Step |
Hyp |
Ref |
Expression |
1 |
|
mapdrn.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
2 |
|
mapdrn.o |
⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 ) |
3 |
|
mapdrn.m |
⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) |
4 |
|
mapdrn.u |
⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
mapdrn.f |
⊢ 𝐹 = ( LFnl ‘ 𝑈 ) |
6 |
|
mapdrn.l |
⊢ 𝐿 = ( LKer ‘ 𝑈 ) |
7 |
|
mapdrn.d |
⊢ 𝐷 = ( LDual ‘ 𝑈 ) |
8 |
|
mapdrn.t |
⊢ 𝑇 = ( LSubSp ‘ 𝐷 ) |
9 |
|
mapdrn.c |
⊢ 𝐶 = { 𝑔 ∈ 𝐹 ∣ ( 𝑂 ‘ ( 𝑂 ‘ ( 𝐿 ‘ 𝑔 ) ) ) = ( 𝐿 ‘ 𝑔 ) } |
10 |
|
mapdrn.k |
⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
11 |
|
eqid |
⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 ) |
12 |
1 2 3 4 11 5 6 7 8 9 10
|
mapd1o |
⊢ ( 𝜑 → 𝑀 : ( LSubSp ‘ 𝑈 ) –1-1-onto→ ( 𝑇 ∩ 𝒫 𝐶 ) ) |
13 |
|
f1ofo |
⊢ ( 𝑀 : ( LSubSp ‘ 𝑈 ) –1-1-onto→ ( 𝑇 ∩ 𝒫 𝐶 ) → 𝑀 : ( LSubSp ‘ 𝑈 ) –onto→ ( 𝑇 ∩ 𝒫 𝐶 ) ) |
14 |
|
forn |
⊢ ( 𝑀 : ( LSubSp ‘ 𝑈 ) –onto→ ( 𝑇 ∩ 𝒫 𝐶 ) → ran 𝑀 = ( 𝑇 ∩ 𝒫 𝐶 ) ) |
15 |
12 13 14
|
3syl |
⊢ ( 𝜑 → ran 𝑀 = ( 𝑇 ∩ 𝒫 𝐶 ) ) |