝐿)𝑏))
49	47, 48	fveq12d 6652	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)))
50	49	anassrs 471	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)))
51	50, 13	eqeq12d 2814	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = (𝑏(·𝑖‘𝐾)𝑎) ↔ ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)) = (𝑏(·𝑖‘𝐿)𝑎)))
52	51	ralbidva 3161	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (∀𝑏 ∈ 𝐵 ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = (𝑏(·𝑖‘𝐾)𝑎) ↔ ∀𝑏 ∈ 𝐵 ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)) = (𝑏(·𝑖‘𝐿)𝑎)))
53	15	raleqdv 3364	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (∀𝑏 ∈ 𝐵 ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = (𝑏(·𝑖‘𝐾)𝑎) ↔ ∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = (𝑏(·𝑖‘𝐾)𝑎)))
54	17	raleqdv 3364	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (∀𝑏 ∈ 𝐵 ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)) = (𝑏(·𝑖‘𝐿)𝑎) ↔ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)) = (𝑏(·𝑖‘𝐿)𝑎)))
55	52, 53, 54	3bitr3d 312	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = (𝑏(·𝑖‘𝐾)𝑎) ↔ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)) = (𝑏(·𝑖‘𝐿)𝑎)))
56	36, 45, 55	3anbi123d 1433	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (((𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖‘𝐾)𝑎)) ∈ (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) ∧ ((𝑎(·𝑖‘𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g‘𝐾)) ∧ ∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = (𝑏(·𝑖‘𝐾)𝑎)) ↔ ((𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)) ∈ (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))) ∧ ((𝑎(·𝑖‘𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g‘𝐿)) ∧ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)) = (𝑏(·𝑖‘𝐿)𝑎))))
57	56	ralbidva 3161	. . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝐵 ((𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖‘𝐾)𝑎)) ∈ (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) ∧ ((𝑎(·𝑖‘𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g‘𝐾)) ∧ ∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = (𝑏(·𝑖‘𝐾)𝑎)) ↔ ∀𝑎 ∈ 𝐵 ((𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)) ∈ (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))) ∧ ((𝑎(·𝑖‘𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g‘𝐿)) ∧ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)) = (𝑏(·𝑖‘𝐿)𝑎))))
58	1	raleqdv 3364	. . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝐵 ((𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖‘𝐾)𝑎)) ∈ (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) ∧ ((𝑎(·𝑖‘𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g‘𝐾)) ∧ ∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = (𝑏(·𝑖‘𝐾)𝑎)) ↔ ∀𝑎 ∈ (Base‘𝐾)((𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖‘𝐾)𝑎)) ∈ (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) ∧ ((𝑎(·𝑖‘𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g‘𝐾)) ∧ ∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = (𝑏(·𝑖‘𝐾)𝑎))))
59	2	raleqdv 3364	. . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝐵 ((𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)) ∈ (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))) ∧ ((𝑎(·𝑖‘𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g‘𝐿)) ∧ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)) = (𝑏(·𝑖‘𝐿)𝑎)) ↔ ∀𝑎 ∈ (Base‘𝐿)((𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)) ∈ (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))) ∧ ((𝑎(·𝑖‘𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g‘𝐿)) ∧ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)) = (𝑏(·𝑖‘𝐿)𝑎))))
60	57, 58, 59	3bitr3d 312	. . 3 ⊢ (𝜑 → (∀𝑎 ∈ (Base‘𝐾)((𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖‘𝐾)𝑎)) ∈ (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) ∧ ((𝑎(·𝑖‘𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g‘𝐾)) ∧ ∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = (𝑏(·𝑖‘𝐾)𝑎)) ↔ ∀𝑎 ∈ (Base‘𝐿)((𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)) ∈ (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))) ∧ ((𝑎(·𝑖‘𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g‘𝐿)) ∧ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)) = (𝑏(·𝑖‘𝐿)𝑎))))
61	8, 10, 60	3anbi123d 1433	. 2 ⊢ (𝜑 → ((𝐾 ∈ LVec ∧ (Scalar‘𝐾) ∈ *-Ring ∧ ∀𝑎 ∈ (Base‘𝐾)((𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖‘𝐾)𝑎)) ∈ (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) ∧ ((𝑎(·𝑖‘𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g‘𝐾)) ∧ ∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = (𝑏(·𝑖‘𝐾)𝑎))) ↔ (𝐿 ∈ LVec ∧ (Scalar‘𝐿) ∈ *-Ring ∧ ∀𝑎 ∈ (Base‘𝐿)((𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)) ∈ (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))) ∧ ((𝑎(·𝑖‘𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g‘𝐿)) ∧ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)) = (𝑏(·𝑖‘𝐿)𝑎)))))
62		eqid 2798	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
63		eqid 2798	. . 3 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
64		eqid 2798	. . 3 ⊢ (·𝑖‘𝐾) = (·𝑖‘𝐾)
65		eqid 2798	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
66		eqid 2798	. . 3 ⊢ (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐾))
67		eqid 2798	. . 3 ⊢ (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐾))
68	62, 63, 64, 65, 66, 67	isphl 20317	. 2 ⊢ (𝐾 ∈ PreHil ↔ (𝐾 ∈ LVec ∧ (Scalar‘𝐾) ∈ *-Ring ∧ ∀𝑎 ∈ (Base‘𝐾)((𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖‘𝐾)𝑎)) ∈ (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) ∧ ((𝑎(·𝑖‘𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g‘𝐾)) ∧ ∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = (𝑏(·𝑖‘𝐾)𝑎))))
69		eqid 2798	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
70		eqid 2798	. . 3 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
71		eqid 2798	. . 3 ⊢ (·𝑖‘𝐿) = (·𝑖‘𝐿)
72		eqid 2798	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
73		eqid 2798	. . 3 ⊢ (*𝑟‘(Scalar‘𝐿)) = (*𝑟‘(Scalar‘𝐿))
74		eqid 2798	. . 3 ⊢ (0g‘(Scalar‘𝐿)) = (0g‘(Scalar‘𝐿))
75	69, 70, 71, 72, 73, 74	isphl 20317	. 2 ⊢ (𝐿 ∈ PreHil ↔ (𝐿 ∈ LVec ∧ (Scalar‘𝐿) ∈ *-Ring ∧ ∀𝑎 ∈ (Base‘𝐿)((𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)) ∈ (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))) ∧ ((𝑎(·𝑖‘𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g‘𝐿)) ∧ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)) = (𝑏(·𝑖‘𝐿)𝑎))))
76	61, 68, 75	3bitr4g 317	1 ⊢ (𝜑 → (𝐾 ∈ PreHil ↔ 𝐿 ∈ PreHil))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ↦ cmpt 5110  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  +_gcplusg 16557  *_𝑟cstv 16559  Scalarcsca 16560   ·_𝑠 cvsca 16561  ·_𝑖cip 16562  0_gc0g 16705  *-Ringcsr 19608   LMHom clmhm 19784  LVecclvec 19867  ringLModcrglmod 19934  PreHilcphl 20313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-sca 16573  df-vsca 16574  df-ip 16575  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-grp 18098  df-ghm 18348  df-mgp 19233  df-ur 19245  df-ring 19292  df-lmod 19629  df-lmhm 19787  df-lvec 19868  df-sra 19937  df-rgmod 19938  df-phl 20315

This theorem is referenced by:  tcphphl  23831