Theorem phlpropd 21059

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 20628	. . 3 ⊢ (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec))
9	4, 5	eqtr3d 2778	. . . 4 ⊢ (𝜑 → (Scalar‘𝐾) = (Scalar‘𝐿))
10	9	eleq1d 2822	. . 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 7384	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵)) → (𝑏(·𝑖‘𝐾)𝑎) = (𝑏(·𝑖‘𝐿)𝑎))
13	12	anass1rs 653	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → (𝑏(·𝑖‘𝐾)𝑎) = (𝑏(·𝑖‘𝐿)𝑎))
14	13	mpteq2dva 5205	. . . . . . . 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 5200	. . . . . . . 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 5200	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ 𝐵 ↦ (𝑏(·𝑖‘𝐿)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)))
19	14, 16, 18	3eqtr3d 2784	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖‘𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖‘𝐿)𝑎)))
20		rlmbas 20664	. . . . . . . . . . . 12 ⊢ (Base‘𝐹) = (Base‘(ringLMod‘𝐹))
21	6, 20	eqtri 2764	. . . . . . . . . . 11 ⊢ 𝑃 = (Base‘(ringLMod‘𝐹))
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 = (Base‘(ringLMod‘𝐹)))
23		fvex 6855	. . . . . . . . . . . 12 ⊢ (Scalar‘𝐾) ∈ V
24	4, 23	eqeltrdi 2846	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ V)
25		rlmsca 20669	. . . . . . . . . . 11 ⊢ (𝐹 ∈ V → 𝐹 = (Scalar‘(ringLMod‘𝐹)))
26	24, 25	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 = (Scalar‘(ringLMod‘𝐹)))
27		eqidd 2737	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) → (𝑥(+g‘(ringLMod‘𝐹))𝑦) = (𝑥(+g‘(ringLMod‘𝐹))𝑦))
28		eqidd 2737	. . . . . . . . . 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 20534	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘𝐹)))
30	4	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐾)))
31	30	oveq2d 7373	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))))
32	5	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐿)))
33	32	oveq2d 7373	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
34	29, 31, 33	3eqtr3d 2784	. . . . . . . 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 2832	. . . . . 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 7384	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑎 ∈ 𝐵)) → (𝑎(·𝑖‘𝐾)𝑎) = (𝑎(·𝑖‘𝐿)𝑎))
38	37	anabsan2 672	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎(·𝑖‘𝐾)𝑎) = (𝑎(·𝑖‘𝐿)𝑎))
39	9	fveq2d 6846	. . . . . . . . 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 2752	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑎(·𝑖‘𝐾)𝑎) = (0g‘(Scalar‘𝐾)) ↔ (𝑎(·𝑖‘𝐿)𝑎) = (0g‘(Scalar‘𝐿))))
42	1, 2, 3	grpidpropd 18517	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
43	42	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (0g‘𝐾) = (0g‘𝐿))
44	43	eqeq2d 2747	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 = (0g‘𝐾) ↔ 𝑎 = (0g‘𝐿)))
45	41, 44	imbi12d 344	. . . . . 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 6846	. . . . . . . . . . . 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 7384	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(·𝑖‘𝐾)𝑏) = (𝑎(·𝑖‘𝐿)𝑏))
49	47, 48	fveq12d 6849	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)))
50	49	anassrs 468	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ 𝑏 ∈ 𝐵) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖‘𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖‘𝐿)𝑏)))
51	50, 13	eqeq12d 2752	. . . . . . . 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 3172	. . . . . . 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 3313	. . . . . . 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 3313	. . . . . . 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 308	. . . . . 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 1436	. . . . 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 3172	. . . 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 3313	. . . 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 3313	. . . 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 308	. . 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 1436	. 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 2736	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
63		eqid 2736	. . 3 ⊢ (Scalar‘𝐾) = (Scalar‘𝐾)
64		eqid 2736	. . 3 ⊢ (·𝑖‘𝐾) = (·𝑖‘𝐾)
65		eqid 2736	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
66		eqid 2736	. . 3 ⊢ (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐾))
67		eqid 2736	. . 3 ⊢ (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐾))
68	62, 63, 64, 65, 66, 67	isphl 21032	. 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 2736	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
70		eqid 2736	. . 3 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿)
71		eqid 2736	. . 3 ⊢ (·𝑖‘𝐿) = (·𝑖‘𝐿)
72		eqid 2736	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
73		eqid 2736	. . 3 ⊢ (*𝑟‘(Scalar‘𝐿)) = (*𝑟‘(Scalar‘𝐿))
74		eqid 2736	. . 3 ⊢ (0g‘(Scalar‘𝐿)) = (0g‘(Scalar‘𝐿))
75	69, 70, 71, 72, 73, 74	isphl 21032	. 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 313	1 ⊢ (𝜑 → (𝐾 ∈ PreHil ↔ 𝐿 ∈ PreHil))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  Vcvv 3445   ↦ cmpt 5188  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  +_gcplusg 17133  *_𝑟cstv 17135  Scalarcsca 17136   ·_𝑠 cvsca 17137  ·_𝑖cip 17138  0_gc0g 17321  *-Ringcsr 20303   LMHom clmhm 20480  LVecclvec 20563  ringLModcrglmod 20630  PreHilcphl 21028

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-sca 17149  df-vsca 17150  df-ip 17151  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-grp 18751  df-ghm 19006  df-mgp 19897  df-ur 19914  df-ring 19966  df-lmod 20324  df-lmhm 20483  df-lvec 20564  df-sra 20633  df-rgmod 20634  df-phl 21030

This theorem is referenced by:  tcphphl  24591