Metamath Proof Explorer


Theorem hgmapcl

Description: Closure of scalar sigma map i.e. the map from the vector space scalar base to the dual space scalar base. (Contributed by NM, 6-Jun-2015)

Ref Expression
Hypotheses hgmapcl.h 𝐻 = ( LHyp ‘ 𝐾 )
hgmapcl.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
hgmapcl.r 𝑅 = ( Scalar ‘ 𝑈 )
hgmapcl.b 𝐵 = ( Base ‘ 𝑅 )
hgmapcl.g 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
hgmapcl.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
hgmapcl.f ( 𝜑𝐹𝐵 )
Assertion hgmapcl ( 𝜑 → ( 𝐺𝐹 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 hgmapcl.h 𝐻 = ( LHyp ‘ 𝐾 )
2 hgmapcl.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 hgmapcl.r 𝑅 = ( Scalar ‘ 𝑈 )
4 hgmapcl.b 𝐵 = ( Base ‘ 𝑅 )
5 hgmapcl.g 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
6 hgmapcl.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
7 hgmapcl.f ( 𝜑𝐹𝐵 )
8 eqid ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
9 eqid ( ·𝑠𝑈 ) = ( ·𝑠𝑈 )
10 eqid ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
11 eqid ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
12 eqid ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
13 1 2 8 9 3 4 10 11 12 5 6 7 hgmapval ( 𝜑 → ( 𝐺𝐹 ) = ( 𝑔𝐵𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐹 ( ·𝑠𝑈 ) 𝑥 ) ) = ( 𝑔 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) ) )
14 1 2 8 9 3 4 10 11 12 6 7 hdmap14lem15 ( 𝜑 → ∃! 𝑔𝐵𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐹 ( ·𝑠𝑈 ) 𝑥 ) ) = ( 𝑔 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) )
15 riotacl ( ∃! 𝑔𝐵𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐹 ( ·𝑠𝑈 ) 𝑥 ) ) = ( 𝑔 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) → ( 𝑔𝐵𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐹 ( ·𝑠𝑈 ) 𝑥 ) ) = ( 𝑔 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) ) ∈ 𝐵 )
16 14 15 syl ( 𝜑 → ( 𝑔𝐵𝑥 ∈ ( Base ‘ 𝑈 ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝐹 ( ·𝑠𝑈 ) 𝑥 ) ) = ( 𝑔 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) ) ∈ 𝐵 )
17 13 16 eqeltrd ( 𝜑 → ( 𝐺𝐹 ) ∈ 𝐵 )