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Theorem phlpropd 21082
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 20673 . . 3 (πœ‘ β†’ (𝐾 ∈ LVec ↔ 𝐿 ∈ LVec))
94, 5eqtr3d 2775 . . . 4 (πœ‘ β†’ (Scalarβ€˜πΎ) = (Scalarβ€˜πΏ))
109eleq1d 2819 . . 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 7388 . . . . . . . . . 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 654 . . . . . . . . 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 5209 . . . . . . . 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 482 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ 𝐡 = (Baseβ€˜πΎ))
1615mpteq1d 5204 . . . . . . . 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 482 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ 𝐡 = (Baseβ€˜πΏ))
1817mpteq1d 5204 . . . . . . . 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 2781 . . . . . . 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 20709 . . . . . . . . . . . 12 (Baseβ€˜πΉ) = (Baseβ€˜(ringLModβ€˜πΉ))
216, 20eqtri 2761 . . . . . . . . . . 11 𝑃 = (Baseβ€˜(ringLModβ€˜πΉ))
2221a1i 11 . . . . . . . . . 10 (πœ‘ β†’ 𝑃 = (Baseβ€˜(ringLModβ€˜πΉ)))
23 fvex 6859 . . . . . . . . . . . 12 (Scalarβ€˜πΎ) ∈ V
244, 23eqeltrdi 2842 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹 ∈ V)
25 rlmsca 20714 . . . . . . . . . . 11 (𝐹 ∈ V β†’ 𝐹 = (Scalarβ€˜(ringLModβ€˜πΉ)))
2624, 25syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐹 = (Scalarβ€˜(ringLModβ€˜πΉ)))
27 eqidd 2734 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝑃)) β†’ (π‘₯(+gβ€˜(ringLModβ€˜πΉ))𝑦) = (π‘₯(+gβ€˜(ringLModβ€˜πΉ))𝑦))
28 eqidd 2734 . . . . . . . . . 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 20578 . . . . . . . . 9 (πœ‘ β†’ (𝐾 LMHom (ringLModβ€˜πΉ)) = (𝐿 LMHom (ringLModβ€˜πΉ)))
304fveq2d 6850 . . . . . . . . . 10 (πœ‘ β†’ (ringLModβ€˜πΉ) = (ringLModβ€˜(Scalarβ€˜πΎ)))
3130oveq2d 7377 . . . . . . . . 9 (πœ‘ β†’ (𝐾 LMHom (ringLModβ€˜πΉ)) = (𝐾 LMHom (ringLModβ€˜(Scalarβ€˜πΎ))))
325fveq2d 6850 . . . . . . . . . 10 (πœ‘ β†’ (ringLModβ€˜πΉ) = (ringLModβ€˜(Scalarβ€˜πΏ)))
3332oveq2d 7377 . . . . . . . . 9 (πœ‘ β†’ (𝐿 LMHom (ringLModβ€˜πΉ)) = (𝐿 LMHom (ringLModβ€˜(Scalarβ€˜πΏ))))
3429, 31, 333eqtr3d 2781 . . . . . . . 8 (πœ‘ β†’ (𝐾 LMHom (ringLModβ€˜(Scalarβ€˜πΎ))) = (𝐿 LMHom (ringLModβ€˜(Scalarβ€˜πΏ))))
3534adantr 482 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (𝐾 LMHom (ringLModβ€˜(Scalarβ€˜πΎ))) = (𝐿 LMHom (ringLModβ€˜(Scalarβ€˜πΏ))))
3619, 35eleq12d 2828 . . . . . 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 7388 . . . . . . . . 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 673 . . . . . . . 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 6850 . . . . . . . . 9 (πœ‘ β†’ (0gβ€˜(Scalarβ€˜πΎ)) = (0gβ€˜(Scalarβ€˜πΏ)))
4039adantr 482 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (0gβ€˜(Scalarβ€˜πΎ)) = (0gβ€˜(Scalarβ€˜πΏ)))
4138, 40eqeq12d 2749 . . . . . . 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 18525 . . . . . . . . 9 (πœ‘ β†’ (0gβ€˜πΎ) = (0gβ€˜πΏ))
4342adantr 482 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (0gβ€˜πΎ) = (0gβ€˜πΏ))
4443eqeq2d 2744 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (π‘Ž = (0gβ€˜πΎ) ↔ π‘Ž = (0gβ€˜πΏ)))
4541, 44imbi12d 345 . . . . . 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 6850 . . . . . . . . . . . 12 (πœ‘ β†’ (*π‘Ÿβ€˜(Scalarβ€˜πΎ)) = (*π‘Ÿβ€˜(Scalarβ€˜πΏ)))
4746adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (*π‘Ÿβ€˜(Scalarβ€˜πΎ)) = (*π‘Ÿβ€˜(Scalarβ€˜πΏ)))
4811oveqrspc2v 7388 . . . . . . . . . . 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 6853 . . . . . . . . . 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 469 . . . . . . . . 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 2749 . . . . . . . 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 3169 . . . . . . 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 3312 . . . . . . 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 3312 . . . . . . 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 309 . . . . . 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 1437 . . . . 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 3169 . . . 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 3312 . . . 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 3312 . . . 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 309 . . 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 1437 . 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 2733 . . 3 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
63 eqid 2733 . . 3 (Scalarβ€˜πΎ) = (Scalarβ€˜πΎ)
64 eqid 2733 . . 3 (Β·π‘–β€˜πΎ) = (Β·π‘–β€˜πΎ)
65 eqid 2733 . . 3 (0gβ€˜πΎ) = (0gβ€˜πΎ)
66 eqid 2733 . . 3 (*π‘Ÿβ€˜(Scalarβ€˜πΎ)) = (*π‘Ÿβ€˜(Scalarβ€˜πΎ))
67 eqid 2733 . . 3 (0gβ€˜(Scalarβ€˜πΎ)) = (0gβ€˜(Scalarβ€˜πΎ))
6862, 63, 64, 65, 66, 67isphl 21055 . 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 2733 . . 3 (Baseβ€˜πΏ) = (Baseβ€˜πΏ)
70 eqid 2733 . . 3 (Scalarβ€˜πΏ) = (Scalarβ€˜πΏ)
71 eqid 2733 . . 3 (Β·π‘–β€˜πΏ) = (Β·π‘–β€˜πΏ)
72 eqid 2733 . . 3 (0gβ€˜πΏ) = (0gβ€˜πΏ)
73 eqid 2733 . . 3 (*π‘Ÿβ€˜(Scalarβ€˜πΏ)) = (*π‘Ÿβ€˜(Scalarβ€˜πΏ))
74 eqid 2733 . . 3 (0gβ€˜(Scalarβ€˜πΏ)) = (0gβ€˜(Scalarβ€˜πΏ))
7569, 70, 71, 72, 73, 74isphl 21055 . 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 314 1 (πœ‘ β†’ (𝐾 ∈ PreHil ↔ 𝐿 ∈ PreHil))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3447   ↦ cmpt 5192  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091  +gcplusg 17141  *π‘Ÿcstv 17143  Scalarcsca 17144   ·𝑠 cvsca 17145  Β·π‘–cip 17146  0gc0g 17329  *-Ringcsr 20346   LMHom clmhm 20524  LVecclvec 20607  ringLModcrglmod 20675  PreHilcphl 21051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-sca 17157  df-vsca 17158  df-ip 17159  df-0g 17331  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-mhm 18609  df-grp 18759  df-ghm 19014  df-mgp 19905  df-ur 19922  df-ring 19974  df-lmod 20367  df-lmhm 20527  df-lvec 20608  df-sra 20678  df-rgmod 20679  df-phl 21053
This theorem is referenced by:  tcphphl  24614
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