Step |
Hyp |
Ref |
Expression |
1 |
|
dia1dim2.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
dia1dim2.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
dia1dim2.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
dva1dim2.u |
⊢ 𝑈 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
dia1dim2.i |
⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 ) |
6 |
|
dva1dim2.n |
⊢ 𝑁 = ( LSpan ‘ 𝑈 ) |
7 |
|
eqid |
⊢ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
eqid |
⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 ) |
9 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) ) |
10 |
1 7 4 8 9
|
dvabase |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
11 |
10
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) |
12 |
11
|
rexeqdv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ∃ 𝑠 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) 𝑔 = ( 𝑠 ( ·𝑠 ‘ 𝑈 ) 𝐹 ) ↔ ∃ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) 𝑔 = ( 𝑠 ( ·𝑠 ‘ 𝑈 ) 𝐹 ) ) ) |
13 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑈 ) = ( ·𝑠 ‘ 𝑈 ) |
14 |
1 2 7 4 13
|
dvavsca |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ) → ( 𝑠 ( ·𝑠 ‘ 𝑈 ) 𝐹 ) = ( 𝑠 ‘ 𝐹 ) ) |
15 |
14
|
anass1rs |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) → ( 𝑠 ( ·𝑠 ‘ 𝑈 ) 𝐹 ) = ( 𝑠 ‘ 𝐹 ) ) |
16 |
15
|
eqeq2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) ∧ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) ) → ( 𝑔 = ( 𝑠 ( ·𝑠 ‘ 𝑈 ) 𝐹 ) ↔ 𝑔 = ( 𝑠 ‘ 𝐹 ) ) ) |
17 |
16
|
rexbidva |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ∃ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) 𝑔 = ( 𝑠 ( ·𝑠 ‘ 𝑈 ) 𝐹 ) ↔ ∃ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) 𝑔 = ( 𝑠 ‘ 𝐹 ) ) ) |
18 |
12 17
|
bitrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( ∃ 𝑠 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) 𝑔 = ( 𝑠 ( ·𝑠 ‘ 𝑈 ) 𝐹 ) ↔ ∃ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) 𝑔 = ( 𝑠 ‘ 𝐹 ) ) ) |
19 |
18
|
abbidv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → { 𝑔 ∣ ∃ 𝑠 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) 𝑔 = ( 𝑠 ( ·𝑠 ‘ 𝑈 ) 𝐹 ) } = { 𝑔 ∣ ∃ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) 𝑔 = ( 𝑠 ‘ 𝐹 ) } ) |
20 |
1 4
|
dvalvec |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑈 ∈ LVec ) |
21 |
20
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝑈 ∈ LVec ) |
22 |
|
lveclmod |
⊢ ( 𝑈 ∈ LVec → 𝑈 ∈ LMod ) |
23 |
21 22
|
syl |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝑈 ∈ LMod ) |
24 |
|
simpr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ 𝑇 ) |
25 |
|
eqid |
⊢ ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 ) |
26 |
1 2 4 25
|
dvavbase |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( Base ‘ 𝑈 ) = 𝑇 ) |
27 |
26
|
adantr |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( Base ‘ 𝑈 ) = 𝑇 ) |
28 |
24 27
|
eleqtrrd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ ( Base ‘ 𝑈 ) ) |
29 |
8 9 25 13 6
|
lspsn |
⊢ ( ( 𝑈 ∈ LMod ∧ 𝐹 ∈ ( Base ‘ 𝑈 ) ) → ( 𝑁 ‘ { 𝐹 } ) = { 𝑔 ∣ ∃ 𝑠 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) 𝑔 = ( 𝑠 ( ·𝑠 ‘ 𝑈 ) 𝐹 ) } ) |
30 |
23 28 29
|
syl2anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑁 ‘ { 𝐹 } ) = { 𝑔 ∣ ∃ 𝑠 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) 𝑔 = ( 𝑠 ( ·𝑠 ‘ 𝑈 ) 𝐹 ) } ) |
31 |
1 2 3 7 5
|
dia1dim |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = { 𝑔 ∣ ∃ 𝑠 ∈ ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 ) 𝑔 = ( 𝑠 ‘ 𝐹 ) } ) |
32 |
19 30 31
|
3eqtr4rd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐼 ‘ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑁 ‘ { 𝐹 } ) ) |