Metamath Proof Explorer


Theorem hgmapvv

Description: Value of a double involution. Part 1.2 of Baer p. 110 line 37. (Contributed by NM, 13-Jun-2015)

Ref Expression
Hypotheses hgmapvv.h 𝐻 = ( LHyp ‘ 𝐾 )
hgmapvv.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
hgmapvv.r 𝑅 = ( Scalar ‘ 𝑈 )
hgmapvv.b 𝐵 = ( Base ‘ 𝑅 )
hgmapvv.g 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
hgmapvv.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
hgmapvv.j ( 𝜑𝑋𝐵 )
Assertion hgmapvv ( 𝜑 → ( 𝐺 ‘ ( 𝐺𝑋 ) ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 hgmapvv.h 𝐻 = ( LHyp ‘ 𝐾 )
2 hgmapvv.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 hgmapvv.r 𝑅 = ( Scalar ‘ 𝑈 )
4 hgmapvv.b 𝐵 = ( Base ‘ 𝑅 )
5 hgmapvv.g 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
6 hgmapvv.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
7 hgmapvv.j ( 𝜑𝑋𝐵 )
8 2fveq3 ( 𝑋 = ( 0g𝑅 ) → ( 𝐺 ‘ ( 𝐺𝑋 ) ) = ( 𝐺 ‘ ( 𝐺 ‘ ( 0g𝑅 ) ) ) )
9 id ( 𝑋 = ( 0g𝑅 ) → 𝑋 = ( 0g𝑅 ) )
10 8 9 eqeq12d ( 𝑋 = ( 0g𝑅 ) → ( ( 𝐺 ‘ ( 𝐺𝑋 ) ) = 𝑋 ↔ ( 𝐺 ‘ ( 𝐺 ‘ ( 0g𝑅 ) ) ) = ( 0g𝑅 ) ) )
11 eqid ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩ = ⟨ ( I ↾ ( Base ‘ 𝐾 ) ) , ( I ↾ ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) ) ⟩
12 eqid ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
13 eqid ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
14 eqid ( ·𝑠𝑈 ) = ( ·𝑠𝑈 )
15 eqid ( .r𝑅 ) = ( .r𝑅 )
16 eqid ( 0g𝑅 ) = ( 0g𝑅 )
17 eqid ( 1r𝑅 ) = ( 1r𝑅 )
18 eqid ( invr𝑅 ) = ( invr𝑅 )
19 eqid ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
20 6 adantr ( ( 𝜑𝑋 ≠ ( 0g𝑅 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
21 7 anim1i ( ( 𝜑𝑋 ≠ ( 0g𝑅 ) ) → ( 𝑋𝐵𝑋 ≠ ( 0g𝑅 ) ) )
22 eldifsn ( 𝑋 ∈ ( 𝐵 ∖ { ( 0g𝑅 ) } ) ↔ ( 𝑋𝐵𝑋 ≠ ( 0g𝑅 ) ) )
23 21 22 sylibr ( ( 𝜑𝑋 ≠ ( 0g𝑅 ) ) → 𝑋 ∈ ( 𝐵 ∖ { ( 0g𝑅 ) } ) )
24 1 11 12 2 13 14 3 4 15 16 17 18 19 5 20 23 hgmapvvlem3 ( ( 𝜑𝑋 ≠ ( 0g𝑅 ) ) → ( 𝐺 ‘ ( 𝐺𝑋 ) ) = 𝑋 )
25 1 2 3 16 5 6 hgmapval0 ( 𝜑 → ( 𝐺 ‘ ( 0g𝑅 ) ) = ( 0g𝑅 ) )
26 25 fveq2d ( 𝜑 → ( 𝐺 ‘ ( 𝐺 ‘ ( 0g𝑅 ) ) ) = ( 𝐺 ‘ ( 0g𝑅 ) ) )
27 26 25 eqtrd ( 𝜑 → ( 𝐺 ‘ ( 𝐺 ‘ ( 0g𝑅 ) ) ) = ( 0g𝑅 ) )
28 10 24 27 pm2.61ne ( 𝜑 → ( 𝐺 ‘ ( 𝐺𝑋 ) ) = 𝑋 )