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 ) |