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Theorem phlpropd 20351
 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 (𝜑𝐵 = (Base‘𝐾))
phlpropd.2 (𝜑𝐵 = (Base‘𝐿))
phlpropd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
phlpropd.4 (𝜑𝐹 = (Scalar‘𝐾))
phlpropd.5 (𝜑𝐹 = (Scalar‘𝐿))
phlpropd.6 𝑃 = (Base‘𝐹)
phlpropd.7 ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
phlpropd.8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(·𝑖𝐾)𝑦) = (𝑥(·𝑖𝐿)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.
StepHypRef Expression
1 phlpropd.1 . . . 4 (𝜑𝐵 = (Base‘𝐾))
2 phlpropd.2 . . . 4 (𝜑𝐵 = (Base‘𝐿))
3 phlpropd.3 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
4 phlpropd.4 . . . 4 (𝜑𝐹 = (Scalar‘𝐾))
5 phlpropd.5 . . . 4 (𝜑𝐹 = (Scalar‘𝐿))
6 phlpropd.6 . . . 4 𝑃 = (Base‘𝐹)
7 phlpropd.7 . . . 4 ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))
81, 2, 3, 4, 5, 6, 7lvecpropd 19939 . . 3 (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec))
94, 5eqtr3d 2861 . . . 4 (𝜑 → (Scalar‘𝐾) = (Scalar‘𝐿))
109eleq1d 2900 . . 3 (𝜑 → ((Scalar‘𝐾) ∈ *-Ring ↔ (Scalar‘𝐿) ∈ *-Ring))
11 phlpropd.8 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(·𝑖𝐾)𝑦) = (𝑥(·𝑖𝐿)𝑦))
1211oveqrspc2v 7176 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑎𝐵)) → (𝑏(·𝑖𝐾)𝑎) = (𝑏(·𝑖𝐿)𝑎))
1312anass1rs 654 . . . . . . . . 9 (((𝜑𝑎𝐵) ∧ 𝑏𝐵) → (𝑏(·𝑖𝐾)𝑎) = (𝑏(·𝑖𝐿)𝑎))
1413mpteq2dva 5147 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑏𝐵 ↦ (𝑏(·𝑖𝐾)𝑎)) = (𝑏𝐵 ↦ (𝑏(·𝑖𝐿)𝑎)))
151adantr 484 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝐵 = (Base‘𝐾))
1615mpteq1d 5141 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑏𝐵 ↦ (𝑏(·𝑖𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)))
172adantr 484 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝐵 = (Base‘𝐿))
1817mpteq1d 5141 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑏𝐵 ↦ (𝑏(·𝑖𝐿)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)))
1914, 16, 183eqtr3d 2867 . . . . . . 7 ((𝜑𝑎𝐵) → (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)))
20 rlmbas 19967 . . . . . . . . . . . 12 (Base‘𝐹) = (Base‘(ringLMod‘𝐹))
216, 20eqtri 2847 . . . . . . . . . . 11 𝑃 = (Base‘(ringLMod‘𝐹))
2221a1i 11 . . . . . . . . . 10 (𝜑𝑃 = (Base‘(ringLMod‘𝐹)))
23 fvex 6674 . . . . . . . . . . . 12 (Scalar‘𝐾) ∈ V
244, 23eqeltrdi 2924 . . . . . . . . . . 11 (𝜑𝐹 ∈ V)
25 rlmsca 19972 . . . . . . . . . . 11 (𝐹 ∈ V → 𝐹 = (Scalar‘(ringLMod‘𝐹)))
2624, 25syl 17 . . . . . . . . . 10 (𝜑𝐹 = (Scalar‘(ringLMod‘𝐹)))
27 eqidd 2825 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(+g‘(ringLMod‘𝐹))𝑦) = (𝑥(+g‘(ringLMod‘𝐹))𝑦))
28 eqidd 2825 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥( ·𝑠 ‘(ringLMod‘𝐹))𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐹))𝑦))
291, 22, 2, 22, 4, 26, 5, 26, 6, 6, 3, 27, 7, 28lmhmpropd 19845 . . . . . . . . 9 (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘𝐹)))
304fveq2d 6665 . . . . . . . . . 10 (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐾)))
3130oveq2d 7165 . . . . . . . . 9 (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))))
325fveq2d 6665 . . . . . . . . . 10 (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐿)))
3332oveq2d 7165 . . . . . . . . 9 (𝜑 → (𝐿 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
3429, 31, 333eqtr3d 2867 . . . . . . . 8 (𝜑 → (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
3534adantr 484 . . . . . . 7 ((𝜑𝑎𝐵) → (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
3619, 35eleq12d 2910 . . . . . 6 ((𝜑𝑎𝐵) → ((𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)) ∈ (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) ↔ (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)) ∈ (𝐿 LMHom (ringLMod‘(Scalar‘𝐿)))))
3711oveqrspc2v 7176 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑎𝐵)) → (𝑎(·𝑖𝐾)𝑎) = (𝑎(·𝑖𝐿)𝑎))
3837anabsan2 673 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑎(·𝑖𝐾)𝑎) = (𝑎(·𝑖𝐿)𝑎))
399fveq2d 6665 . . . . . . . . 9 (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
4039adantr 484 . . . . . . . 8 ((𝜑𝑎𝐵) → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
4138, 40eqeq12d 2840 . . . . . . 7 ((𝜑𝑎𝐵) → ((𝑎(·𝑖𝐾)𝑎) = (0g‘(Scalar‘𝐾)) ↔ (𝑎(·𝑖𝐿)𝑎) = (0g‘(Scalar‘𝐿))))
421, 2, 3grpidpropd 17872 . . . . . . . . 9 (𝜑 → (0g𝐾) = (0g𝐿))
4342adantr 484 . . . . . . . 8 ((𝜑𝑎𝐵) → (0g𝐾) = (0g𝐿))
4443eqeq2d 2835 . . . . . . 7 ((𝜑𝑎𝐵) → (𝑎 = (0g𝐾) ↔ 𝑎 = (0g𝐿)))
4541, 44imbi12d 348 . . . . . 6 ((𝜑𝑎𝐵) → (((𝑎(·𝑖𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g𝐾)) ↔ ((𝑎(·𝑖𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g𝐿))))
469fveq2d 6665 . . . . . . . . . . . 12 (𝜑 → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
4746adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
4811oveqrspc2v 7176 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(·𝑖𝐾)𝑏) = (𝑎(·𝑖𝐿)𝑏))
4947, 48fveq12d 6668 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)))
5049anassrs 471 . . . . . . . . 9 (((𝜑𝑎𝐵) ∧ 𝑏𝐵) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)))
5150, 13eqeq12d 2840 . . . . . . . 8 (((𝜑𝑎𝐵) ∧ 𝑏𝐵) → (((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎) ↔ ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎)))
5251ralbidva 3191 . . . . . . 7 ((𝜑𝑎𝐵) → (∀𝑏𝐵 ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎) ↔ ∀𝑏𝐵 ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎)))
5315raleqdv 3402 . . . . . . 7 ((𝜑𝑎𝐵) → (∀𝑏𝐵 ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎) ↔ ∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎)))
5417raleqdv 3402 . . . . . . 7 ((𝜑𝑎𝐵) → (∀𝑏𝐵 ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎) ↔ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎)))
5552, 53, 543bitr3d 312 . . . . . 6 ((𝜑𝑎𝐵) → (∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎) ↔ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎)))
5636, 45, 553anbi123d 1433 . . . . 5 ((𝜑𝑎𝐵) → (((𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)) ∈ (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) ∧ ((𝑎(·𝑖𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g𝐾)) ∧ ∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎)) ↔ ((𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)) ∈ (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))) ∧ ((𝑎(·𝑖𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g𝐿)) ∧ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎))))
5756ralbidva 3191 . . . 4 (𝜑 → (∀𝑎𝐵 ((𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)) ∈ (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) ∧ ((𝑎(·𝑖𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g𝐾)) ∧ ∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎)) ↔ ∀𝑎𝐵 ((𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)) ∈ (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))) ∧ ((𝑎(·𝑖𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g𝐿)) ∧ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎))))
581raleqdv 3402 . . . 4 (𝜑 → (∀𝑎𝐵 ((𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)) ∈ (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) ∧ ((𝑎(·𝑖𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g𝐾)) ∧ ∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎)) ↔ ∀𝑎 ∈ (Base‘𝐾)((𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)) ∈ (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) ∧ ((𝑎(·𝑖𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g𝐾)) ∧ ∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎))))
592raleqdv 3402 . . . 4 (𝜑 → (∀𝑎𝐵 ((𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)) ∈ (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))) ∧ ((𝑎(·𝑖𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g𝐿)) ∧ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎)) ↔ ∀𝑎 ∈ (Base‘𝐿)((𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)) ∈ (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))) ∧ ((𝑎(·𝑖𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g𝐿)) ∧ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎))))
6057, 58, 593bitr3d 312 . . 3 (𝜑 → (∀𝑎 ∈ (Base‘𝐾)((𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)) ∈ (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) ∧ ((𝑎(·𝑖𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g𝐾)) ∧ ∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎)) ↔ ∀𝑎 ∈ (Base‘𝐿)((𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)) ∈ (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))) ∧ ((𝑎(·𝑖𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g𝐿)) ∧ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎))))
618, 10, 603anbi123d 1433 . 2 (𝜑 → ((𝐾 ∈ LVec ∧ (Scalar‘𝐾) ∈ *-Ring ∧ ∀𝑎 ∈ (Base‘𝐾)((𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)) ∈ (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) ∧ ((𝑎(·𝑖𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g𝐾)) ∧ ∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎))) ↔ (𝐿 ∈ LVec ∧ (Scalar‘𝐿) ∈ *-Ring ∧ ∀𝑎 ∈ (Base‘𝐿)((𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)) ∈ (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))) ∧ ((𝑎(·𝑖𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g𝐿)) ∧ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎)))))
62 eqid 2824 . . 3 (Base‘𝐾) = (Base‘𝐾)
63 eqid 2824 . . 3 (Scalar‘𝐾) = (Scalar‘𝐾)
64 eqid 2824 . . 3 (·𝑖𝐾) = (·𝑖𝐾)
65 eqid 2824 . . 3 (0g𝐾) = (0g𝐾)
66 eqid 2824 . . 3 (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐾))
67 eqid 2824 . . 3 (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐾))
6862, 63, 64, 65, 66, 67isphl 20324 . 2 (𝐾 ∈ PreHil ↔ (𝐾 ∈ LVec ∧ (Scalar‘𝐾) ∈ *-Ring ∧ ∀𝑎 ∈ (Base‘𝐾)((𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)) ∈ (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) ∧ ((𝑎(·𝑖𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g𝐾)) ∧ ∀𝑏 ∈ (Base‘𝐾)((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎))))
69 eqid 2824 . . 3 (Base‘𝐿) = (Base‘𝐿)
70 eqid 2824 . . 3 (Scalar‘𝐿) = (Scalar‘𝐿)
71 eqid 2824 . . 3 (·𝑖𝐿) = (·𝑖𝐿)
72 eqid 2824 . . 3 (0g𝐿) = (0g𝐿)
73 eqid 2824 . . 3 (*𝑟‘(Scalar‘𝐿)) = (*𝑟‘(Scalar‘𝐿))
74 eqid 2824 . . 3 (0g‘(Scalar‘𝐿)) = (0g‘(Scalar‘𝐿))
7569, 70, 71, 72, 73, 74isphl 20324 . 2 (𝐿 ∈ PreHil ↔ (𝐿 ∈ LVec ∧ (Scalar‘𝐿) ∈ *-Ring ∧ ∀𝑎 ∈ (Base‘𝐿)((𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)) ∈ (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))) ∧ ((𝑎(·𝑖𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g𝐿)) ∧ ∀𝑏 ∈ (Base‘𝐿)((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎))))
7661, 68, 753bitr4g 317 1 (𝜑 → (𝐾 ∈ PreHil ↔ 𝐿 ∈ PreHil))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∀wral 3133  Vcvv 3480   ↦ cmpt 5132  ‘cfv 6343  (class class class)co 7149  Basecbs 16483  +gcplusg 16565  *𝑟cstv 16567  Scalarcsca 16568   ·𝑠 cvsca 16569  ·𝑖cip 16570  0gc0g 16713  *-Ringcsr 19615   LMHom clmhm 19791  LVecclvec 19874  ringLModcrglmod 19941  PreHilcphl 20320 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-sca 16581  df-vsca 16582  df-ip 16583  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-grp 18106  df-ghm 18356  df-mgp 19240  df-ur 19252  df-ring 19299  df-lmod 19636  df-lmhm 19794  df-lvec 19875  df-sra 19944  df-rgmod 19945  df-phl 20322 This theorem is referenced by:  tcphphl  23837
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