Step |
Hyp |
Ref |
Expression |
1 |
|
efgh.1 |
⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ ( exp ‘ ( 𝐴 · 𝑥 ) ) ) |
2 |
|
simp1l |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → 𝐴 ∈ ℂ ) |
3 |
|
simp1r |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) |
4 |
|
cnfldbas |
⊢ ℂ = ( Base ‘ ℂfld ) |
5 |
4
|
subgss |
⊢ ( 𝑋 ∈ ( SubGrp ‘ ℂfld ) → 𝑋 ⊆ ℂ ) |
6 |
3 5
|
syl |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → 𝑋 ⊆ ℂ ) |
7 |
|
simp2 |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → 𝐵 ∈ 𝑋 ) |
8 |
6 7
|
sseldd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → 𝐵 ∈ ℂ ) |
9 |
|
simp3 |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → 𝐶 ∈ 𝑋 ) |
10 |
6 9
|
sseldd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → 𝐶 ∈ ℂ ) |
11 |
2 8 10
|
adddid |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐴 · ( 𝐵 + 𝐶 ) ) = ( ( 𝐴 · 𝐵 ) + ( 𝐴 · 𝐶 ) ) ) |
12 |
11
|
fveq2d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( exp ‘ ( 𝐴 · ( 𝐵 + 𝐶 ) ) ) = ( exp ‘ ( ( 𝐴 · 𝐵 ) + ( 𝐴 · 𝐶 ) ) ) ) |
13 |
2 8
|
mulcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐴 · 𝐵 ) ∈ ℂ ) |
14 |
2 10
|
mulcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐴 · 𝐶 ) ∈ ℂ ) |
15 |
|
efadd |
⊢ ( ( ( 𝐴 · 𝐵 ) ∈ ℂ ∧ ( 𝐴 · 𝐶 ) ∈ ℂ ) → ( exp ‘ ( ( 𝐴 · 𝐵 ) + ( 𝐴 · 𝐶 ) ) ) = ( ( exp ‘ ( 𝐴 · 𝐵 ) ) · ( exp ‘ ( 𝐴 · 𝐶 ) ) ) ) |
16 |
13 14 15
|
syl2anc |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( exp ‘ ( ( 𝐴 · 𝐵 ) + ( 𝐴 · 𝐶 ) ) ) = ( ( exp ‘ ( 𝐴 · 𝐵 ) ) · ( exp ‘ ( 𝐴 · 𝐶 ) ) ) ) |
17 |
12 16
|
eqtrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( exp ‘ ( 𝐴 · ( 𝐵 + 𝐶 ) ) ) = ( ( exp ‘ ( 𝐴 · 𝐵 ) ) · ( exp ‘ ( 𝐴 · 𝐶 ) ) ) ) |
18 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐴 · 𝑥 ) = ( 𝐴 · 𝑦 ) ) |
19 |
18
|
fveq2d |
⊢ ( 𝑥 = 𝑦 → ( exp ‘ ( 𝐴 · 𝑥 ) ) = ( exp ‘ ( 𝐴 · 𝑦 ) ) ) |
20 |
19
|
cbvmptv |
⊢ ( 𝑥 ∈ 𝑋 ↦ ( exp ‘ ( 𝐴 · 𝑥 ) ) ) = ( 𝑦 ∈ 𝑋 ↦ ( exp ‘ ( 𝐴 · 𝑦 ) ) ) |
21 |
1 20
|
eqtri |
⊢ 𝐹 = ( 𝑦 ∈ 𝑋 ↦ ( exp ‘ ( 𝐴 · 𝑦 ) ) ) |
22 |
|
oveq2 |
⊢ ( 𝑦 = ( 𝐵 + 𝐶 ) → ( 𝐴 · 𝑦 ) = ( 𝐴 · ( 𝐵 + 𝐶 ) ) ) |
23 |
22
|
fveq2d |
⊢ ( 𝑦 = ( 𝐵 + 𝐶 ) → ( exp ‘ ( 𝐴 · 𝑦 ) ) = ( exp ‘ ( 𝐴 · ( 𝐵 + 𝐶 ) ) ) ) |
24 |
|
cnfldadd |
⊢ + = ( +g ‘ ℂfld ) |
25 |
24
|
subgcl |
⊢ ( ( 𝑋 ∈ ( SubGrp ‘ ℂfld ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐵 + 𝐶 ) ∈ 𝑋 ) |
26 |
25
|
3adant1l |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐵 + 𝐶 ) ∈ 𝑋 ) |
27 |
|
fvexd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( exp ‘ ( 𝐴 · ( 𝐵 + 𝐶 ) ) ) ∈ V ) |
28 |
21 23 26 27
|
fvmptd3 |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝐵 + 𝐶 ) ) = ( exp ‘ ( 𝐴 · ( 𝐵 + 𝐶 ) ) ) ) |
29 |
|
oveq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝐴 · 𝑦 ) = ( 𝐴 · 𝐵 ) ) |
30 |
29
|
fveq2d |
⊢ ( 𝑦 = 𝐵 → ( exp ‘ ( 𝐴 · 𝑦 ) ) = ( exp ‘ ( 𝐴 · 𝐵 ) ) ) |
31 |
|
fvexd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( exp ‘ ( 𝐴 · 𝐵 ) ) ∈ V ) |
32 |
21 30 7 31
|
fvmptd3 |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐹 ‘ 𝐵 ) = ( exp ‘ ( 𝐴 · 𝐵 ) ) ) |
33 |
|
oveq2 |
⊢ ( 𝑦 = 𝐶 → ( 𝐴 · 𝑦 ) = ( 𝐴 · 𝐶 ) ) |
34 |
33
|
fveq2d |
⊢ ( 𝑦 = 𝐶 → ( exp ‘ ( 𝐴 · 𝑦 ) ) = ( exp ‘ ( 𝐴 · 𝐶 ) ) ) |
35 |
|
fvexd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( exp ‘ ( 𝐴 · 𝐶 ) ) ∈ V ) |
36 |
21 34 9 35
|
fvmptd3 |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐹 ‘ 𝐶 ) = ( exp ‘ ( 𝐴 · 𝐶 ) ) ) |
37 |
32 36
|
oveq12d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( ( 𝐹 ‘ 𝐵 ) · ( 𝐹 ‘ 𝐶 ) ) = ( ( exp ‘ ( 𝐴 · 𝐵 ) ) · ( exp ‘ ( 𝐴 · 𝐶 ) ) ) ) |
38 |
17 28 37
|
3eqtr4d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝑋 ∈ ( SubGrp ‘ ℂfld ) ) ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐹 ‘ ( 𝐵 + 𝐶 ) ) = ( ( 𝐹 ‘ 𝐵 ) · ( 𝐹 ‘ 𝐶 ) ) ) |