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Theorem phlpropd 19919
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 19086 . . 3 (𝜑 → (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec))
94, 5eqtr3d 2657 . . . 4 (𝜑 → (Scalar‘𝐾) = (Scalar‘𝐿))
109eleq1d 2683 . . 3 (𝜑 → ((Scalar‘𝐾) ∈ *-Ring ↔ (Scalar‘𝐿) ∈ *-Ring))
11 phlpropd.8 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(·𝑖𝐾)𝑦) = (𝑥(·𝑖𝐿)𝑦))
1211oveqrspc2v 6627 . . . . . . . . . 10 ((𝜑 ∧ (𝑏𝐵𝑎𝐵)) → (𝑏(·𝑖𝐾)𝑎) = (𝑏(·𝑖𝐿)𝑎))
1312anass1rs 848 . . . . . . . . 9 (((𝜑𝑎𝐵) ∧ 𝑏𝐵) → (𝑏(·𝑖𝐾)𝑎) = (𝑏(·𝑖𝐿)𝑎))
1413mpteq2dva 4704 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑏𝐵 ↦ (𝑏(·𝑖𝐾)𝑎)) = (𝑏𝐵 ↦ (𝑏(·𝑖𝐿)𝑎)))
151adantr 481 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝐵 = (Base‘𝐾))
1615mpteq1d 4698 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑏𝐵 ↦ (𝑏(·𝑖𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)))
172adantr 481 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝐵 = (Base‘𝐿))
1817mpteq1d 4698 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑏𝐵 ↦ (𝑏(·𝑖𝐿)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)))
1914, 16, 183eqtr3d 2663 . . . . . . 7 ((𝜑𝑎𝐵) → (𝑏 ∈ (Base‘𝐾) ↦ (𝑏(·𝑖𝐾)𝑎)) = (𝑏 ∈ (Base‘𝐿) ↦ (𝑏(·𝑖𝐿)𝑎)))
20 rlmbas 19114 . . . . . . . . . . . 12 (Base‘𝐹) = (Base‘(ringLMod‘𝐹))
216, 20eqtri 2643 . . . . . . . . . . 11 𝑃 = (Base‘(ringLMod‘𝐹))
2221a1i 11 . . . . . . . . . 10 (𝜑𝑃 = (Base‘(ringLMod‘𝐹)))
23 fvex 6158 . . . . . . . . . . . 12 (Scalar‘𝐾) ∈ V
244, 23syl6eqel 2706 . . . . . . . . . . 11 (𝜑𝐹 ∈ V)
25 rlmsca 19119 . . . . . . . . . . 11 (𝐹 ∈ V → 𝐹 = (Scalar‘(ringLMod‘𝐹)))
2624, 25syl 17 . . . . . . . . . 10 (𝜑𝐹 = (Scalar‘(ringLMod‘𝐹)))
27 eqidd 2622 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑃𝑦𝑃)) → (𝑥(+g‘(ringLMod‘𝐹))𝑦) = (𝑥(+g‘(ringLMod‘𝐹))𝑦))
28 eqidd 2622 . . . . . . . . . 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 18992 . . . . . . . . 9 (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘𝐹)))
304fveq2d 6152 . . . . . . . . . 10 (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐾)))
3130oveq2d 6620 . . . . . . . . 9 (𝜑 → (𝐾 LMHom (ringLMod‘𝐹)) = (𝐾 LMHom (ringLMod‘(Scalar‘𝐾))))
325fveq2d 6152 . . . . . . . . . 10 (𝜑 → (ringLMod‘𝐹) = (ringLMod‘(Scalar‘𝐿)))
3332oveq2d 6620 . . . . . . . . 9 (𝜑 → (𝐿 LMHom (ringLMod‘𝐹)) = (𝐿 LMHom (ringLMod‘(Scalar‘𝐿))))
3429, 31, 333eqtr3d 2663 . . . . . . . 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 2692 . . . . . 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 6627 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝐵𝑎𝐵)) → (𝑎(·𝑖𝐾)𝑎) = (𝑎(·𝑖𝐿)𝑎))
3837anabsan2 862 . . . . . . . 8 ((𝜑𝑎𝐵) → (𝑎(·𝑖𝐾)𝑎) = (𝑎(·𝑖𝐿)𝑎))
399fveq2d 6152 . . . . . . . . 9 (𝜑 → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
4039adantr 481 . . . . . . . 8 ((𝜑𝑎𝐵) → (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐿)))
4138, 40eqeq12d 2636 . . . . . . 7 ((𝜑𝑎𝐵) → ((𝑎(·𝑖𝐾)𝑎) = (0g‘(Scalar‘𝐾)) ↔ (𝑎(·𝑖𝐿)𝑎) = (0g‘(Scalar‘𝐿))))
421, 2, 3grpidpropd 17182 . . . . . . . . 9 (𝜑 → (0g𝐾) = (0g𝐿))
4342adantr 481 . . . . . . . 8 ((𝜑𝑎𝐵) → (0g𝐾) = (0g𝐿))
4443eqeq2d 2631 . . . . . . 7 ((𝜑𝑎𝐵) → (𝑎 = (0g𝐾) ↔ 𝑎 = (0g𝐿)))
4541, 44imbi12d 334 . . . . . 6 ((𝜑𝑎𝐵) → (((𝑎(·𝑖𝐾)𝑎) = (0g‘(Scalar‘𝐾)) → 𝑎 = (0g𝐾)) ↔ ((𝑎(·𝑖𝐿)𝑎) = (0g‘(Scalar‘𝐿)) → 𝑎 = (0g𝐿))))
469fveq2d 6152 . . . . . . . . . . . 12 (𝜑 → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
4746adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐿)))
4811oveqrspc2v 6627 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(·𝑖𝐾)𝑏) = (𝑎(·𝑖𝐿)𝑏))
4947, 48fveq12d 6154 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)))
5049anassrs 679 . . . . . . . . 9 (((𝜑𝑎𝐵) ∧ 𝑏𝐵) → ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)))
5150, 13eqeq12d 2636 . . . . . . . 8 (((𝜑𝑎𝐵) ∧ 𝑏𝐵) → (((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎) ↔ ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎)))
5251ralbidva 2979 . . . . . . 7 ((𝜑𝑎𝐵) → (∀𝑏𝐵 ((*𝑟‘(Scalar‘𝐾))‘(𝑎(·𝑖𝐾)𝑏)) = (𝑏(·𝑖𝐾)𝑎) ↔ ∀𝑏𝐵 ((*𝑟‘(Scalar‘𝐿))‘(𝑎(·𝑖𝐿)𝑏)) = (𝑏(·𝑖𝐿)𝑎)))
5315raleqdv 3133 . . . . . . 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 3133 . . . . . . 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 1396 . . . . 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 2979 . . . 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 3133 . . . 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 3133 . . . 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 1396 . 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 2621 . . 3 (Base‘𝐾) = (Base‘𝐾)
63 eqid 2621 . . 3 (Scalar‘𝐾) = (Scalar‘𝐾)
64 eqid 2621 . . 3 (·𝑖𝐾) = (·𝑖𝐾)
65 eqid 2621 . . 3 (0g𝐾) = (0g𝐾)
66 eqid 2621 . . 3 (*𝑟‘(Scalar‘𝐾)) = (*𝑟‘(Scalar‘𝐾))
67 eqid 2621 . . 3 (0g‘(Scalar‘𝐾)) = (0g‘(Scalar‘𝐾))
6862, 63, 64, 65, 66, 67isphl 19892 . 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 2621 . . 3 (Base‘𝐿) = (Base‘𝐿)
70 eqid 2621 . . 3 (Scalar‘𝐿) = (Scalar‘𝐿)
71 eqid 2621 . . 3 (·𝑖𝐿) = (·𝑖𝐿)
72 eqid 2621 . . 3 (0g𝐿) = (0g𝐿)
73 eqid 2621 . . 3 (*𝑟‘(Scalar‘𝐿)) = (*𝑟‘(Scalar‘𝐿))
74 eqid 2621 . . 3 (0g‘(Scalar‘𝐿)) = (0g‘(Scalar‘𝐿))
7569, 70, 71, 72, 73, 74isphl 19892 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  cmpt 4673  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  *𝑟cstv 15864  Scalarcsca 15865   ·𝑠 cvsca 15866  ·𝑖cip 15867  0gc0g 16021  *-Ringcsr 18765   LMHom clmhm 18938  LVecclvec 19021  ringLModcrglmod 19088  PreHilcphl 19888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-sca 15878  df-vsca 15879  df-ip 15880  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-grp 17346  df-ghm 17579  df-mgp 18411  df-ur 18423  df-ring 18470  df-lmod 18786  df-lmhm 18941  df-lvec 19022  df-sra 19091  df-rgmod 19092  df-phl 19890
This theorem is referenced by:  tchphl  22934
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