Metamath Proof Explorer

Theorem hgmapval1

Description: Value of the scalar sigma map at one. (Contributed by NM, 12-Jun-2015)

Ref Expression
Hypotheses hgmapval1.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
hgmapval1.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
hgmapval1.r ⊢ 𝑅 = ( Scalar ‘ 𝑈 )
hgmapval1.i ⊢ 1 = ( 1r ‘ 𝑅 )
hgmapval1.g ⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
hgmapval1.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
Assertion hgmapval1 ( 𝜑 → ( 𝐺 ‘ 1 ) = 1 )

Proof

Step Hyp Ref Expression
1 hgmapval1.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 hgmapval1.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 hgmapval1.r ⊢ 𝑅 = ( Scalar ‘ 𝑈 )
4 hgmapval1.i ⊢ 1 = ( 1r ‘ 𝑅 )
5 hgmapval1.g ⊢ 𝐺 = ( ( HGMap ‘ 𝐾 ) ‘ 𝑊 )
6 hgmapval1.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7 eqid ⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
8 eqid ⊢ ( 0g ‘ 𝑈 ) = ( 0g ‘ 𝑈 )
9 1 2 7 8 6 dvh1dim ⊢ ( 𝜑 → ∃ 𝑥 ∈ ( Base ‘ 𝑈 ) 𝑥 ≠ ( 0g ‘ 𝑈 ) )
10 eqid ⊢ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
11 eqid ⊢ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
12 eqid ⊢ ( 1r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( 1r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) )
13 1 2 3 4 10 11 12 6 lcd1 ⊢ ( 𝜑 → ( 1r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) = 1 )
14 13 oveq1d ⊢ ( 𝜑 → ( ( 1r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) = ( 1 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) )
15 14 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → ( ( 1r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) = ( 1 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) )
16 1 10 6 lcdlmod ⊢ ( 𝜑 → ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∈ LMod )
17 16 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∈ LMod )
18 eqid ⊢ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
19 eqid ⊢ ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
20 6 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
21 simp2 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → 𝑥 ∈ ( Base ‘ 𝑈 ) )
22 1 2 7 10 18 19 20 21 hdmapcl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) )
23 eqid ⊢ ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
24 18 11 23 12 lmodvs1 ⊢ ( ( ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∈ LMod ∧ ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ∈ ( Base ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) → ( ( 1r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) = ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) )
25 17 22 24 syl2anc ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → ( ( 1r ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) = ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) )
26 15 25 eqtr3d ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → ( 1 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) = ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) )
27 1 2 6 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
28 27 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → 𝑈 ∈ LMod )
29 eqid ⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 )
30 7 3 29 4 lmodvs1 ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ) → ( 1 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) = 𝑥 )
31 28 21 30 syl2anc ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → ( 1 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) = 𝑥 )
32 31 fveq2d ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 1 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) )
33 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
34 3 lmodring ⊢ ( 𝑈 ∈ LMod → 𝑅 ∈ Ring )
35 33 4 ringidcl ⊢ ( 𝑅 ∈ Ring → 1 ∈ ( Base ‘ 𝑅 ) )
36 27 34 35 3syl ⊢ ( 𝜑 → 1 ∈ ( Base ‘ 𝑅 ) )
37 36 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → 1 ∈ ( Base ‘ 𝑅 ) )
38 1 2 7 29 3 33 10 23 19 5 20 21 37 hgmapvs ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 1 ( ·𝑠 ‘ 𝑈 ) 𝑥 ) ) = ( ( 𝐺 ‘ 1 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) )
39 26 32 38 3eqtr2rd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → ( ( 𝐺 ‘ 1 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) = ( 1 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) )
40 eqid ⊢ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) )
41 eqid ⊢ ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) = ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) )
42 1 10 6 lcdlvec ⊢ ( 𝜑 → ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∈ LVec )
43 42 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∈ LVec )
44 1 2 3 33 5 6 36 hgmapcl ⊢ ( 𝜑 → ( 𝐺 ‘ 1 ) ∈ ( Base ‘ 𝑅 ) )
45 1 2 3 33 10 11 40 6 lcdsbase ⊢ ( 𝜑 → ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) = ( Base ‘ 𝑅 ) )
46 44 45 eleqtrrd ⊢ ( 𝜑 → ( 𝐺 ‘ 1 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) )
47 46 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐺 ‘ 1 ) ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) )
48 36 45 eleqtrrd ⊢ ( 𝜑 → 1 ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) )
49 48 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → 1 ∈ ( Base ‘ ( Scalar ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ) )
50 simp3 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → 𝑥 ≠ ( 0g ‘ 𝑈 ) )
51 1 2 7 8 10 41 19 20 21 hdmapeq0 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) = ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ↔ 𝑥 = ( 0g ‘ 𝑈 ) ) )
52 51 necon3bid ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ≠ ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ↔ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) )
53 50 52 mpbird ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ≠ ( 0g ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) )
54 18 23 11 40 41 43 47 49 22 53 lvecvscan2 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → ( ( ( 𝐺 ‘ 1 ) ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) = ( 1 ( ·𝑠 ‘ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) ( ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ) ↔ ( 𝐺 ‘ 1 ) = 1 ) )
55 39 54 mpbid ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝑈 ) ∧ 𝑥 ≠ ( 0g ‘ 𝑈 ) ) → ( 𝐺 ‘ 1 ) = 1 )
56 55 rexlimdv3a ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ( Base ‘ 𝑈 ) 𝑥 ≠ ( 0g ‘ 𝑈 ) → ( 𝐺 ‘ 1 ) = 1 ) )
57 9 56 mpd ⊢ ( 𝜑 → ( 𝐺 ‘ 1 ) = 1 )