Theorem phlpropd 19819

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 18988	. . 3 ⊢ (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec))
9	4, 5	eqtr3d 2646	. . . 4 ⊢ (𝜑 → (Scalar‘𝐾) = (Scalar‘𝐿))
10	9	eleq1d 2672	. . 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 6572	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵)) → (𝑏(·𝑖‘𝐾)𝑎) = (𝑏(·𝑖‘𝐿)𝑎))
13	12	anass1rs 845	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑏(·𝑖‘𝐾)𝑎) = (𝑏(·𝑖‘𝐿)𝑎))
14	13	mpteq2dva 4672	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ 𝐵 ↦ (𝑏(·𝑖‘𝐾)𝑎)) = (𝑏 ∈ 𝐵 ↦ (𝑏(·𝑖‘𝐿)𝑎)))
15	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 = (Base‘𝐾))
16	15	mpteq1d 4666	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ 𝐵 ↦ (𝑏(·𝑖‘𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖‘𝐾)𝑎)))
17	2	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 = (Base‘𝐿))
18	17	mpteq1d 4666	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ 𝐵 ↦ (𝑏(·𝑖‘𝐿)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)))
19	14, 16, 18	3eqtr3d 2652	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖‘𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)))
20		rlmbas 19016	. . . . . . . . . . . 12 ⊢ (Base‘𝐹) = (Base‘(ringLMod‘𝐹))
21	6, 20	eqtri 2632	. . . . . . . . . . 11 ⊢ 𝑃 = (Base‘(ringLMod‘𝐹))
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 = (Base‘(ringLMod‘𝐹)))
23		fvex 6113	. . . . . . . . . . . 12 ⊢ (Scalar‘𝐾) ∈ V
24	4, 23	syl6eqel 2696	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ V)
25		rlmsca 19021	. . . . . . . . . . 11 ⊢ (𝐹 ∈ V → 𝐹 = (Scalar‘(ringLMod‘𝐹)))
26	24, 25	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (Scalar‘(ringLMod‘𝐹)))
27		eqidd 2611	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘(ringLMod‘𝐹))𝑦) = (𝑥(+g‘(ringLMod‘𝐹))𝑦))
28		eqidd 2611	. . . . . . . . . 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 18894	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘𝐹)))
30	4	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐾)))
31	30	oveq2d 6565	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))))
32	5	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐿)))
33	32	oveq2d 6565	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
34	29, 31, 33	3eqtr3d 2652	. . . . . . . 8 ⊢ (𝜑 → (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
35	34	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
36	19, 35	eleq12d 2682	. . . . . 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 6572	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵)) → (𝑎(·𝑖‘𝐾)𝑎) = (𝑎(·𝑖‘𝐿)𝑎))
38	37	anabsan2 859	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎(·𝑖‘𝐾)𝑎) = (𝑎(·𝑖‘𝐿)𝑎))
39	9	fveq2d 6107	. . . . . . . . 9 ⊢ (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
40	39	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
41	38, 40	eqeq12d 2625	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑎(·𝑖‘𝐾)𝑎) = (0g‘(Scalar‘𝐾)) ↔ (𝑎(·𝑖‘𝐿)𝑎) = (0g‘(Scalar‘𝐿))))
42	1, 2, 3	grpidpropd 17084	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
43	42	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (0g‘𝐾) = (0g‘𝐿))
44	43	eqeq2d 2620	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 = (0g‘𝐾) ↔ 𝑎 = (0g‘𝐿)))
45	41, 44	imbi12d 333	. . . . . 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 6107	. . . . . . . . . . . 12 ⊢ (𝜑 → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
47	46	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
48	11	oveqrspc2v 6572	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(·𝑖‘𝐾)𝑏) = (𝑎(·𝑖‘𝐿)𝑏))
49	47, 48	fveq12d 6109	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)))
50	49	anassrs 678	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)))
51	50, 13	eqeq12d 2625	. . . . . . . 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 2968	. . . . . . 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 3121	. . . . . . 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 3121	. . . . . . 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 297	. . . . . 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 1391	. . . . 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 2968	. . . 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 3121	. . . 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 3121	. . . 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 297	. . 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 1391	. 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 2610	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
63		eqid 2610	. . 3 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
64		eqid 2610	. . 3 ⊢ (·𝑖‘𝐾) = (·𝑖‘𝐾)
65		eqid 2610	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
66		eqid 2610	. . 3 ⊢ (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐾))
67		eqid 2610	. . 3 ⊢ (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐾))
68	62, 63, 64, 65, 66, 67	isphl 19792	. 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 2610	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
70		eqid 2610	. . 3 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
71		eqid 2610	. . 3 ⊢ (·𝑖‘𝐿) = (·𝑖‘𝐿)
72		eqid 2610	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
73		eqid 2610	. . 3 ⊢ (*𝑟‘(Scalar‘𝐿)) = (*𝑟‘(Scalar‘𝐿))
74		eqid 2610	. . 3 ⊢ (0g‘(Scalar‘𝐿)) = (0g‘(Scalar‘𝐿))
75	69, 70, 71, 72, 73, 74	isphl 19792	. 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 302	1 ⊢ (𝜑 → (𝐾 ∈ PreHil ↔ 𝐿 ∈ PreHil))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +_gcplusg 15768  *_𝑟cstv 15770  Scalarcsca 15771   ·_𝑠 cvsca 15772  ·_𝑖cip 15773  0_gc0g 15923  *-Ringcsr 18667   LMHom clmhm 18840  LVecclvec 18923  ringLModcrglmod 18990  PreHilcphl 19788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-sca 15784  df-vsca 15785  df-ip 15786  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-ghm 17481  df-mgp 18313  df-ur 18325  df-ring 18372  df-lmod 18688  df-lmhm 18843  df-lvec 18924  df-sra 18993  df-rgmod 18994  df-phl 19790

This theorem is referenced by:  tchphl  22834