Step |
Hyp |
Ref |
Expression |
1 |
|
hdmap1val.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
hdmap1fval.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
hdmap1fval.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
4 |
|
hdmap1fval.s |
⊢ − = ( -g ‘ 𝑈 ) |
5 |
|
hdmap1fval.o |
⊢ 0 = ( 0g ‘ 𝑈 ) |
6 |
|
hdmap1fval.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
7 |
|
hdmap1fval.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
hdmap1fval.d |
⊢ 𝐷 = ( Base ‘ 𝐶 ) |
9 |
|
hdmap1fval.r |
⊢ 𝑅 = ( -g ‘ 𝐶 ) |
10 |
|
hdmap1fval.q |
⊢ 𝑄 = ( 0g ‘ 𝐶 ) |
11 |
|
hdmap1fval.j |
⊢ 𝐽 = ( LSpan ‘ 𝐶 ) |
12 |
|
hdmap1fval.m |
⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) |
13 |
|
hdmap1fval.i |
⊢ 𝐼 = ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) |
14 |
|
hdmap1fval.k |
⊢ ( 𝜑 → ( 𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻 ) ) |
15 |
1
|
hdmap1ffval |
⊢ ( 𝐾 ∈ 𝐴 → ( HDMap1 ‘ 𝐾 ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑎 ∣ [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( LSpan ‘ 𝑢 ) / 𝑛 ] [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) } ) ) |
16 |
15
|
fveq1d |
⊢ ( 𝐾 ∈ 𝐴 → ( ( HDMap1 ‘ 𝐾 ) ‘ 𝑊 ) = ( ( 𝑤 ∈ 𝐻 ↦ { 𝑎 ∣ [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( LSpan ‘ 𝑢 ) / 𝑛 ] [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) } ) ‘ 𝑊 ) ) |
17 |
13 16
|
syl5eq |
⊢ ( 𝐾 ∈ 𝐴 → 𝐼 = ( ( 𝑤 ∈ 𝐻 ↦ { 𝑎 ∣ [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( LSpan ‘ 𝑢 ) / 𝑛 ] [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) } ) ‘ 𝑊 ) ) |
18 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) |
19 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) |
20 |
|
fveq2 |
⊢ ( 𝑤 = 𝑊 → ( ( mapd ‘ 𝐾 ) ‘ 𝑤 ) = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ) |
21 |
20
|
sbceq1d |
⊢ ( 𝑤 = 𝑊 → ( [ ( ( mapd ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ [ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ) ) |
22 |
21
|
sbcbidv |
⊢ ( 𝑤 = 𝑊 → ( [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ) ) |
23 |
22
|
sbcbidv |
⊢ ( 𝑤 = 𝑊 → ( [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ) ) |
24 |
19 23
|
sbceqbid |
⊢ ( 𝑤 = 𝑊 → ( [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ) ) |
25 |
24
|
sbcbidv |
⊢ ( 𝑤 = 𝑊 → ( [ ( LSpan ‘ 𝑢 ) / 𝑛 ] [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ [ ( LSpan ‘ 𝑢 ) / 𝑛 ] [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ) ) |
26 |
25
|
sbcbidv |
⊢ ( 𝑤 = 𝑊 → ( [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( LSpan ‘ 𝑢 ) / 𝑛 ] [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( LSpan ‘ 𝑢 ) / 𝑛 ] [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ) ) |
27 |
18 26
|
sbceqbid |
⊢ ( 𝑤 = 𝑊 → ( [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( LSpan ‘ 𝑢 ) / 𝑛 ] [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( LSpan ‘ 𝑢 ) / 𝑛 ] [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ) ) |
28 |
|
fvex |
⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ∈ V |
29 |
|
fvex |
⊢ ( Base ‘ 𝑢 ) ∈ V |
30 |
|
fvex |
⊢ ( LSpan ‘ 𝑢 ) ∈ V |
31 |
2
|
eqeq2i |
⊢ ( 𝑢 = 𝑈 ↔ 𝑢 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) |
32 |
31
|
biimpri |
⊢ ( 𝑢 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) → 𝑢 = 𝑈 ) |
33 |
32
|
3ad2ant1 |
⊢ ( ( 𝑢 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑣 = ( Base ‘ 𝑢 ) ∧ 𝑛 = ( LSpan ‘ 𝑢 ) ) → 𝑢 = 𝑈 ) |
34 |
|
simp2 |
⊢ ( ( 𝑢 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑣 = ( Base ‘ 𝑢 ) ∧ 𝑛 = ( LSpan ‘ 𝑢 ) ) → 𝑣 = ( Base ‘ 𝑢 ) ) |
35 |
32
|
fveq2d |
⊢ ( 𝑢 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) → ( Base ‘ 𝑢 ) = ( Base ‘ 𝑈 ) ) |
36 |
35
|
3ad2ant1 |
⊢ ( ( 𝑢 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑣 = ( Base ‘ 𝑢 ) ∧ 𝑛 = ( LSpan ‘ 𝑢 ) ) → ( Base ‘ 𝑢 ) = ( Base ‘ 𝑈 ) ) |
37 |
34 36
|
eqtrd |
⊢ ( ( 𝑢 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑣 = ( Base ‘ 𝑢 ) ∧ 𝑛 = ( LSpan ‘ 𝑢 ) ) → 𝑣 = ( Base ‘ 𝑈 ) ) |
38 |
37 3
|
eqtr4di |
⊢ ( ( 𝑢 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑣 = ( Base ‘ 𝑢 ) ∧ 𝑛 = ( LSpan ‘ 𝑢 ) ) → 𝑣 = 𝑉 ) |
39 |
|
simp3 |
⊢ ( ( 𝑢 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑣 = ( Base ‘ 𝑢 ) ∧ 𝑛 = ( LSpan ‘ 𝑢 ) ) → 𝑛 = ( LSpan ‘ 𝑢 ) ) |
40 |
33
|
fveq2d |
⊢ ( ( 𝑢 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑣 = ( Base ‘ 𝑢 ) ∧ 𝑛 = ( LSpan ‘ 𝑢 ) ) → ( LSpan ‘ 𝑢 ) = ( LSpan ‘ 𝑈 ) ) |
41 |
39 40
|
eqtrd |
⊢ ( ( 𝑢 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑣 = ( Base ‘ 𝑢 ) ∧ 𝑛 = ( LSpan ‘ 𝑢 ) ) → 𝑛 = ( LSpan ‘ 𝑈 ) ) |
42 |
41 6
|
eqtr4di |
⊢ ( ( 𝑢 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑣 = ( Base ‘ 𝑢 ) ∧ 𝑛 = ( LSpan ‘ 𝑢 ) ) → 𝑛 = 𝑁 ) |
43 |
|
fvex |
⊢ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∈ V |
44 |
|
fvex |
⊢ ( Base ‘ 𝑐 ) ∈ V |
45 |
|
fvex |
⊢ ( LSpan ‘ 𝑐 ) ∈ V |
46 |
|
id |
⊢ ( 𝑐 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) → 𝑐 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ) |
47 |
46 7
|
eqtr4di |
⊢ ( 𝑐 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) → 𝑐 = 𝐶 ) |
48 |
47
|
3ad2ant1 |
⊢ ( ( 𝑐 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑑 = ( Base ‘ 𝑐 ) ∧ 𝑗 = ( LSpan ‘ 𝑐 ) ) → 𝑐 = 𝐶 ) |
49 |
|
simp2 |
⊢ ( ( 𝑐 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑑 = ( Base ‘ 𝑐 ) ∧ 𝑗 = ( LSpan ‘ 𝑐 ) ) → 𝑑 = ( Base ‘ 𝑐 ) ) |
50 |
48
|
fveq2d |
⊢ ( ( 𝑐 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑑 = ( Base ‘ 𝑐 ) ∧ 𝑗 = ( LSpan ‘ 𝑐 ) ) → ( Base ‘ 𝑐 ) = ( Base ‘ 𝐶 ) ) |
51 |
50 8
|
eqtr4di |
⊢ ( ( 𝑐 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑑 = ( Base ‘ 𝑐 ) ∧ 𝑗 = ( LSpan ‘ 𝑐 ) ) → ( Base ‘ 𝑐 ) = 𝐷 ) |
52 |
49 51
|
eqtrd |
⊢ ( ( 𝑐 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑑 = ( Base ‘ 𝑐 ) ∧ 𝑗 = ( LSpan ‘ 𝑐 ) ) → 𝑑 = 𝐷 ) |
53 |
|
simp3 |
⊢ ( ( 𝑐 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑑 = ( Base ‘ 𝑐 ) ∧ 𝑗 = ( LSpan ‘ 𝑐 ) ) → 𝑗 = ( LSpan ‘ 𝑐 ) ) |
54 |
48
|
fveq2d |
⊢ ( ( 𝑐 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑑 = ( Base ‘ 𝑐 ) ∧ 𝑗 = ( LSpan ‘ 𝑐 ) ) → ( LSpan ‘ 𝑐 ) = ( LSpan ‘ 𝐶 ) ) |
55 |
54 11
|
eqtr4di |
⊢ ( ( 𝑐 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑑 = ( Base ‘ 𝑐 ) ∧ 𝑗 = ( LSpan ‘ 𝑐 ) ) → ( LSpan ‘ 𝑐 ) = 𝐽 ) |
56 |
53 55
|
eqtrd |
⊢ ( ( 𝑐 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑑 = ( Base ‘ 𝑐 ) ∧ 𝑗 = ( LSpan ‘ 𝑐 ) ) → 𝑗 = 𝐽 ) |
57 |
|
fvex |
⊢ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ∈ V |
58 |
|
id |
⊢ ( 𝑚 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) → 𝑚 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) ) |
59 |
58 12
|
eqtr4di |
⊢ ( 𝑚 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) → 𝑚 = 𝑀 ) |
60 |
|
fveq1 |
⊢ ( 𝑚 = 𝑀 → ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) ) |
61 |
60
|
eqeq1d |
⊢ ( 𝑚 = 𝑀 → ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ↔ ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ) ) |
62 |
|
fveq1 |
⊢ ( 𝑚 = 𝑀 → ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) ) |
63 |
62
|
eqeq1d |
⊢ ( 𝑚 = 𝑀 → ( ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ↔ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) |
64 |
61 63
|
anbi12d |
⊢ ( 𝑚 = 𝑀 → ( ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ↔ ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) |
65 |
64
|
riotabidv |
⊢ ( 𝑚 = 𝑀 → ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) = ( ℩ ℎ ∈ 𝑑 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) |
66 |
65
|
ifeq2d |
⊢ ( 𝑚 = 𝑀 → if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) = if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) |
67 |
66
|
mpteq2dv |
⊢ ( 𝑚 = 𝑀 → ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) = ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ) |
68 |
67
|
eleq2d |
⊢ ( 𝑚 = 𝑀 → ( 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ) ) |
69 |
59 68
|
syl |
⊢ ( 𝑚 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) → ( 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ) ) |
70 |
57 69
|
sbcie |
⊢ ( [ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ) |
71 |
|
simp2 |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → 𝑑 = 𝐷 ) |
72 |
|
xpeq2 |
⊢ ( 𝑑 = 𝐷 → ( 𝑣 × 𝑑 ) = ( 𝑣 × 𝐷 ) ) |
73 |
72
|
xpeq1d |
⊢ ( 𝑑 = 𝐷 → ( ( 𝑣 × 𝑑 ) × 𝑣 ) = ( ( 𝑣 × 𝐷 ) × 𝑣 ) ) |
74 |
71 73
|
syl |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → ( ( 𝑣 × 𝑑 ) × 𝑣 ) = ( ( 𝑣 × 𝐷 ) × 𝑣 ) ) |
75 |
|
simp1 |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → 𝑐 = 𝐶 ) |
76 |
75
|
fveq2d |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → ( 0g ‘ 𝑐 ) = ( 0g ‘ 𝐶 ) ) |
77 |
76 10
|
eqtr4di |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → ( 0g ‘ 𝑐 ) = 𝑄 ) |
78 |
|
simp3 |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → 𝑗 = 𝐽 ) |
79 |
78
|
fveq1d |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → ( 𝑗 ‘ { ℎ } ) = ( 𝐽 ‘ { ℎ } ) ) |
80 |
79
|
eqeq2d |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ↔ ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ) ) |
81 |
75
|
fveq2d |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → ( -g ‘ 𝑐 ) = ( -g ‘ 𝐶 ) ) |
82 |
81 9
|
eqtr4di |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → ( -g ‘ 𝑐 ) = 𝑅 ) |
83 |
82
|
oveqd |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) = ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) ) |
84 |
83
|
sneqd |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } = { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) |
85 |
78 84
|
fveq12d |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) |
86 |
85
|
eqeq2d |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → ( ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ↔ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) |
87 |
80 86
|
anbi12d |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → ( ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ↔ ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) |
88 |
71 87
|
riotaeqbidv |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → ( ℩ ℎ ∈ 𝑑 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) = ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) |
89 |
77 88
|
ifeq12d |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) = if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) ) |
90 |
74 89
|
mpteq12dv |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) = ( 𝑥 ∈ ( ( 𝑣 × 𝐷 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) ) ) |
91 |
90
|
eleq2d |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → ( 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝐷 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) ) ) ) |
92 |
70 91
|
syl5bb |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑑 = 𝐷 ∧ 𝑗 = 𝐽 ) → ( [ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝐷 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) ) ) ) |
93 |
48 52 56 92
|
syl3anc |
⊢ ( ( 𝑐 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑑 = ( Base ‘ 𝑐 ) ∧ 𝑗 = ( LSpan ‘ 𝑐 ) ) → ( [ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝐷 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) ) ) ) |
94 |
43 44 45 93
|
sbc3ie |
⊢ ( [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝐷 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) ) ) |
95 |
|
simp2 |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → 𝑣 = 𝑉 ) |
96 |
95
|
xpeq1d |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → ( 𝑣 × 𝐷 ) = ( 𝑉 × 𝐷 ) ) |
97 |
96 95
|
xpeq12d |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → ( ( 𝑣 × 𝐷 ) × 𝑣 ) = ( ( 𝑉 × 𝐷 ) × 𝑉 ) ) |
98 |
|
simp1 |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → 𝑢 = 𝑈 ) |
99 |
98
|
fveq2d |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → ( 0g ‘ 𝑢 ) = ( 0g ‘ 𝑈 ) ) |
100 |
99 5
|
eqtr4di |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → ( 0g ‘ 𝑢 ) = 0 ) |
101 |
100
|
eqeq2d |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) ↔ ( 2nd ‘ 𝑥 ) = 0 ) ) |
102 |
|
simp3 |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → 𝑛 = 𝑁 ) |
103 |
102
|
fveq1d |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) = ( 𝑁 ‘ { ( 2nd ‘ 𝑥 ) } ) ) |
104 |
103
|
fveqeq2d |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ↔ ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ) ) |
105 |
98
|
fveq2d |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → ( -g ‘ 𝑢 ) = ( -g ‘ 𝑈 ) ) |
106 |
105 4
|
eqtr4di |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → ( -g ‘ 𝑢 ) = − ) |
107 |
106
|
oveqd |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) = ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) − ( 2nd ‘ 𝑥 ) ) ) |
108 |
107
|
sneqd |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } = { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) − ( 2nd ‘ 𝑥 ) ) } ) |
109 |
102 108
|
fveq12d |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) = ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) − ( 2nd ‘ 𝑥 ) ) } ) ) |
110 |
109
|
fveqeq2d |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → ( ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ↔ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) − ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) |
111 |
104 110
|
anbi12d |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → ( ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) − ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) |
112 |
111
|
riotabidv |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) = ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) − ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) |
113 |
101 112
|
ifbieq2d |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) = 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 ) } ) ) ) ) ) |
114 |
97 113
|
mpteq12dv |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → ( 𝑥 ∈ ( ( 𝑣 × 𝐷 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) ) = ( 𝑥 ∈ ( ( 𝑉 × 𝐷 ) × 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 ) } ) ) ) ) ) ) |
115 |
114
|
eleq2d |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → ( 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝐷 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , 𝑄 , ( ℩ ℎ ∈ 𝐷 ( ( 𝑀 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) 𝑅 ℎ ) } ) ) ) ) ) ↔ 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑉 × 𝐷 ) × 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 ) } ) ) ) ) ) ) ) |
116 |
94 115
|
syl5bb |
⊢ ( ( 𝑢 = 𝑈 ∧ 𝑣 = 𝑉 ∧ 𝑛 = 𝑁 ) → ( [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑉 × 𝐷 ) × 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 ) } ) ) ) ) ) ) ) |
117 |
33 38 42 116
|
syl3anc |
⊢ ( ( 𝑢 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝑣 = ( Base ‘ 𝑢 ) ∧ 𝑛 = ( LSpan ‘ 𝑢 ) ) → ( [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑉 × 𝐷 ) × 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 ) } ) ) ) ) ) ) ) |
118 |
28 29 30 117
|
sbc3ie |
⊢ ( [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( LSpan ‘ 𝑢 ) / 𝑛 ] [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑉 × 𝐷 ) × 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 ) } ) ) ) ) ) ) |
119 |
27 118
|
syl6bb |
⊢ ( 𝑤 = 𝑊 → ( [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( LSpan ‘ 𝑢 ) / 𝑛 ] [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) ↔ 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑉 × 𝐷 ) × 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 ) } ) ) ) ) ) ) ) |
120 |
119
|
abbi1dv |
⊢ ( 𝑤 = 𝑊 → { 𝑎 ∣ [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( LSpan ‘ 𝑢 ) / 𝑛 ] [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) } = ( 𝑥 ∈ ( ( 𝑉 × 𝐷 ) × 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 ) } ) ) ) ) ) ) |
121 |
|
eqid |
⊢ ( 𝑤 ∈ 𝐻 ↦ { 𝑎 ∣ [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( LSpan ‘ 𝑢 ) / 𝑛 ] [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) } ) = ( 𝑤 ∈ 𝐻 ↦ { 𝑎 ∣ [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( LSpan ‘ 𝑢 ) / 𝑛 ] [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) } ) |
122 |
3
|
fvexi |
⊢ 𝑉 ∈ V |
123 |
8
|
fvexi |
⊢ 𝐷 ∈ V |
124 |
122 123
|
xpex |
⊢ ( 𝑉 × 𝐷 ) ∈ V |
125 |
124 122
|
xpex |
⊢ ( ( 𝑉 × 𝐷 ) × 𝑉 ) ∈ V |
126 |
125
|
mptex |
⊢ ( 𝑥 ∈ ( ( 𝑉 × 𝐷 ) × 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 ) } ) ) ) ) ) ∈ V |
127 |
120 121 126
|
fvmpt |
⊢ ( 𝑊 ∈ 𝐻 → ( ( 𝑤 ∈ 𝐻 ↦ { 𝑎 ∣ [ ( ( DVecH ‘ 𝐾 ) ‘ 𝑤 ) / 𝑢 ] [ ( Base ‘ 𝑢 ) / 𝑣 ] [ ( LSpan ‘ 𝑢 ) / 𝑛 ] [ ( ( LCDual ‘ 𝐾 ) ‘ 𝑤 ) / 𝑐 ] [ ( Base ‘ 𝑐 ) / 𝑑 ] [ ( LSpan ‘ 𝑐 ) / 𝑗 ] [ ( ( mapd ‘ 𝐾 ) ‘ 𝑤 ) / 𝑚 ] 𝑎 ∈ ( 𝑥 ∈ ( ( 𝑣 × 𝑑 ) × 𝑣 ) ↦ if ( ( 2nd ‘ 𝑥 ) = ( 0g ‘ 𝑢 ) , ( 0g ‘ 𝑐 ) , ( ℩ ℎ ∈ 𝑑 ( ( 𝑚 ‘ ( 𝑛 ‘ { ( 2nd ‘ 𝑥 ) } ) ) = ( 𝑗 ‘ { ℎ } ) ∧ ( 𝑚 ‘ ( 𝑛 ‘ { ( ( 1st ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑢 ) ( 2nd ‘ 𝑥 ) ) } ) ) = ( 𝑗 ‘ { ( ( 2nd ‘ ( 1st ‘ 𝑥 ) ) ( -g ‘ 𝑐 ) ℎ ) } ) ) ) ) ) } ) ‘ 𝑊 ) = ( 𝑥 ∈ ( ( 𝑉 × 𝐷 ) × 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 ) } ) ) ) ) ) ) |
128 |
17 127
|
sylan9eq |
⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐻 ) → 𝐼 = ( 𝑥 ∈ ( ( 𝑉 × 𝐷 ) × 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 ) } ) ) ) ) ) ) |
129 |
14 128
|
syl |
⊢ ( 𝜑 → 𝐼 = ( 𝑥 ∈ ( ( 𝑉 × 𝐷 ) × 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 ) } ) ) ) ) ) ) |