| Step |
Hyp |
Ref |
Expression |
| 1 |
|
rrxval.r |
⊢ 𝐻 = ( ℝ^ ‘ 𝐼 ) |
| 2 |
|
rrxbase.b |
⊢ 𝐵 = ( Base ‘ 𝐻 ) |
| 3 |
|
rrxvsca.r |
⊢ ∙ = ( ·𝑠 ‘ 𝐻 ) |
| 4 |
|
rrxvsca.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑉 ) |
| 5 |
|
rrxvsca.j |
⊢ ( 𝜑 → 𝐽 ∈ 𝐼 ) |
| 6 |
|
rrxvsca.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
| 7 |
|
rrxvsca.x |
⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐻 ) ) |
| 8 |
1
|
rrxval |
⊢ ( 𝐼 ∈ 𝑉 → 𝐻 = ( toℂPreHil ‘ ( ℝfld freeLMod 𝐼 ) ) ) |
| 9 |
4 8
|
syl |
⊢ ( 𝜑 → 𝐻 = ( toℂPreHil ‘ ( ℝfld freeLMod 𝐼 ) ) ) |
| 10 |
9
|
fveq2d |
⊢ ( 𝜑 → ( ·𝑠 ‘ 𝐻 ) = ( ·𝑠 ‘ ( toℂPreHil ‘ ( ℝfld freeLMod 𝐼 ) ) ) ) |
| 11 |
3 10
|
syl5eq |
⊢ ( 𝜑 → ∙ = ( ·𝑠 ‘ ( toℂPreHil ‘ ( ℝfld freeLMod 𝐼 ) ) ) ) |
| 12 |
11
|
oveqd |
⊢ ( 𝜑 → ( 𝐴 ∙ 𝑋 ) = ( 𝐴 ( ·𝑠 ‘ ( toℂPreHil ‘ ( ℝfld freeLMod 𝐼 ) ) ) 𝑋 ) ) |
| 13 |
12
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝐴 ∙ 𝑋 ) ‘ 𝐽 ) = ( ( 𝐴 ( ·𝑠 ‘ ( toℂPreHil ‘ ( ℝfld freeLMod 𝐼 ) ) ) 𝑋 ) ‘ 𝐽 ) ) |
| 14 |
|
eqid |
⊢ ( ℝfld freeLMod 𝐼 ) = ( ℝfld freeLMod 𝐼 ) |
| 15 |
|
eqid |
⊢ ( Base ‘ ( ℝfld freeLMod 𝐼 ) ) = ( Base ‘ ( ℝfld freeLMod 𝐼 ) ) |
| 16 |
|
rebase |
⊢ ℝ = ( Base ‘ ℝfld ) |
| 17 |
9
|
fveq2d |
⊢ ( 𝜑 → ( Base ‘ 𝐻 ) = ( Base ‘ ( toℂPreHil ‘ ( ℝfld freeLMod 𝐼 ) ) ) ) |
| 18 |
|
eqid |
⊢ ( toℂPreHil ‘ ( ℝfld freeLMod 𝐼 ) ) = ( toℂPreHil ‘ ( ℝfld freeLMod 𝐼 ) ) |
| 19 |
18 15
|
tcphbas |
⊢ ( Base ‘ ( ℝfld freeLMod 𝐼 ) ) = ( Base ‘ ( toℂPreHil ‘ ( ℝfld freeLMod 𝐼 ) ) ) |
| 20 |
17 19
|
eqtr4di |
⊢ ( 𝜑 → ( Base ‘ 𝐻 ) = ( Base ‘ ( ℝfld freeLMod 𝐼 ) ) ) |
| 21 |
7 20
|
eleqtrd |
⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ ( ℝfld freeLMod 𝐼 ) ) ) |
| 22 |
|
eqid |
⊢ ( ·𝑠 ‘ ( ℝfld freeLMod 𝐼 ) ) = ( ·𝑠 ‘ ( ℝfld freeLMod 𝐼 ) ) |
| 23 |
18 22
|
tcphvsca |
⊢ ( ·𝑠 ‘ ( ℝfld freeLMod 𝐼 ) ) = ( ·𝑠 ‘ ( toℂPreHil ‘ ( ℝfld freeLMod 𝐼 ) ) ) |
| 24 |
23
|
eqcomi |
⊢ ( ·𝑠 ‘ ( toℂPreHil ‘ ( ℝfld freeLMod 𝐼 ) ) ) = ( ·𝑠 ‘ ( ℝfld freeLMod 𝐼 ) ) |
| 25 |
|
remulr |
⊢ · = ( .r ‘ ℝfld ) |
| 26 |
14 15 16 4 6 21 5 24 25
|
frlmvscaval |
⊢ ( 𝜑 → ( ( 𝐴 ( ·𝑠 ‘ ( toℂPreHil ‘ ( ℝfld freeLMod 𝐼 ) ) ) 𝑋 ) ‘ 𝐽 ) = ( 𝐴 · ( 𝑋 ‘ 𝐽 ) ) ) |
| 27 |
13 26
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝐴 ∙ 𝑋 ) ‘ 𝐽 ) = ( 𝐴 · ( 𝑋 ‘ 𝐽 ) ) ) |