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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 (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 20628 . . 3 (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec))
94, 5eqtr3d 2778 . . . 4 (𝜑 → (Scalar‘𝐾) = (Scalar‘𝐿))
109eleq1d 2822 . . 3 (𝜑 → ((Scalar‘𝐾) ∈ *-Ring ↔ (Scalar‘𝐿) ∈ *-Ring))
11 phlpropd.8 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(·𝑖𝐾)𝑦) = (𝑥(·𝑖𝐿)𝑦))
1211oveqrspc2v 7384 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑎𝐵)) → (𝑏(·𝑖𝐾)𝑎) = (𝑏(·𝑖𝐿)𝑎))
1312anass1rs 653 . . . . . . . . 9 (((𝜑𝑎𝐵) ∧ 𝑏𝐵) → (𝑏(·𝑖𝐾)𝑎) = (𝑏(·𝑖𝐿)𝑎))
1413mpteq2dva 5205 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑏𝐵 ↦ (𝑏(·𝑖𝐾)𝑎)) = (𝑏𝐵 ↦ (𝑏(·𝑖𝐿)𝑎)))
151adantr 481 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝐵 = (Base‘𝐾))
1615mpteq1d 5200 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑏𝐵 ↦ (𝑏(·𝑖𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)))
172adantr 481 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝐵 = (Base‘𝐿))
1817mpteq1d 5200 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑏𝐵 ↦ (𝑏(·𝑖𝐿)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)))
1914, 16, 183eqtr3d 2784 . . . . . . 7 ((𝜑𝑎𝐵) → (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)))
20 rlmbas 20664 . . . . . . . . . . . 12 (Base‘𝐹) = (Base‘(ringLMod‘𝐹))
216, 20eqtri 2764 . . . . . . . . . . 11 𝑃 = (Base‘(ringLMod‘𝐹))
2221a1i 11 . . . . . . . . . 10 (𝜑𝑃 = (Base‘(ringLMod‘𝐹)))
23 fvex 6855 . . . . . . . . . . . 12 (Scalar‘𝐾) ∈ V
244, 23eqeltrdi 2846 . . . . . . . . . . 11 (𝜑𝐹 ∈ V)
25 rlmsca 20669 . . . . . . . . . . 11 (𝐹 ∈ V → 𝐹 = (Scalar‘(ringLMod‘𝐹)))
2624, 25syl 17 . . . . . . . . . 10 (𝜑𝐹 = (Scalar‘(ringLMod‘𝐹)))
27 eqidd 2737 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(+g‘(ringLMod‘𝐹))𝑦) = (𝑥(+g‘(ringLMod‘𝐹))𝑦))
28 eqidd 2737 . . . . . . . . . 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 20534 . . . . . . . . 9 (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘𝐹)))
304fveq2d 6846 . . . . . . . . . 10 (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐾)))
3130oveq2d 7373 . . . . . . . . 9 (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))))
325fveq2d 6846 . . . . . . . . . 10 (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐿)))
3332oveq2d 7373 . . . . . . . . 9 (𝜑 → (𝐿 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
3429, 31, 333eqtr3d 2784 . . . . . . . 8 (𝜑 → (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
3534adantr 481 . . . . . . 7 ((𝜑𝑎𝐵) → (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
3619, 35eleq12d 2832 . . . . . 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 7384 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑎𝐵)) → (𝑎(·𝑖𝐾)𝑎) = (𝑎(·𝑖𝐿)𝑎))
3837anabsan2 672 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑎(·𝑖𝐾)𝑎) = (𝑎(·𝑖𝐿)𝑎))
399fveq2d 6846 . . . . . . . . 9 (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
4039adantr 481 . . . . . . . 8 ((𝜑𝑎𝐵) → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
4138, 40eqeq12d 2752 . . . . . . 7 ((𝜑𝑎𝐵) → ((𝑎(·𝑖𝐾)𝑎) = (0g‘(Scalar‘𝐾)) ↔ (𝑎(·𝑖𝐿)𝑎) = (0g‘(Scalar‘𝐿))))
421, 2, 3grpidpropd 18517 . . . . . . . . 9 (𝜑 → (0g𝐾) = (0g𝐿))
4342adantr 481 . . . . . . . 8 ((𝜑𝑎𝐵) → (0g𝐾) = (0g𝐿))
4443eqeq2d 2747 . . . . . . 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 6846 . . . . . . . . . . . 12 (𝜑 → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
4746adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
4811oveqrspc2v 7384 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(·𝑖𝐾)𝑏) = (𝑎(·𝑖𝐿)𝑏))
4947, 48fveq12d 6849 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)))
5049anassrs 468 . . . . . . . . 9 (((𝜑𝑎𝐵) ∧ 𝑏𝐵) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)))
5150, 13eqeq12d 2752 . . . . . . . 8 (((𝜑𝑎𝐵) ∧ 𝑏𝐵) → (((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎) ↔ ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎)))
5251ralbidva 3172 . . . . . . 7 ((𝜑𝑎𝐵) → (∀𝑏𝐵 ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎) ↔ ∀𝑏𝐵 ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎)))
5315raleqdv 3313 . . . . . . 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 3313 . . . . . . 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 308 . . . . . 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 1436 . . . . 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 3172 . . . 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 3313 . . . 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 3313 . . . 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 308 . . 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 1436 . 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 2736 . . 3 (Base‘𝐾) = (Base‘𝐾)
63 eqid 2736 . . 3 (Scalar‘𝐾) = (Scalar‘𝐾)
64 eqid 2736 . . 3 (·𝑖𝐾) = (·𝑖𝐾)
65 eqid 2736 . . 3 (0g𝐾) = (0g𝐾)
66 eqid 2736 . . 3 (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐾))
67 eqid 2736 . . 3 (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐾))
6862, 63, 64, 65, 66, 67isphl 21032 . 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 2736 . . 3 (Base‘𝐿) = (Base‘𝐿)
70 eqid 2736 . . 3 (Scalar‘𝐿) = (Scalar‘𝐿)
71 eqid 2736 . . 3 (·𝑖𝐿) = (·𝑖𝐿)
72 eqid 2736 . . 3 (0g𝐿) = (0g𝐿)
73 eqid 2736 . . 3 (*𝑟‘(Scalar‘𝐿)) = (*𝑟‘(Scalar‘𝐿))
74 eqid 2736 . . 3 (0g‘(Scalar‘𝐿)) = (0g‘(Scalar‘𝐿))
7569, 70, 71, 72, 73, 74isphl 21032 . 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 313 1 (𝜑 → (𝐾 ∈ PreHil ↔ 𝐿 ∈ PreHil))
Colors of variables: wff setvar class
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  0gc0g 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
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