Step |
Hyp |
Ref |
Expression |
1 |
|
trnset.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
2 |
|
trnset.s |
⊢ 𝑆 = ( PSubSp ‘ 𝐾 ) |
3 |
|
trnset.p |
⊢ + = ( +𝑃 ‘ 𝐾 ) |
4 |
|
trnset.o |
⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 ) |
5 |
|
trnset.w |
⊢ 𝑊 = ( WAtoms ‘ 𝐾 ) |
6 |
|
trnset.m |
⊢ 𝑀 = ( PAut ‘ 𝐾 ) |
7 |
|
trnset.l |
⊢ 𝐿 = ( Dil ‘ 𝐾 ) |
8 |
|
trnset.t |
⊢ 𝑇 = ( Trn ‘ 𝐾 ) |
9 |
1 2 3 4 5 6 7 8
|
trnsetN |
⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴 ) → ( 𝑇 ‘ 𝐷 ) = { 𝑓 ∈ ( 𝐿 ‘ 𝐷 ) ∣ ∀ 𝑞 ∈ ( 𝑊 ‘ 𝐷 ) ∀ 𝑟 ∈ ( 𝑊 ‘ 𝐷 ) ( ( 𝑞 + ( 𝑓 ‘ 𝑞 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) = ( ( 𝑟 + ( 𝑓 ‘ 𝑟 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) } ) |
10 |
9
|
eleq2d |
⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴 ) → ( 𝐹 ∈ ( 𝑇 ‘ 𝐷 ) ↔ 𝐹 ∈ { 𝑓 ∈ ( 𝐿 ‘ 𝐷 ) ∣ ∀ 𝑞 ∈ ( 𝑊 ‘ 𝐷 ) ∀ 𝑟 ∈ ( 𝑊 ‘ 𝐷 ) ( ( 𝑞 + ( 𝑓 ‘ 𝑞 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) = ( ( 𝑟 + ( 𝑓 ‘ 𝑟 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) } ) ) |
11 |
|
fveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑞 ) = ( 𝐹 ‘ 𝑞 ) ) |
12 |
11
|
oveq2d |
⊢ ( 𝑓 = 𝐹 → ( 𝑞 + ( 𝑓 ‘ 𝑞 ) ) = ( 𝑞 + ( 𝐹 ‘ 𝑞 ) ) ) |
13 |
12
|
ineq1d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑞 + ( 𝑓 ‘ 𝑞 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) = ( ( 𝑞 + ( 𝐹 ‘ 𝑞 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) ) |
14 |
|
fveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑟 ) = ( 𝐹 ‘ 𝑟 ) ) |
15 |
14
|
oveq2d |
⊢ ( 𝑓 = 𝐹 → ( 𝑟 + ( 𝑓 ‘ 𝑟 ) ) = ( 𝑟 + ( 𝐹 ‘ 𝑟 ) ) ) |
16 |
15
|
ineq1d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑟 + ( 𝑓 ‘ 𝑟 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) = ( ( 𝑟 + ( 𝐹 ‘ 𝑟 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) ) |
17 |
13 16
|
eqeq12d |
⊢ ( 𝑓 = 𝐹 → ( ( ( 𝑞 + ( 𝑓 ‘ 𝑞 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) = ( ( 𝑟 + ( 𝑓 ‘ 𝑟 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) ↔ ( ( 𝑞 + ( 𝐹 ‘ 𝑞 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) = ( ( 𝑟 + ( 𝐹 ‘ 𝑟 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) ) ) |
18 |
17
|
2ralbidv |
⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑞 ∈ ( 𝑊 ‘ 𝐷 ) ∀ 𝑟 ∈ ( 𝑊 ‘ 𝐷 ) ( ( 𝑞 + ( 𝑓 ‘ 𝑞 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) = ( ( 𝑟 + ( 𝑓 ‘ 𝑟 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) ↔ ∀ 𝑞 ∈ ( 𝑊 ‘ 𝐷 ) ∀ 𝑟 ∈ ( 𝑊 ‘ 𝐷 ) ( ( 𝑞 + ( 𝐹 ‘ 𝑞 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) = ( ( 𝑟 + ( 𝐹 ‘ 𝑟 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) ) ) |
19 |
18
|
elrab |
⊢ ( 𝐹 ∈ { 𝑓 ∈ ( 𝐿 ‘ 𝐷 ) ∣ ∀ 𝑞 ∈ ( 𝑊 ‘ 𝐷 ) ∀ 𝑟 ∈ ( 𝑊 ‘ 𝐷 ) ( ( 𝑞 + ( 𝑓 ‘ 𝑞 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) = ( ( 𝑟 + ( 𝑓 ‘ 𝑟 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) } ↔ ( 𝐹 ∈ ( 𝐿 ‘ 𝐷 ) ∧ ∀ 𝑞 ∈ ( 𝑊 ‘ 𝐷 ) ∀ 𝑟 ∈ ( 𝑊 ‘ 𝐷 ) ( ( 𝑞 + ( 𝐹 ‘ 𝑞 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) = ( ( 𝑟 + ( 𝐹 ‘ 𝑟 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) ) ) |
20 |
10 19
|
bitrdi |
⊢ ( ( 𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴 ) → ( 𝐹 ∈ ( 𝑇 ‘ 𝐷 ) ↔ ( 𝐹 ∈ ( 𝐿 ‘ 𝐷 ) ∧ ∀ 𝑞 ∈ ( 𝑊 ‘ 𝐷 ) ∀ 𝑟 ∈ ( 𝑊 ‘ 𝐷 ) ( ( 𝑞 + ( 𝐹 ‘ 𝑞 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) = ( ( 𝑟 + ( 𝐹 ‘ 𝑟 ) ) ∩ ( ⊥ ‘ { 𝐷 } ) ) ) ) ) |