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 )