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Theorem phlpropd 21691
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 21187 . . 3 (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec))
94, 5eqtr3d 2777 . . . 4 (𝜑 → (Scalar‘𝐾) = (Scalar‘𝐿))
109eleq1d 2824 . . 3 (𝜑 → ((Scalar‘𝐾) ∈ *-Ring ↔ (Scalar‘𝐿) ∈ *-Ring))
11 phlpropd.8 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(·𝑖𝐾)𝑦) = (𝑥(·𝑖𝐿)𝑦))
1211oveqrspc2v 7458 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑎𝐵)) → (𝑏(·𝑖𝐾)𝑎) = (𝑏(·𝑖𝐿)𝑎))
1312anass1rs 655 . . . . . . . . 9 (((𝜑𝑎𝐵) ∧ 𝑏𝐵) → (𝑏(·𝑖𝐾)𝑎) = (𝑏(·𝑖𝐿)𝑎))
1413mpteq2dva 5248 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑏𝐵 ↦ (𝑏(·𝑖𝐾)𝑎)) = (𝑏𝐵 ↦ (𝑏(·𝑖𝐿)𝑎)))
151adantr 480 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝐵 = (Base‘𝐾))
1615mpteq1d 5243 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑏𝐵 ↦ (𝑏(·𝑖𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)))
172adantr 480 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝐵 = (Base‘𝐿))
1817mpteq1d 5243 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑏𝐵 ↦ (𝑏(·𝑖𝐿)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)))
1914, 16, 183eqtr3d 2783 . . . . . . 7 ((𝜑𝑎𝐵) → (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)))
20 rlmbas 21218 . . . . . . . . . . . 12 (Base‘𝐹) = (Base‘(ringLMod‘𝐹))
216, 20eqtri 2763 . . . . . . . . . . 11 𝑃 = (Base‘(ringLMod‘𝐹))
2221a1i 11 . . . . . . . . . 10 (𝜑𝑃 = (Base‘(ringLMod‘𝐹)))
23 fvex 6920 . . . . . . . . . . . 12 (Scalar‘𝐾) ∈ V
244, 23eqeltrdi 2847 . . . . . . . . . . 11 (𝜑𝐹 ∈ V)
25 rlmsca 21223 . . . . . . . . . . 11 (𝐹 ∈ V → 𝐹 = (Scalar‘(ringLMod‘𝐹)))
2624, 25syl 17 . . . . . . . . . 10 (𝜑𝐹 = (Scalar‘(ringLMod‘𝐹)))
27 eqidd 2736 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(+g‘(ringLMod‘𝐹))𝑦) = (𝑥(+g‘(ringLMod‘𝐹))𝑦))
28 eqidd 2736 . . . . . . . . . 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 21090 . . . . . . . . 9 (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘𝐹)))
304fveq2d 6911 . . . . . . . . . 10 (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐾)))
3130oveq2d 7447 . . . . . . . . 9 (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))))
325fveq2d 6911 . . . . . . . . . 10 (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐿)))
3332oveq2d 7447 . . . . . . . . 9 (𝜑 → (𝐿 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
3429, 31, 333eqtr3d 2783 . . . . . . . 8 (𝜑 → (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
3534adantr 480 . . . . . . 7 ((𝜑𝑎𝐵) → (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
3619, 35eleq12d 2833 . . . . . 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 7458 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑎𝐵)) → (𝑎(·𝑖𝐾)𝑎) = (𝑎(·𝑖𝐿)𝑎))
3837anabsan2 674 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑎(·𝑖𝐾)𝑎) = (𝑎(·𝑖𝐿)𝑎))
399fveq2d 6911 . . . . . . . . 9 (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
4039adantr 480 . . . . . . . 8 ((𝜑𝑎𝐵) → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
4138, 40eqeq12d 2751 . . . . . . 7 ((𝜑𝑎𝐵) → ((𝑎(·𝑖𝐾)𝑎) = (0g‘(Scalar‘𝐾)) ↔ (𝑎(·𝑖𝐿)𝑎) = (0g‘(Scalar‘𝐿))))
421, 2, 3grpidpropd 18688 . . . . . . . . 9 (𝜑 → (0g𝐾) = (0g𝐿))
4342adantr 480 . . . . . . . 8 ((𝜑𝑎𝐵) → (0g𝐾) = (0g𝐿))
4443eqeq2d 2746 . . . . . . 7 ((𝜑𝑎𝐵) → (𝑎 = (0g𝐾) ↔ 𝑎 = (0g𝐿)))
4541, 44imbi12d 344 . . . . . 6 ((𝜑𝑎𝐵) → (((𝑎(·𝑖𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g𝐾)) ↔ ((𝑎(·𝑖𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g𝐿))))
469fveq2d 6911 . . . . . . . . . . . 12 (𝜑 → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
4746adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
4811oveqrspc2v 7458 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(·𝑖𝐾)𝑏) = (𝑎(·𝑖𝐿)𝑏))
4947, 48fveq12d 6914 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)))
5049anassrs 467 . . . . . . . . 9 (((𝜑𝑎𝐵) ∧ 𝑏𝐵) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)))
5150, 13eqeq12d 2751 . . . . . . . 8 (((𝜑𝑎𝐵) ∧ 𝑏𝐵) → (((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎) ↔ ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎)))
5251ralbidva 3174 . . . . . . 7 ((𝜑𝑎𝐵) → (∀𝑏𝐵 ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎) ↔ ∀𝑏𝐵 ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎)))
5315raleqdv 3324 . . . . . . 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 3324 . . . . . . 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 309 . . . . . 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 1435 . . . . 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 3174 . . . 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 3324 . . . 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 3324 . . . 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 309 . . 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 1435 . 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 2735 . . 3 (Base‘𝐾) = (Base‘𝐾)
63 eqid 2735 . . 3 (Scalar‘𝐾) = (Scalar‘𝐾)
64 eqid 2735 . . 3 (·𝑖𝐾) = (·𝑖𝐾)
65 eqid 2735 . . 3 (0g𝐾) = (0g𝐾)
66 eqid 2735 . . 3 (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐾))
67 eqid 2735 . . 3 (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐾))
6862, 63, 64, 65, 66, 67isphl 21664 . 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 2735 . . 3 (Base‘𝐿) = (Base‘𝐿)
70 eqid 2735 . . 3 (Scalar‘𝐿) = (Scalar‘𝐿)
71 eqid 2735 . . 3 (·𝑖𝐿) = (·𝑖𝐿)
72 eqid 2735 . . 3 (0g𝐿) = (0g𝐿)
73 eqid 2735 . . 3 (*𝑟‘(Scalar‘𝐿)) = (*𝑟‘(Scalar‘𝐿))
74 eqid 2735 . . 3 (0g‘(Scalar‘𝐿)) = (0g‘(Scalar‘𝐿))
7569, 70, 71, 72, 73, 74isphl 21664 . 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 314 1 (𝜑 → (𝐾 ∈ PreHil ↔ 𝐿 ∈ PreHil))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cmpt 5231  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  *𝑟cstv 17300  Scalarcsca 17301   ·𝑠 cvsca 17302  ·𝑖cip 17303  0gc0g 17486  *-Ringcsr 20856   LMHom clmhm 21036  LVecclvec 21119  ringLModcrglmod 21189  PreHilcphl 21660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-sca 17314  df-vsca 17315  df-ip 17316  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-grp 18967  df-ghm 19244  df-mgp 20153  df-ur 20200  df-ring 20253  df-lmod 20877  df-lmhm 21039  df-lvec 21120  df-sra 21190  df-rgmod 21191  df-phl 21662
This theorem is referenced by:  tcphphl  25275
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