Theorem phlpropd 20000

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 19167	. . 3 ⊢ (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec))
9	4, 5	eqtr3d 2658	. . . 4 ⊢ (𝜑 → (Scalar‘𝐾) = (Scalar‘𝐿))
10	9	eleq1d 2686	. . 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 6673	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵)) → (𝑏(·𝑖‘𝐾)𝑎) = (𝑏(·𝑖‘𝐿)𝑎))
13	12	anass1rs 849	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑏(·𝑖‘𝐾)𝑎) = (𝑏(·𝑖‘𝐿)𝑎))
14	13	mpteq2dva 4744	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ 𝐵 ↦ (𝑏(·𝑖‘𝐾)𝑎)) = (𝑏 ∈ 𝐵 ↦ (𝑏(·𝑖‘𝐿)𝑎)))
15	1	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 = (Base‘𝐾))
16	15	mpteq1d 4738	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ 𝐵 ↦ (𝑏(·𝑖‘𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖‘𝐾)𝑎)))
17	2	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 = (Base‘𝐿))
18	17	mpteq1d 4738	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ 𝐵 ↦ (𝑏(·𝑖‘𝐿)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)))
19	14, 16, 18	3eqtr3d 2664	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖‘𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)))
20		rlmbas 19195	. . . . . . . . . . . 12 ⊢ (Base‘𝐹) = (Base‘(ringLMod‘𝐹))
21	6, 20	eqtri 2644	. . . . . . . . . . 11 ⊢ 𝑃 = (Base‘(ringLMod‘𝐹))
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 = (Base‘(ringLMod‘𝐹)))
23		fvex 6201	. . . . . . . . . . . 12 ⊢ (Scalar‘𝐾) ∈ V
24	4, 23	syl6eqel 2709	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ V)
25		rlmsca 19200	. . . . . . . . . . 11 ⊢ (𝐹 ∈ V → 𝐹 = (Scalar‘(ringLMod‘𝐹)))
26	24, 25	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (Scalar‘(ringLMod‘𝐹)))
27		eqidd 2623	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘(ringLMod‘𝐹))𝑦) = (𝑥(+g‘(ringLMod‘𝐹))𝑦))
28		eqidd 2623	. . . . . . . . . 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 19073	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘𝐹)))
30	4	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐾)))
31	30	oveq2d 6666	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))))
32	5	fveq2d 6195	. . . . . . . . . 10 ⊢ (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐿)))
33	32	oveq2d 6666	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
34	29, 31, 33	3eqtr3d 2664	. . . . . . . 8 ⊢ (𝜑 → (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
35	34	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
36	19, 35	eleq12d 2695	. . . . . 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 6673	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵)) → (𝑎(·𝑖‘𝐾)𝑎) = (𝑎(·𝑖‘𝐿)𝑎))
38	37	anabsan2 863	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎(·𝑖‘𝐾)𝑎) = (𝑎(·𝑖‘𝐿)𝑎))
39	9	fveq2d 6195	. . . . . . . . 9 ⊢ (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
40	39	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
41	38, 40	eqeq12d 2637	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑎(·𝑖‘𝐾)𝑎) = (0g‘(Scalar‘𝐾)) ↔ (𝑎(·𝑖‘𝐿)𝑎) = (0g‘(Scalar‘𝐿))))
42	1, 2, 3	grpidpropd 17261	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
43	42	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (0g‘𝐾) = (0g‘𝐿))
44	43	eqeq2d 2632	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 = (0g‘𝐾) ↔ 𝑎 = (0g‘𝐿)))
45	41, 44	imbi12d 334	. . . . . 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 6195	. . . . . . . . . . . 12 ⊢ (𝜑 → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
47	46	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
48	11	oveqrspc2v 6673	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(·𝑖‘𝐾)𝑏) = (𝑎(·𝑖‘𝐿)𝑏))
49	47, 48	fveq12d 6197	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)))
50	49	anassrs 680	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)))
51	50, 13	eqeq12d 2637	. . . . . . . 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 2985	. . . . . . 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 3144	. . . . . . 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 3144	. . . . . . 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 298	. . . . . 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 1399	. . . . 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 2985	. . . 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 3144	. . . 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 3144	. . . 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 298	. . 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 1399	. 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 2622	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
63		eqid 2622	. . 3 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
64		eqid 2622	. . 3 ⊢ (·𝑖‘𝐾) = (·𝑖‘𝐾)
65		eqid 2622	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
66		eqid 2622	. . 3 ⊢ (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐾))
67		eqid 2622	. . 3 ⊢ (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐾))
68	62, 63, 64, 65, 66, 67	isphl 19973	. 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 2622	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
70		eqid 2622	. . 3 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
71		eqid 2622	. . 3 ⊢ (·𝑖‘𝐿) = (·𝑖‘𝐿)
72		eqid 2622	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
73		eqid 2622	. . 3 ⊢ (*𝑟‘(Scalar‘𝐿)) = (*𝑟‘(Scalar‘𝐿))
74		eqid 2622	. . 3 ⊢ (0g‘(Scalar‘𝐿)) = (0g‘(Scalar‘𝐿))
75	69, 70, 71, 72, 73, 74	isphl 19973	. 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 303	1 ⊢ (𝜑 → (𝐾 ∈ PreHil ↔ 𝐿 ∈ PreHil))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  *_𝑟cstv 15943  Scalarcsca 15944   ·_𝑠 cvsca 15945  ·_𝑖cip 15946  0_gc0g 16100  *-Ringcsr 18844   LMHom clmhm 19019  LVecclvec 19102  ringLModcrglmod 19169  PreHilcphl 19969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-sca 15957  df-vsca 15958  df-ip 15959  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lmhm 19022  df-lvec 19103  df-sra 19172  df-rgmod 19173  df-phl 19971

This theorem is referenced by:  tchphl  23026