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Theorem phlpropd 21207
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 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))
phlpropd.2 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))
phlpropd.3 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))
phlpropd.4 (πœ‘ β†’ 𝐹 = (Scalarβ€˜πΎ))
phlpropd.5 (πœ‘ β†’ 𝐹 = (Scalarβ€˜πΏ))
phlpropd.6 𝑃 = (Baseβ€˜πΉ)
phlpropd.7 ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))
phlpropd.8 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(Β·π‘–β€˜πΎ)𝑦) = (π‘₯(Β·π‘–β€˜πΏ)𝑦))
Assertion
Ref Expression
phlpropd (πœ‘ β†’ (𝐾 ∈ PreHil ↔ 𝐿 ∈ PreHil))
Distinct variable groups:   π‘₯,𝑦,𝐡   π‘₯,𝐹,𝑦   π‘₯,𝐾,𝑦   π‘₯,𝐿,𝑦   π‘₯,𝑃,𝑦   πœ‘,π‘₯,𝑦

Proof of Theorem phlpropd
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 phlpropd.1 . . . 4 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))
2 phlpropd.2 . . . 4 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))
3 phlpropd.3 . . . 4 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))
4 phlpropd.4 . . . 4 (πœ‘ β†’ 𝐹 = (Scalarβ€˜πΎ))
5 phlpropd.5 . . . 4 (πœ‘ β†’ 𝐹 = (Scalarβ€˜πΏ))
6 phlpropd.6 . . . 4 𝑃 = (Baseβ€˜πΉ)
7 phlpropd.7 . . . 4 ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))
81, 2, 3, 4, 5, 6, 7lvecpropd 20779 . . 3 (πœ‘ β†’ (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec))
94, 5eqtr3d 2774 . . . 4 (πœ‘ β†’ (Scalarβ€˜πΎ) = (Scalarβ€˜πΏ))
109eleq1d 2818 . . 3 (πœ‘ β†’ ((Scalarβ€˜πΎ) ∈ *-Ring ↔ (Scalarβ€˜πΏ) ∈ *-Ring))
11 phlpropd.8 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(Β·π‘–β€˜πΎ)𝑦) = (π‘₯(Β·π‘–β€˜πΏ)𝑦))
1211oveqrspc2v 7435 . . . . . . . . . 10 ((πœ‘ ∧ (𝑏 ∈ 𝐡 ∧ π‘Ž ∈ 𝐡)) β†’ (𝑏(Β·π‘–β€˜πΎ)π‘Ž) = (𝑏(Β·π‘–β€˜πΏ)π‘Ž))
1312anass1rs 653 . . . . . . . . 9 (((πœ‘ ∧ π‘Ž ∈ 𝐡) ∧ 𝑏 ∈ 𝐡) β†’ (𝑏(Β·π‘–β€˜πΎ)π‘Ž) = (𝑏(Β·π‘–β€˜πΏ)π‘Ž))
1413mpteq2dva 5248 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (𝑏 ∈ 𝐡 ↦ (𝑏(Β·π‘–β€˜πΎ)π‘Ž)) = (𝑏 ∈ 𝐡 ↦ (𝑏(Β·π‘–β€˜πΏ)π‘Ž)))
151adantr 481 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ 𝐡 = (Baseβ€˜πΎ))
1615mpteq1d 5243 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (𝑏 ∈ 𝐡 ↦ (𝑏(Β·π‘–β€˜πΎ)π‘Ž)) = (𝑏 ∈ (Baseβ€˜πΎ) ↦ (𝑏(Β·π‘–β€˜πΎ)π‘Ž)))
172adantr 481 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ 𝐡 = (Baseβ€˜πΏ))
1817mpteq1d 5243 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (𝑏 ∈ 𝐡 ↦ (𝑏(Β·π‘–β€˜πΏ)π‘Ž)) = (𝑏 ∈ (Baseβ€˜πΏ) ↦ (𝑏(Β·π‘–β€˜πΏ)π‘Ž)))
1914, 16, 183eqtr3d 2780 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (𝑏 ∈ (Baseβ€˜πΎ) ↦ (𝑏(Β·π‘–β€˜πΎ)π‘Ž)) = (𝑏 ∈ (Baseβ€˜πΏ) ↦ (𝑏(Β·π‘–β€˜πΏ)π‘Ž)))
20 rlmbas 20816 . . . . . . . . . . . 12 (Baseβ€˜πΉ) = (Baseβ€˜(ringLModβ€˜πΉ))
216, 20eqtri 2760 . . . . . . . . . . 11 𝑃 = (Baseβ€˜(ringLModβ€˜πΉ))
2221a1i 11 . . . . . . . . . 10 (πœ‘ β†’ 𝑃 = (Baseβ€˜(ringLModβ€˜πΉ)))
23 fvex 6904 . . . . . . . . . . . 12 (Scalarβ€˜πΎ) ∈ V
244, 23eqeltrdi 2841 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹 ∈ V)
25 rlmsca 20821 . . . . . . . . . . 11 (𝐹 ∈ V β†’ 𝐹 = (Scalarβ€˜(ringLModβ€˜πΉ)))
2624, 25syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐹 = (Scalarβ€˜(ringLModβ€˜πΉ)))
27 eqidd 2733 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) β†’ (π‘₯(+gβ€˜(ringLModβ€˜πΉ))𝑦) = (π‘₯(+gβ€˜(ringLModβ€˜πΉ))𝑦))
28 eqidd 2733 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) β†’ (π‘₯( ·𝑠 β€˜(ringLModβ€˜πΉ))𝑦) = (π‘₯( ·𝑠 β€˜(ringLModβ€˜πΉ))𝑦))
291, 22, 2, 22, 4, 26, 5, 26, 6, 6, 3, 27, 7, 28lmhmpropd 20683 . . . . . . . . 9 (πœ‘ β†’ (𝐾 LMHom (ringLModβ€˜πΉ)) = (𝐿 LMHom (ringLModβ€˜πΉ)))
304fveq2d 6895 . . . . . . . . . 10 (πœ‘ β†’ (ringLModβ€˜πΉ) = (ringLModβ€˜(Scalarβ€˜πΎ)))
3130oveq2d 7424 . . . . . . . . 9 (πœ‘ β†’ (𝐾 LMHom (ringLModβ€˜πΉ)) = (𝐾 LMHom (ringLModβ€˜(Scalarβ€˜πΎ))))
325fveq2d 6895 . . . . . . . . . 10 (πœ‘ β†’ (ringLModβ€˜πΉ) = (ringLModβ€˜(Scalarβ€˜πΏ)))
3332oveq2d 7424 . . . . . . . . 9 (πœ‘ β†’ (𝐿 LMHom (ringLModβ€˜πΉ)) = (𝐿 LMHom (ringLModβ€˜(Scalarβ€˜πΏ))))
3429, 31, 333eqtr3d 2780 . . . . . . . 8 (πœ‘ β†’ (𝐾 LMHom (ringLModβ€˜(Scalarβ€˜πΎ))) = (𝐿 LMHom (ringLModβ€˜(Scalarβ€˜πΏ))))
3534adantr 481 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (𝐾 LMHom (ringLModβ€˜(Scalarβ€˜πΎ))) = (𝐿 LMHom (ringLModβ€˜(Scalarβ€˜πΏ))))
3619, 35eleq12d 2827 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ ((𝑏 ∈ (Baseβ€˜πΎ) ↦ (𝑏(Β·π‘–β€˜πΎ)π‘Ž)) ∈ (𝐾 LMHom (ringLModβ€˜(Scalarβ€˜πΎ))) ↔ (𝑏 ∈ (Baseβ€˜πΏ) ↦ (𝑏(Β·π‘–β€˜πΏ)π‘Ž)) ∈ (𝐿 LMHom (ringLModβ€˜(Scalarβ€˜πΏ)))))
3711oveqrspc2v 7435 . . . . . . . . 9 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ π‘Ž ∈ 𝐡)) β†’ (π‘Ž(Β·π‘–β€˜πΎ)π‘Ž) = (π‘Ž(Β·π‘–β€˜πΏ)π‘Ž))
3837anabsan2 672 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (π‘Ž(Β·π‘–β€˜πΎ)π‘Ž) = (π‘Ž(Β·π‘–β€˜πΏ)π‘Ž))
399fveq2d 6895 . . . . . . . . 9 (πœ‘ β†’ (0gβ€˜(Scalarβ€˜πΎ)) = (0gβ€˜(Scalarβ€˜πΏ)))
4039adantr 481 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (0gβ€˜(Scalarβ€˜πΎ)) = (0gβ€˜(Scalarβ€˜πΏ)))
4138, 40eqeq12d 2748 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ ((π‘Ž(Β·π‘–β€˜πΎ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΎ)) ↔ (π‘Ž(Β·π‘–β€˜πΏ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΏ))))
421, 2, 3grpidpropd 18580 . . . . . . . . 9 (πœ‘ β†’ (0gβ€˜πΎ) = (0gβ€˜πΏ))
4342adantr 481 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (0gβ€˜πΎ) = (0gβ€˜πΏ))
4443eqeq2d 2743 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (π‘Ž = (0gβ€˜πΎ) ↔ π‘Ž = (0gβ€˜πΏ)))
4541, 44imbi12d 344 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (((π‘Ž(Β·π‘–β€˜πΎ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΎ)) β†’ π‘Ž = (0gβ€˜πΎ)) ↔ ((π‘Ž(Β·π‘–β€˜πΏ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΏ)) β†’ π‘Ž = (0gβ€˜πΏ))))
469fveq2d 6895 . . . . . . . . . . . 12 (πœ‘ β†’ (*π‘Ÿβ€˜(Scalarβ€˜πΎ)) = (*π‘Ÿβ€˜(Scalarβ€˜πΏ)))
4746adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (*π‘Ÿβ€˜(Scalarβ€˜πΎ)) = (*π‘Ÿβ€˜(Scalarβ€˜πΏ)))
4811oveqrspc2v 7435 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž(Β·π‘–β€˜πΎ)𝑏) = (π‘Ž(Β·π‘–β€˜πΏ)𝑏))
4947, 48fveq12d 6898 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ((*π‘Ÿβ€˜(Scalarβ€˜πΎ))β€˜(π‘Ž(Β·π‘–β€˜πΎ)𝑏)) = ((*π‘Ÿβ€˜(Scalarβ€˜πΏ))β€˜(π‘Ž(Β·π‘–β€˜πΏ)𝑏)))
5049anassrs 468 . . . . . . . . 9 (((πœ‘ ∧ π‘Ž ∈ 𝐡) ∧ 𝑏 ∈ 𝐡) β†’ ((*π‘Ÿβ€˜(Scalarβ€˜πΎ))β€˜(π‘Ž(Β·π‘–β€˜πΎ)𝑏)) = ((*π‘Ÿβ€˜(Scalarβ€˜πΏ))β€˜(π‘Ž(Β·π‘–β€˜πΏ)𝑏)))
5150, 13eqeq12d 2748 . . . . . . . 8 (((πœ‘ ∧ π‘Ž ∈ 𝐡) ∧ 𝑏 ∈ 𝐡) β†’ (((*π‘Ÿβ€˜(Scalarβ€˜πΎ))β€˜(π‘Ž(Β·π‘–β€˜πΎ)𝑏)) = (𝑏(Β·π‘–β€˜πΎ)π‘Ž) ↔ ((*π‘Ÿβ€˜(Scalarβ€˜πΏ))β€˜(π‘Ž(Β·π‘–β€˜πΏ)𝑏)) = (𝑏(Β·π‘–β€˜πΏ)π‘Ž)))
5251ralbidva 3175 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (βˆ€π‘ ∈ 𝐡 ((*π‘Ÿβ€˜(Scalarβ€˜πΎ))β€˜(π‘Ž(Β·π‘–β€˜πΎ)𝑏)) = (𝑏(Β·π‘–β€˜πΎ)π‘Ž) ↔ βˆ€π‘ ∈ 𝐡 ((*π‘Ÿβ€˜(Scalarβ€˜πΏ))β€˜(π‘Ž(Β·π‘–β€˜πΏ)𝑏)) = (𝑏(Β·π‘–β€˜πΏ)π‘Ž)))
5315raleqdv 3325 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (βˆ€π‘ ∈ 𝐡 ((*π‘Ÿβ€˜(Scalarβ€˜πΎ))β€˜(π‘Ž(Β·π‘–β€˜πΎ)𝑏)) = (𝑏(Β·π‘–β€˜πΎ)π‘Ž) ↔ βˆ€π‘ ∈ (Baseβ€˜πΎ)((*π‘Ÿβ€˜(Scalarβ€˜πΎ))β€˜(π‘Ž(Β·π‘–β€˜πΎ)𝑏)) = (𝑏(Β·π‘–β€˜πΎ)π‘Ž)))
5417raleqdv 3325 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (βˆ€π‘ ∈ 𝐡 ((*π‘Ÿβ€˜(Scalarβ€˜πΏ))β€˜(π‘Ž(Β·π‘–β€˜πΏ)𝑏)) = (𝑏(Β·π‘–β€˜πΏ)π‘Ž) ↔ βˆ€π‘ ∈ (Baseβ€˜πΏ)((*π‘Ÿβ€˜(Scalarβ€˜πΏ))β€˜(π‘Ž(Β·π‘–β€˜πΏ)𝑏)) = (𝑏(Β·π‘–β€˜πΏ)π‘Ž)))
5552, 53, 543bitr3d 308 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (βˆ€π‘ ∈ (Baseβ€˜πΎ)((*π‘Ÿβ€˜(Scalarβ€˜πΎ))β€˜(π‘Ž(Β·π‘–β€˜πΎ)𝑏)) = (𝑏(Β·π‘–β€˜πΎ)π‘Ž) ↔ βˆ€π‘ ∈ (Baseβ€˜πΏ)((*π‘Ÿβ€˜(Scalarβ€˜πΏ))β€˜(π‘Ž(Β·π‘–β€˜πΏ)𝑏)) = (𝑏(Β·π‘–β€˜πΏ)π‘Ž)))
5636, 45, 553anbi123d 1436 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (((𝑏 ∈ (Baseβ€˜πΎ) ↦ (𝑏(Β·π‘–β€˜πΎ)π‘Ž)) ∈ (𝐾 LMHom (ringLModβ€˜(Scalarβ€˜πΎ))) ∧ ((π‘Ž(Β·π‘–β€˜πΎ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΎ)) β†’ π‘Ž = (0gβ€˜πΎ)) ∧ βˆ€π‘ ∈ (Baseβ€˜πΎ)((*π‘Ÿβ€˜(Scalarβ€˜πΎ))β€˜(π‘Ž(Β·π‘–β€˜πΎ)𝑏)) = (𝑏(Β·π‘–β€˜πΎ)π‘Ž)) ↔ ((𝑏 ∈ (Baseβ€˜πΏ) ↦ (𝑏(Β·π‘–β€˜πΏ)π‘Ž)) ∈ (𝐿 LMHom (ringLModβ€˜(Scalarβ€˜πΏ))) ∧ ((π‘Ž(Β·π‘–β€˜πΏ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΏ)) β†’ π‘Ž = (0gβ€˜πΏ)) ∧ βˆ€π‘ ∈ (Baseβ€˜πΏ)((*π‘Ÿβ€˜(Scalarβ€˜πΏ))β€˜(π‘Ž(Β·π‘–β€˜πΏ)𝑏)) = (𝑏(Β·π‘–β€˜πΏ)π‘Ž))))
5756ralbidva 3175 . . . 4 (πœ‘ β†’ (βˆ€π‘Ž ∈ 𝐡 ((𝑏 ∈ (Baseβ€˜πΎ) ↦ (𝑏(Β·π‘–β€˜πΎ)π‘Ž)) ∈ (𝐾 LMHom (ringLModβ€˜(Scalarβ€˜πΎ))) ∧ ((π‘Ž(Β·π‘–β€˜πΎ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΎ)) β†’ π‘Ž = (0gβ€˜πΎ)) ∧ βˆ€π‘ ∈ (Baseβ€˜πΎ)((*π‘Ÿβ€˜(Scalarβ€˜πΎ))β€˜(π‘Ž(Β·π‘–β€˜πΎ)𝑏)) = (𝑏(Β·π‘–β€˜πΎ)π‘Ž)) ↔ βˆ€π‘Ž ∈ 𝐡 ((𝑏 ∈ (Baseβ€˜πΏ) ↦ (𝑏(Β·π‘–β€˜πΏ)π‘Ž)) ∈ (𝐿 LMHom (ringLModβ€˜(Scalarβ€˜πΏ))) ∧ ((π‘Ž(Β·π‘–β€˜πΏ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΏ)) β†’ π‘Ž = (0gβ€˜πΏ)) ∧ βˆ€π‘ ∈ (Baseβ€˜πΏ)((*π‘Ÿβ€˜(Scalarβ€˜πΏ))β€˜(π‘Ž(Β·π‘–β€˜πΏ)𝑏)) = (𝑏(Β·π‘–β€˜πΏ)π‘Ž))))
581raleqdv 3325 . . . 4 (πœ‘ β†’ (βˆ€π‘Ž ∈ 𝐡 ((𝑏 ∈ (Baseβ€˜πΎ) ↦ (𝑏(Β·π‘–β€˜πΎ)π‘Ž)) ∈ (𝐾 LMHom (ringLModβ€˜(Scalarβ€˜πΎ))) ∧ ((π‘Ž(Β·π‘–β€˜πΎ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΎ)) β†’ π‘Ž = (0gβ€˜πΎ)) ∧ βˆ€π‘ ∈ (Baseβ€˜πΎ)((*π‘Ÿβ€˜(Scalarβ€˜πΎ))β€˜(π‘Ž(Β·π‘–β€˜πΎ)𝑏)) = (𝑏(Β·π‘–β€˜πΎ)π‘Ž)) ↔ βˆ€π‘Ž ∈ (Baseβ€˜πΎ)((𝑏 ∈ (Baseβ€˜πΎ) ↦ (𝑏(Β·π‘–β€˜πΎ)π‘Ž)) ∈ (𝐾 LMHom (ringLModβ€˜(Scalarβ€˜πΎ))) ∧ ((π‘Ž(Β·π‘–β€˜πΎ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΎ)) β†’ π‘Ž = (0gβ€˜πΎ)) ∧ βˆ€π‘ ∈ (Baseβ€˜πΎ)((*π‘Ÿβ€˜(Scalarβ€˜πΎ))β€˜(π‘Ž(Β·π‘–β€˜πΎ)𝑏)) = (𝑏(Β·π‘–β€˜πΎ)π‘Ž))))
592raleqdv 3325 . . . 4 (πœ‘ β†’ (βˆ€π‘Ž ∈ 𝐡 ((𝑏 ∈ (Baseβ€˜πΏ) ↦ (𝑏(Β·π‘–β€˜πΏ)π‘Ž)) ∈ (𝐿 LMHom (ringLModβ€˜(Scalarβ€˜πΏ))) ∧ ((π‘Ž(Β·π‘–β€˜πΏ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΏ)) β†’ π‘Ž = (0gβ€˜πΏ)) ∧ βˆ€π‘ ∈ (Baseβ€˜πΏ)((*π‘Ÿβ€˜(Scalarβ€˜πΏ))β€˜(π‘Ž(Β·π‘–β€˜πΏ)𝑏)) = (𝑏(Β·π‘–β€˜πΏ)π‘Ž)) ↔ βˆ€π‘Ž ∈ (Baseβ€˜πΏ)((𝑏 ∈ (Baseβ€˜πΏ) ↦ (𝑏(Β·π‘–β€˜πΏ)π‘Ž)) ∈ (𝐿 LMHom (ringLModβ€˜(Scalarβ€˜πΏ))) ∧ ((π‘Ž(Β·π‘–β€˜πΏ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΏ)) β†’ π‘Ž = (0gβ€˜πΏ)) ∧ βˆ€π‘ ∈ (Baseβ€˜πΏ)((*π‘Ÿβ€˜(Scalarβ€˜πΏ))β€˜(π‘Ž(Β·π‘–β€˜πΏ)𝑏)) = (𝑏(Β·π‘–β€˜πΏ)π‘Ž))))
6057, 58, 593bitr3d 308 . . 3 (πœ‘ β†’ (βˆ€π‘Ž ∈ (Baseβ€˜πΎ)((𝑏 ∈ (Baseβ€˜πΎ) ↦ (𝑏(Β·π‘–β€˜πΎ)π‘Ž)) ∈ (𝐾 LMHom (ringLModβ€˜(Scalarβ€˜πΎ))) ∧ ((π‘Ž(Β·π‘–β€˜πΎ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΎ)) β†’ π‘Ž = (0gβ€˜πΎ)) ∧ βˆ€π‘ ∈ (Baseβ€˜πΎ)((*π‘Ÿβ€˜(Scalarβ€˜πΎ))β€˜(π‘Ž(Β·π‘–β€˜πΎ)𝑏)) = (𝑏(Β·π‘–β€˜πΎ)π‘Ž)) ↔ βˆ€π‘Ž ∈ (Baseβ€˜πΏ)((𝑏 ∈ (Baseβ€˜πΏ) ↦ (𝑏(Β·π‘–β€˜πΏ)π‘Ž)) ∈ (𝐿 LMHom (ringLModβ€˜(Scalarβ€˜πΏ))) ∧ ((π‘Ž(Β·π‘–β€˜πΏ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΏ)) β†’ π‘Ž = (0gβ€˜πΏ)) ∧ βˆ€π‘ ∈ (Baseβ€˜πΏ)((*π‘Ÿβ€˜(Scalarβ€˜πΏ))β€˜(π‘Ž(Β·π‘–β€˜πΏ)𝑏)) = (𝑏(Β·π‘–β€˜πΏ)π‘Ž))))
618, 10, 603anbi123d 1436 . 2 (πœ‘ β†’ ((𝐾 ∈ LVec ∧ (Scalarβ€˜πΎ) ∈ *-Ring ∧ βˆ€π‘Ž ∈ (Baseβ€˜πΎ)((𝑏 ∈ (Baseβ€˜πΎ) ↦ (𝑏(Β·π‘–β€˜πΎ)π‘Ž)) ∈ (𝐾 LMHom (ringLModβ€˜(Scalarβ€˜πΎ))) ∧ ((π‘Ž(Β·π‘–β€˜πΎ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΎ)) β†’ π‘Ž = (0gβ€˜πΎ)) ∧ βˆ€π‘ ∈ (Baseβ€˜πΎ)((*π‘Ÿβ€˜(Scalarβ€˜πΎ))β€˜(π‘Ž(Β·π‘–β€˜πΎ)𝑏)) = (𝑏(Β·π‘–β€˜πΎ)π‘Ž))) ↔ (𝐿 ∈ LVec ∧ (Scalarβ€˜πΏ) ∈ *-Ring ∧ βˆ€π‘Ž ∈ (Baseβ€˜πΏ)((𝑏 ∈ (Baseβ€˜πΏ) ↦ (𝑏(Β·π‘–β€˜πΏ)π‘Ž)) ∈ (𝐿 LMHom (ringLModβ€˜(Scalarβ€˜πΏ))) ∧ ((π‘Ž(Β·π‘–β€˜πΏ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΏ)) β†’ π‘Ž = (0gβ€˜πΏ)) ∧ βˆ€π‘ ∈ (Baseβ€˜πΏ)((*π‘Ÿβ€˜(Scalarβ€˜πΏ))β€˜(π‘Ž(Β·π‘–β€˜πΏ)𝑏)) = (𝑏(Β·π‘–β€˜πΏ)π‘Ž)))))
62 eqid 2732 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
63 eqid 2732 . . 3 (Scalarβ€˜πΎ) = (Scalarβ€˜πΎ)
64 eqid 2732 . . 3 (Β·π‘–β€˜πΎ) = (Β·π‘–β€˜πΎ)
65 eqid 2732 . . 3 (0gβ€˜πΎ) = (0gβ€˜πΎ)
66 eqid 2732 . . 3 (*π‘Ÿβ€˜(Scalarβ€˜πΎ)) = (*π‘Ÿβ€˜(Scalarβ€˜πΎ))
67 eqid 2732 . . 3 (0gβ€˜(Scalarβ€˜πΎ)) = (0gβ€˜(Scalarβ€˜πΎ))
6862, 63, 64, 65, 66, 67isphl 21180 . 2 (𝐾 ∈ PreHil ↔ (𝐾 ∈ LVec ∧ (Scalarβ€˜πΎ) ∈ *-Ring ∧ βˆ€π‘Ž ∈ (Baseβ€˜πΎ)((𝑏 ∈ (Baseβ€˜πΎ) ↦ (𝑏(Β·π‘–β€˜πΎ)π‘Ž)) ∈ (𝐾 LMHom (ringLModβ€˜(Scalarβ€˜πΎ))) ∧ ((π‘Ž(Β·π‘–β€˜πΎ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΎ)) β†’ π‘Ž = (0gβ€˜πΎ)) ∧ βˆ€π‘ ∈ (Baseβ€˜πΎ)((*π‘Ÿβ€˜(Scalarβ€˜πΎ))β€˜(π‘Ž(Β·π‘–β€˜πΎ)𝑏)) = (𝑏(Β·π‘–β€˜πΎ)π‘Ž))))
69 eqid 2732 . . 3 (Baseβ€˜πΏ) = (Baseβ€˜πΏ)
70 eqid 2732 . . 3 (Scalarβ€˜πΏ) = (Scalarβ€˜πΏ)
71 eqid 2732 . . 3 (Β·π‘–β€˜πΏ) = (Β·π‘–β€˜πΏ)
72 eqid 2732 . . 3 (0gβ€˜πΏ) = (0gβ€˜πΏ)
73 eqid 2732 . . 3 (*π‘Ÿβ€˜(Scalarβ€˜πΏ)) = (*π‘Ÿβ€˜(Scalarβ€˜πΏ))
74 eqid 2732 . . 3 (0gβ€˜(Scalarβ€˜πΏ)) = (0gβ€˜(Scalarβ€˜πΏ))
7569, 70, 71, 72, 73, 74isphl 21180 . 2 (𝐿 ∈ PreHil ↔ (𝐿 ∈ LVec ∧ (Scalarβ€˜πΏ) ∈ *-Ring ∧ βˆ€π‘Ž ∈ (Baseβ€˜πΏ)((𝑏 ∈ (Baseβ€˜πΏ) ↦ (𝑏(Β·π‘–β€˜πΏ)π‘Ž)) ∈ (𝐿 LMHom (ringLModβ€˜(Scalarβ€˜πΏ))) ∧ ((π‘Ž(Β·π‘–β€˜πΏ)π‘Ž) = (0gβ€˜(Scalarβ€˜πΏ)) β†’ π‘Ž = (0gβ€˜πΏ)) ∧ βˆ€π‘ ∈ (Baseβ€˜πΏ)((*π‘Ÿβ€˜(Scalarβ€˜πΏ))β€˜(π‘Ž(Β·π‘–β€˜πΏ)𝑏)) = (𝑏(Β·π‘–β€˜πΏ)π‘Ž))))
7661, 68, 753bitr4g 313 1 (πœ‘ β†’ (𝐾 ∈ PreHil ↔ 𝐿 ∈ PreHil))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   ↦ cmpt 5231  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  +gcplusg 17196  *π‘Ÿcstv 17198  Scalarcsca 17199   ·𝑠 cvsca 17200  Β·π‘–cip 17201  0gc0g 17384  *-Ringcsr 20451   LMHom clmhm 20629  LVecclvec 20712  ringLModcrglmod 20781  PreHilcphl 21176
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-sca 17212  df-vsca 17213  df-ip 17214  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-mhm 18670  df-grp 18821  df-ghm 19089  df-mgp 19987  df-ur 20004  df-ring 20057  df-lmod 20472  df-lmhm 20632  df-lvec 20713  df-sra 20784  df-rgmod 20785  df-phl 21178
This theorem is referenced by:  tcphphl  24743
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