Step |
Hyp |
Ref |
Expression |
1 |
|
mapdh.q |
⊢ 𝑄 = ( 0g ‘ 𝐶 ) |
2 |
|
mapdh.i |
⊢ 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2nd ‘ 𝑥 ) = 0 , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) − ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) ) |
3 |
|
mapdh.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
mapdh.m |
⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
mapdh.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
mapdh.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
7 |
|
mapdh.s |
⊢ − = ( -g ‘ 𝑈 ) |
8 |
|
mapdhc.o |
⊢ 0 = ( 0g ‘ 𝑈 ) |
9 |
|
mapdh.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
10 |
|
mapdh.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
11 |
|
mapdh.d |
⊢ 𝐷 = ( Base ‘ 𝐶 ) |
12 |
|
mapdh.r |
⊢ 𝑅 = ( -g ‘ 𝐶 ) |
13 |
|
mapdh.j |
⊢ 𝐽 = ( LSpan ‘ 𝐶 ) |
14 |
|
mapdh.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
15 |
|
mapdhc.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐷 ) |
16 |
|
mapdh.mn |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) ) |
17 |
|
mapdhcl.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) |
18 |
|
mapdhe.y |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ) |
19 |
|
mapdhe.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐷 ) |
20 |
|
mapdh.ne2 |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) |
21 |
1 2 17 15 18
|
mapdhval2 |
⊢ ( 𝜑 → ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) = ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) ) |
22 |
21
|
eqeq1d |
⊢ ( 𝜑 → ( ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) = 𝐺 ↔ ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) = 𝐺 ) ) |
23 |
3 4 5 6 7 8 9 10 11 12 13 14 17 18 15 20 16
|
mapdpg |
⊢ ( 𝜑 → ∃! ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) |
24 |
|
nfv |
⊢ Ⅎ ℎ 𝜑 |
25 |
|
nfcvd |
⊢ ( 𝜑 → Ⅎ ℎ 𝐺 ) |
26 |
|
nfvd |
⊢ ( 𝜑 → Ⅎ ℎ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) |
27 |
|
sneq |
⊢ ( ℎ = 𝐺 → { ℎ } = { 𝐺 } ) |
28 |
27
|
fveq2d |
⊢ ( ℎ = 𝐺 → ( 𝐽 ‘ { ℎ } ) = ( 𝐽 ‘ { 𝐺 } ) ) |
29 |
28
|
eqeq2d |
⊢ ( ℎ = 𝐺 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ↔ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ) ) |
30 |
|
oveq2 |
⊢ ( ℎ = 𝐺 → ( 𝐹 𝑅 ℎ ) = ( 𝐹 𝑅 𝐺 ) ) |
31 |
30
|
sneqd |
⊢ ( ℎ = 𝐺 → { ( 𝐹 𝑅 ℎ ) } = { ( 𝐹 𝑅 𝐺 ) } ) |
32 |
31
|
fveq2d |
⊢ ( ℎ = 𝐺 → ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) |
33 |
32
|
eqeq2d |
⊢ ( ℎ = 𝐺 → ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ↔ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) |
34 |
29 33
|
anbi12d |
⊢ ( ℎ = 𝐺 → ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) ) |
35 |
34
|
adantl |
⊢ ( ( 𝜑 ∧ ℎ = 𝐺 ) → ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) ) |
36 |
24 25 26 19 35
|
riota2df |
⊢ ( ( 𝜑 ∧ ∃! ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) → ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ↔ ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) = 𝐺 ) ) |
37 |
23 36
|
mpdan |
⊢ ( 𝜑 → ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ↔ ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ℎ ) } ) ) ) = 𝐺 ) ) |
38 |
22 37
|
bitr4d |
⊢ ( 𝜑 → ( ( 𝐼 ‘ 〈 𝑋 , 𝐹 , 𝑌 〉 ) = 𝐺 ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) ) |