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Theorem phlpropd 20217
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 19382 . . 3 (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec))
94, 5eqtr3d 2807 . . . 4 (𝜑 → (Scalar‘𝐾) = (Scalar‘𝐿))
109eleq1d 2835 . . 3 (𝜑 → ((Scalar‘𝐾) ∈ *-Ring ↔ (Scalar‘𝐿) ∈ *-Ring))
11 phlpropd.8 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(·𝑖𝐾)𝑦) = (𝑥(·𝑖𝐿)𝑦))
1211oveqrspc2v 6822 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑎𝐵)) → (𝑏(·𝑖𝐾)𝑎) = (𝑏(·𝑖𝐿)𝑎))
1312anass1rs 634 . . . . . . . . 9 (((𝜑𝑎𝐵) ∧ 𝑏𝐵) → (𝑏(·𝑖𝐾)𝑎) = (𝑏(·𝑖𝐿)𝑎))
1413mpteq2dva 4879 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑏𝐵 ↦ (𝑏(·𝑖𝐾)𝑎)) = (𝑏𝐵 ↦ (𝑏(·𝑖𝐿)𝑎)))
151adantr 466 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝐵 = (Base‘𝐾))
1615mpteq1d 4873 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑏𝐵 ↦ (𝑏(·𝑖𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)))
172adantr 466 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝐵 = (Base‘𝐿))
1817mpteq1d 4873 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑏𝐵 ↦ (𝑏(·𝑖𝐿)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)))
1914, 16, 183eqtr3d 2813 . . . . . . 7 ((𝜑𝑎𝐵) → (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)))
20 rlmbas 19410 . . . . . . . . . . . 12 (Base‘𝐹) = (Base‘(ringLMod‘𝐹))
216, 20eqtri 2793 . . . . . . . . . . 11 𝑃 = (Base‘(ringLMod‘𝐹))
2221a1i 11 . . . . . . . . . 10 (𝜑𝑃 = (Base‘(ringLMod‘𝐹)))
23 fvex 6344 . . . . . . . . . . . 12 (Scalar‘𝐾) ∈ V
244, 23syl6eqel 2858 . . . . . . . . . . 11 (𝜑𝐹 ∈ V)
25 rlmsca 19415 . . . . . . . . . . 11 (𝐹 ∈ V → 𝐹 = (Scalar‘(ringLMod‘𝐹)))
2624, 25syl 17 . . . . . . . . . 10 (𝜑𝐹 = (Scalar‘(ringLMod‘𝐹)))
27 eqidd 2772 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(+g‘(ringLMod‘𝐹))𝑦) = (𝑥(+g‘(ringLMod‘𝐹))𝑦))
28 eqidd 2772 . . . . . . . . . 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 19286 . . . . . . . . 9 (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘𝐹)))
304fveq2d 6337 . . . . . . . . . 10 (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐾)))
3130oveq2d 6812 . . . . . . . . 9 (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))))
325fveq2d 6337 . . . . . . . . . 10 (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐿)))
3332oveq2d 6812 . . . . . . . . 9 (𝜑 → (𝐿 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
3429, 31, 333eqtr3d 2813 . . . . . . . 8 (𝜑 → (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
3534adantr 466 . . . . . . 7 ((𝜑𝑎𝐵) → (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
3619, 35eleq12d 2844 . . . . . 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 6822 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑎𝐵)) → (𝑎(·𝑖𝐾)𝑎) = (𝑎(·𝑖𝐿)𝑎))
3837anabsan2 653 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑎(·𝑖𝐾)𝑎) = (𝑎(·𝑖𝐿)𝑎))
399fveq2d 6337 . . . . . . . . 9 (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
4039adantr 466 . . . . . . . 8 ((𝜑𝑎𝐵) → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
4138, 40eqeq12d 2786 . . . . . . 7 ((𝜑𝑎𝐵) → ((𝑎(·𝑖𝐾)𝑎) = (0g‘(Scalar‘𝐾)) ↔ (𝑎(·𝑖𝐿)𝑎) = (0g‘(Scalar‘𝐿))))
421, 2, 3grpidpropd 17469 . . . . . . . . 9 (𝜑 → (0g𝐾) = (0g𝐿))
4342adantr 466 . . . . . . . 8 ((𝜑𝑎𝐵) → (0g𝐾) = (0g𝐿))
4443eqeq2d 2781 . . . . . . 7 ((𝜑𝑎𝐵) → (𝑎 = (0g𝐾) ↔ 𝑎 = (0g𝐿)))
4541, 44imbi12d 333 . . . . . 6 ((𝜑𝑎𝐵) → (((𝑎(·𝑖𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g𝐾)) ↔ ((𝑎(·𝑖𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g𝐿))))
469fveq2d 6337 . . . . . . . . . . . 12 (𝜑 → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
4746adantr 466 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
4811oveqrspc2v 6822 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(·𝑖𝐾)𝑏) = (𝑎(·𝑖𝐿)𝑏))
4947, 48fveq12d 6340 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)))
5049anassrs 453 . . . . . . . . 9 (((𝜑𝑎𝐵) ∧ 𝑏𝐵) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)))
5150, 13eqeq12d 2786 . . . . . . . 8 (((𝜑𝑎𝐵) ∧ 𝑏𝐵) → (((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎) ↔ ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎)))
5251ralbidva 3134 . . . . . . 7 ((𝜑𝑎𝐵) → (∀𝑏𝐵 ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎) ↔ ∀𝑏𝐵 ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎)))
5315raleqdv 3293 . . . . . . 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 3293 . . . . . . 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 298 . . . . . 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 1547 . . . . 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 3134 . . . 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 3293 . . . 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 3293 . . . 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 298 . . 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 1547 . 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 2771 . . 3 (Base‘𝐾) = (Base‘𝐾)
63 eqid 2771 . . 3 (Scalar‘𝐾) = (Scalar‘𝐾)
64 eqid 2771 . . 3 (·𝑖𝐾) = (·𝑖𝐾)
65 eqid 2771 . . 3 (0g𝐾) = (0g𝐾)
66 eqid 2771 . . 3 (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐾))
67 eqid 2771 . . 3 (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐾))
6862, 63, 64, 65, 66, 67isphl 20190 . 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 2771 . . 3 (Base‘𝐿) = (Base‘𝐿)
70 eqid 2771 . . 3 (Scalar‘𝐿) = (Scalar‘𝐿)
71 eqid 2771 . . 3 (·𝑖𝐿) = (·𝑖𝐿)
72 eqid 2771 . . 3 (0g𝐿) = (0g𝐿)
73 eqid 2771 . . 3 (*𝑟‘(Scalar‘𝐿)) = (*𝑟‘(Scalar‘𝐿))
74 eqid 2771 . . 3 (0g‘(Scalar‘𝐿)) = (0g‘(Scalar‘𝐿))
7569, 70, 71, 72, 73, 74isphl 20190 . 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 303 1 (𝜑 → (𝐾 ∈ PreHil ↔ 𝐿 ∈ PreHil))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cmpt 4864  cfv 6030  (class class class)co 6796  Basecbs 16064  +gcplusg 16149  *𝑟cstv 16151  Scalarcsca 16152   ·𝑠 cvsca 16153  ·𝑖cip 16154  0gc0g 16308  *-Ringcsr 19054   LMHom clmhm 19232  LVecclvec 19315  ringLModcrglmod 19384  PreHilcphl 20186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-er 7900  df-map 8015  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-3 11286  df-4 11287  df-5 11288  df-6 11289  df-7 11290  df-8 11291  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-sca 16165  df-vsca 16166  df-ip 16167  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-mhm 17543  df-grp 17633  df-ghm 17866  df-mgp 18698  df-ur 18710  df-ring 18757  df-lmod 19075  df-lmhm 19235  df-lvec 19316  df-sra 19387  df-rgmod 19388  df-phl 20188
This theorem is referenced by:  tchphl  23245
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