Theorem phlpropd 21855

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 21341	. . 3 ⊢ (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec))
9	4, 5	eqtr3d 2802	. . . 4 ⊢ (𝜑 → (Scalar‘𝐾) = (Scalar‘𝐿))
10	9	eleq1d 2850	. . 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 7446	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵)) → (𝑏(·𝑖‘𝐾)𝑎) = (𝑏(·𝑖‘𝐿)𝑎))
13	12	anass1rs 668	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑏(·𝑖‘𝐾)𝑎) = (𝑏(·𝑖‘𝐿)𝑎))
14	13	mpteq2dva 5206	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ 𝐵 ↦ (𝑏(·𝑖‘𝐾)𝑎)) = (𝑏 ∈ 𝐵 ↦ (𝑏(·𝑖‘𝐿)𝑎)))
15	1	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 = (Base‘𝐾))
16	15	mpteq1d 5203	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ 𝐵 ↦ (𝑏(·𝑖‘𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖‘𝐾)𝑎)))
17	2	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 = (Base‘𝐿))
18	17	mpteq1d 5203	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ 𝐵 ↦ (𝑏(·𝑖‘𝐿)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)))
19	14, 16, 18	3eqtr3d 2808	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖‘𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)))
20		rlmbas 21364	. . . . . . . . . . . 12 ⊢ (Base‘𝐹) = (Base‘(ringLMod‘𝐹))
21	6, 20	eqtri 2788	. . . . . . . . . . 11 ⊢ 𝑃 = (Base‘(ringLMod‘𝐹))
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 = (Base‘(ringLMod‘𝐹)))
23		fvex 6898	. . . . . . . . . . . 12 ⊢ (Scalar‘𝐾) ∈ V
24	4, 23	eqeltrdi 2873	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ V)
25		rlmsca 21369	. . . . . . . . . . 11 ⊢ (𝐹 ∈ V → 𝐹 = (Scalar‘(ringLMod‘𝐹)))
26	24, 25	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (Scalar‘(ringLMod‘𝐹)))
27		eqidd 2766	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘(ringLMod‘𝐹))𝑦) = (𝑥(+g‘(ringLMod‘𝐹))𝑦))
28		eqidd 2766	. . . . . . . . . 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 21244	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘𝐹)))
30	4	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐾)))
31	30	oveq2d 7435	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))))
32	5	fveq2d 6889	. . . . . . . . . 10 ⊢ (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐿)))
33	32	oveq2d 7435	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
34	29, 31, 33	3eqtr3d 2808	. . . . . . . 8 ⊢ (𝜑 → (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
35	34	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
36	19, 35	eleq12d 2859	. . . . . 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 7446	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵)) → (𝑎(·𝑖‘𝐾)𝑎) = (𝑎(·𝑖‘𝐿)𝑎))
38	37	anabsan2 687	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎(·𝑖‘𝐾)𝑎) = (𝑎(·𝑖‘𝐿)𝑎))
39	9	fveq2d 6889	. . . . . . . . 9 ⊢ (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
40	39	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
41	38, 40	eqeq12d 2781	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑎(·𝑖‘𝐾)𝑎) = (0g‘(Scalar‘𝐾)) ↔ (𝑎(·𝑖‘𝐿)𝑎) = (0g‘(Scalar‘𝐿))))
42	1, 2, 3	grpidpropd 18745	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
43	42	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (0g‘𝐾) = (0g‘𝐿))
44	43	eqeq2d 2776	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 = (0g‘𝐾) ↔ 𝑎 = (0g‘𝐿)))
45	41, 44	imbi12d 347	. . . . . 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 6889	. . . . . . . . . . . 12 ⊢ (𝜑 → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
47	46	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
48	11	oveqrspc2v 7446	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(·𝑖‘𝐾)𝑏) = (𝑎(·𝑖‘𝐿)𝑏))
49	47, 48	fveq12d 6892	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)))
50	49	anassrs 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)))
51	50, 13	eqeq12d 2781	. . . . . . . 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 3188	. . . . . . 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 3325	. . . . . . 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 3325	. . . . . . 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 1464	. . . . 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 3188	. . . 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 3325	. . . 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 3325	. . . 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 1464	. 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 2765	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
63		eqid 2765	. . 3 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
64		eqid 2765	. . 3 ⊢ (·𝑖‘𝐾) = (·𝑖‘𝐾)
65		eqid 2765	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
66		eqid 2765	. . 3 ⊢ (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐾))
67		eqid 2765	. . 3 ⊢ (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐾))
68	62, 63, 64, 65, 66, 67	isphl 21828	. 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 2765	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
70		eqid 2765	. . 3 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
71		eqid 2765	. . 3 ⊢ (·𝑖‘𝐿) = (·𝑖‘𝐿)
72		eqid 2765	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
73		eqid 2765	. . 3 ⊢ (*𝑟‘(Scalar‘𝐿)) = (*𝑟‘(Scalar‘𝐿))
74		eqid 2765	. . 3 ⊢ (0g‘(Scalar‘𝐿)) = (0g‘(Scalar‘𝐿))
75	69, 70, 71, 72, 73, 74	isphl 21828	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  Vcvv 3457   ↦ cmpt 5194  ‘cfv 6540  (class class class)co 7419  Basecbs 17291  +_gcplusg 17332  *_𝑟cstv 17334  Scalarcsca 17335   ·_𝑠 cvsca 17336  ·_𝑖cip 17337  0_gc0g 17514  *-Ringcsr 20991   LMHom clmhm 21190  LVecclvec 21273  ringLModcrglmod 21343  PreHilcphl 21824

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-map 8832  df-en 8950  df-dom 8951  df-sdom 8952  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-nn 12249  df-2 12318  df-3 12319  df-4 12320  df-5 12321  df-6 12322  df-7 12323  df-8 12324  df-sets 17246  df-slot 17264  df-ndx 17276  df-base 17292  df-ress 17313  df-plusg 17345  df-sca 17348  df-vsca 17349  df-ip 17350  df-0g 17516  df-mgm 18720  df-sgrp 18809  df-mnd 18825  df-mhm 18878  df-grp 19047  df-ghm 19328  df-mgp 20261  df-ur 20308  df-ring 20361  df-lmod 21033  df-lmhm 21193  df-lvec 21274  df-sra 21344  df-rgmod 21345  df-phl 21826

This theorem is used by:  tcphphl  25437