Step |
Hyp |
Ref |
Expression |
1 |
|
mapdpg.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
mapdpg.m |
⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
mapdpg.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
mapdpg.v |
⊢ 𝑉 = ( Base ‘ 𝑈 ) |
5 |
|
mapdpg.s |
⊢ − = ( -g ‘ 𝑈 ) |
6 |
|
mapdpg.z |
⊢ 0 = ( 0g ‘ 𝑈 ) |
7 |
|
mapdpg.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
8 |
|
mapdpg.c |
⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
mapdpg.f |
⊢ 𝐹 = ( Base ‘ 𝐶 ) |
10 |
|
mapdpg.r |
⊢ 𝑅 = ( -g ‘ 𝐶 ) |
11 |
|
mapdpg.j |
⊢ 𝐽 = ( LSpan ‘ 𝐶 ) |
12 |
|
mapdpg.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
13 |
|
mapdpg.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) ) |
14 |
|
mapdpg.y |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ) |
15 |
|
mapdpg.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐹 ) |
16 |
|
mapdpg.ne |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) |
17 |
|
mapdpg.e |
⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ) |
18 |
13
|
eldifad |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
19 |
14
|
eldifad |
⊢ ( 𝜑 → 𝑌 ∈ 𝑉 ) |
20 |
|
eqid |
⊢ ( LSSum ‘ 𝐶 ) = ( LSSum ‘ 𝐶 ) |
21 |
1 2 3 4 5 7 8 12 18 19 20 11
|
mapdpglem2 |
⊢ ( 𝜑 → ∃ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) |
22 |
12
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
23 |
18
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) → 𝑋 ∈ 𝑉 ) |
24 |
19
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) → 𝑌 ∈ 𝑉 ) |
25 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) → 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) |
26 |
|
eqid |
⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 ) |
27 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) ) |
28 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝐶 ) = ( ·𝑠 ‘ 𝐶 ) |
29 |
15
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) → 𝐺 ∈ 𝐹 ) |
30 |
17
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ) |
31 |
1 2 3 4 5 7 8 22 23 24 20 11 9 25 26 27 28 10 29 30
|
mapdpglem3 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) → ∃ 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) ) |
32 |
22
|
3ad2ant1 |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ∧ ( 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
33 |
23
|
3ad2ant1 |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ∧ ( 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) ) → 𝑋 ∈ 𝑉 ) |
34 |
24
|
3ad2ant1 |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ∧ ( 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) ) → 𝑌 ∈ 𝑉 ) |
35 |
|
simp12 |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ∧ ( 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) ) → 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ) |
36 |
29
|
3ad2ant1 |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ∧ ( 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) ) → 𝐺 ∈ 𝐹 ) |
37 |
30
|
3ad2ant1 |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ∧ ( 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ) |
38 |
16
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) |
39 |
38
|
3ad2ant1 |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ∧ ( 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) |
40 |
|
simp13 |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ∧ ( 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) |
41 |
|
eqid |
⊢ ( 0g ‘ ( Scalar ‘ 𝑈 ) ) = ( 0g ‘ ( Scalar ‘ 𝑈 ) ) |
42 |
|
simp2l |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ∧ ( 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) ) → 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ) |
43 |
|
simp2r |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ∧ ( 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) ) → 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) |
44 |
|
simp3 |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ∧ ( 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) ) → 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) ) |
45 |
|
eldifsni |
⊢ ( 𝑋 ∈ ( 𝑉 ∖ { 0 } ) → 𝑋 ≠ 0 ) |
46 |
13 45
|
syl |
⊢ ( 𝜑 → 𝑋 ≠ 0 ) |
47 |
46
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) → 𝑋 ≠ 0 ) |
48 |
47
|
3ad2ant1 |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ∧ ( 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) ) → 𝑋 ≠ 0 ) |
49 |
|
eldifsni |
⊢ ( 𝑌 ∈ ( 𝑉 ∖ { 0 } ) → 𝑌 ≠ 0 ) |
50 |
14 49
|
syl |
⊢ ( 𝜑 → 𝑌 ≠ 0 ) |
51 |
50
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) → 𝑌 ≠ 0 ) |
52 |
51
|
3ad2ant1 |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ∧ ( 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) ) → 𝑌 ≠ 0 ) |
53 |
|
eqid |
⊢ ( ( ( invr ‘ ( Scalar ‘ 𝑈 ) ) ‘ 𝑔 ) ( ·𝑠 ‘ 𝐶 ) 𝑧 ) = ( ( ( invr ‘ ( Scalar ‘ 𝑈 ) ) ‘ 𝑔 ) ( ·𝑠 ‘ 𝐶 ) 𝑧 ) |
54 |
1 2 3 4 5 7 8 32 33 34 20 11 9 35 26 27 28 10 36 37 6 39 40 41 42 43 44 48 52 53
|
mapdpglem23 |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) ∧ ( 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) ) → ∃ ℎ ∈ 𝐹 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ) |
55 |
54
|
3exp |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) → ( ( 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∧ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) → ( 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) → ∃ ℎ ∈ 𝐹 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ) ) ) |
56 |
55
|
rexlimdvv |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) → ( ∃ 𝑔 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∃ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) 𝑡 = ( ( 𝑔 ( ·𝑠 ‘ 𝐶 ) 𝐺 ) 𝑅 𝑧 ) → ∃ ℎ ∈ 𝐹 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ) ) |
57 |
31 56
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) ) → ∃ ℎ ∈ 𝐹 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ) |
58 |
57
|
rexlimdv3a |
⊢ ( 𝜑 → ( ∃ 𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) → ∃ ℎ ∈ 𝐹 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ) ) |
59 |
21 58
|
mpd |
⊢ ( 𝜑 → ∃ ℎ ∈ 𝐹 ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ) |