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 ∧ 𝑊 ∈ 𝐻 ) )
hgmapvv.j ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion hgmapvv ( 𝜑 → ( 𝐺 ‘ ( 𝐺 ‘ 𝑋 ) ) = 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 ∧ 𝑊 ∈ 𝐻 ) )
7 hgmapvv.j ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
8 2fveq3 ⊢ ( 𝑋 = ( 0g ‘ 𝑅 ) → ( 𝐺 ‘ ( 𝐺 ‘ 𝑋 ) ) = ( 𝐺 ‘ ( 𝐺 ‘ ( 0g ‘ 𝑅 ) ) ) )
9 id ⊢ ( 𝑋 = ( 0g ‘ 𝑅 ) → 𝑋 = ( 0g ‘ 𝑅 ) )
10 8 9 eqeq12d ⊢ ( 𝑋 = ( 0g ‘ 𝑅 ) → ( ( 𝐺 ‘ ( 𝐺 ‘ 𝑋 ) ) = 𝑋 ↔ ( 𝐺 ‘ ( 𝐺 ‘ ( 0g ‘ 𝑅 ) ) ) = ( 0g ‘ 𝑅 ) ) )
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 ⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 )
15 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
16 eqid ⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
17 eqid ⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
18 eqid ⊢ ( invr ‘ 𝑅 ) = ( invr ‘ 𝑅 )
19 eqid ⊢ ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
20 6 adantr ⊢ ( ( 𝜑 ∧ 𝑋 ≠ ( 0g ‘ 𝑅 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
21 7 anim1i ⊢ ( ( 𝜑 ∧ 𝑋 ≠ ( 0g ‘ 𝑅 ) ) → ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ ( 0g ‘ 𝑅 ) ) )
22 eldifsn ⊢ ( 𝑋 ∈ ( 𝐵 ∖ { ( 0g ‘ 𝑅 ) } ) ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ ( 0g ‘ 𝑅 ) ) )
23 21 22 sylibr ⊢ ( ( 𝜑 ∧ 𝑋 ≠ ( 0g ‘ 𝑅 ) ) → 𝑋 ∈ ( 𝐵 ∖ { ( 0g ‘ 𝑅 ) } ) )
24 1 11 12 2 13 14 3 4 15 16 17 18 19 5 20 23 hgmapvvlem3 ⊢ ( ( 𝜑 ∧ 𝑋 ≠ ( 0g ‘ 𝑅 ) ) → ( 𝐺 ‘ ( 𝐺 ‘ 𝑋 ) ) = 𝑋 )
25 1 2 3 16 5 6 hgmapval0 ⊢ ( 𝜑 → ( 𝐺 ‘ ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
26 25 fveq2d ⊢ ( 𝜑 → ( 𝐺 ‘ ( 𝐺 ‘ ( 0g ‘ 𝑅 ) ) ) = ( 𝐺 ‘ ( 0g ‘ 𝑅 ) ) )
27 26 25 eqtrd ⊢ ( 𝜑 → ( 𝐺 ‘ ( 𝐺 ‘ ( 0g ‘ 𝑅 ) ) ) = ( 0g ‘ 𝑅 ) )
28 10 24 27 pm2.61ne ⊢ ( 𝜑 → ( 𝐺 ‘ ( 𝐺 ‘ 𝑋 ) ) = 𝑋 )