| Step |
Hyp |
Ref |
Expression |
| 1 |
|
mhphf4.q |
⊢ 𝑄 = ( 𝐼 eval 𝑆 ) |
| 2 |
|
mhphf4.h |
⊢ 𝐻 = ( 𝐼 mHomP 𝑆 ) |
| 3 |
|
mhphf4.k |
⊢ 𝐾 = ( Base ‘ 𝑆 ) |
| 4 |
|
mhphf4.f |
⊢ 𝐹 = ( 𝑆 freeLMod 𝐼 ) |
| 5 |
|
mhphf4.m |
⊢ 𝑀 = ( Base ‘ 𝐹 ) |
| 6 |
|
mhphf4.b |
⊢ ∙ = ( ·𝑠 ‘ 𝐹 ) |
| 7 |
|
mhphf4.x |
⊢ · = ( .r ‘ 𝑆 ) |
| 8 |
|
mhphf4.e |
⊢ ↑ = ( .g ‘ ( mulGrp ‘ 𝑆 ) ) |
| 9 |
|
mhphf4.s |
⊢ ( 𝜑 → 𝑆 ∈ CRing ) |
| 10 |
|
mhphf4.l |
⊢ ( 𝜑 → 𝐿 ∈ 𝐾 ) |
| 11 |
|
mhphf4.p |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝐻 ‘ 𝑁 ) ) |
| 12 |
|
mhphf4.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑀 ) |
| 13 |
1 3
|
evlval |
⊢ 𝑄 = ( ( 𝐼 evalSub 𝑆 ) ‘ 𝐾 ) |
| 14 |
|
eqid |
⊢ ( 𝐼 mHomP ( 𝑆 ↾s 𝐾 ) ) = ( 𝐼 mHomP ( 𝑆 ↾s 𝐾 ) ) |
| 15 |
|
eqid |
⊢ ( 𝑆 ↾s 𝐾 ) = ( 𝑆 ↾s 𝐾 ) |
| 16 |
9
|
crngringd |
⊢ ( 𝜑 → 𝑆 ∈ Ring ) |
| 17 |
3
|
subrgid |
⊢ ( 𝑆 ∈ Ring → 𝐾 ∈ ( SubRing ‘ 𝑆 ) ) |
| 18 |
16 17
|
syl |
⊢ ( 𝜑 → 𝐾 ∈ ( SubRing ‘ 𝑆 ) ) |
| 19 |
3
|
ressid |
⊢ ( 𝑆 ∈ CRing → ( 𝑆 ↾s 𝐾 ) = 𝑆 ) |
| 20 |
9 19
|
syl |
⊢ ( 𝜑 → ( 𝑆 ↾s 𝐾 ) = 𝑆 ) |
| 21 |
20
|
eqcomd |
⊢ ( 𝜑 → 𝑆 = ( 𝑆 ↾s 𝐾 ) ) |
| 22 |
21
|
oveq2d |
⊢ ( 𝜑 → ( 𝐼 mHomP 𝑆 ) = ( 𝐼 mHomP ( 𝑆 ↾s 𝐾 ) ) ) |
| 23 |
2 22
|
eqtrid |
⊢ ( 𝜑 → 𝐻 = ( 𝐼 mHomP ( 𝑆 ↾s 𝐾 ) ) ) |
| 24 |
23
|
fveq1d |
⊢ ( 𝜑 → ( 𝐻 ‘ 𝑁 ) = ( ( 𝐼 mHomP ( 𝑆 ↾s 𝐾 ) ) ‘ 𝑁 ) ) |
| 25 |
11 24
|
eleqtrd |
⊢ ( 𝜑 → 𝑋 ∈ ( ( 𝐼 mHomP ( 𝑆 ↾s 𝐾 ) ) ‘ 𝑁 ) ) |
| 26 |
13 14 15 3 4 5 6 7 8 9 18 10 25 12
|
mhphf3 |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 𝑋 ) ‘ ( 𝐿 ∙ 𝐴 ) ) = ( ( 𝑁 ↑ 𝐿 ) · ( ( 𝑄 ‘ 𝑋 ) ‘ 𝐴 ) ) ) |