| Step | Hyp | Ref
| Expression |
| 1 | | iprodefisumlem.1 |
. . . 4
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 2 | | iprodefisumlem.2 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 3 | | iprodefisumlem.3 |
. . . . . 6
⊢ (𝜑 → 𝐹:𝑍⟶ℂ) |
| 4 | | fvco3 7008 |
. . . . . 6
⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑘 ∈ 𝑍) → ((exp ∘ 𝐹)‘𝑘) = (exp‘(𝐹‘𝑘))) |
| 5 | 3, 4 | sylan 580 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ 𝐹)‘𝑘) = (exp‘(𝐹‘𝑘))) |
| 6 | 3 | ffvelcdmda 7104 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
| 7 | | efcl 16118 |
. . . . . 6
⊢ ((𝐹‘𝑘) ∈ ℂ → (exp‘(𝐹‘𝑘)) ∈ ℂ) |
| 8 | 6, 7 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (exp‘(𝐹‘𝑘)) ∈ ℂ) |
| 9 | 5, 8 | eqeltrd 2841 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ 𝐹)‘𝑘) ∈ ℂ) |
| 10 | 1, 2, 9 | prodf 15923 |
. . 3
⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)):𝑍⟶ℂ) |
| 11 | 10 | ffnd 6737 |
. 2
⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) Fn 𝑍) |
| 12 | | eff 16117 |
. . . 4
⊢
exp:ℂ⟶ℂ |
| 13 | | ffn 6736 |
. . . 4
⊢
(exp:ℂ⟶ℂ → exp Fn ℂ) |
| 14 | 12, 13 | ax-mp 5 |
. . 3
⊢ exp Fn
ℂ |
| 15 | 1, 2, 6 | serf 14071 |
. . 3
⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ) |
| 16 | | fnfco 6773 |
. . 3
⊢ ((exp Fn
ℂ ∧ seq𝑀( + ,
𝐹):𝑍⟶ℂ) → (exp ∘
seq𝑀( + , 𝐹)) Fn 𝑍) |
| 17 | 14, 15, 16 | sylancr 587 |
. 2
⊢ (𝜑 → (exp ∘ seq𝑀( + , 𝐹)) Fn 𝑍) |
| 18 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑗 = 𝑀 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑀)) |
| 19 | | 2fveq3 6911 |
. . . . . . . 8
⊢ (𝑗 = 𝑀 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑀))) |
| 20 | 18, 19 | eqeq12d 2753 |
. . . . . . 7
⊢ (𝑗 = 𝑀 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))) |
| 21 | 20 | imbi2d 340 |
. . . . . 6
⊢ (𝑗 = 𝑀 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀))))) |
| 22 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑗 = 𝑛 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑛)) |
| 23 | | 2fveq3 6911 |
. . . . . . . 8
⊢ (𝑗 = 𝑛 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) |
| 24 | 22, 23 | eqeq12d 2753 |
. . . . . . 7
⊢ (𝑗 = 𝑛 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)))) |
| 25 | 24 | imbi2d 340 |
. . . . . 6
⊢ (𝑗 = 𝑛 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))))) |
| 26 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑗 = (𝑛 + 1) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1))) |
| 27 | | 2fveq3 6911 |
. . . . . . . 8
⊢ (𝑗 = (𝑛 + 1) → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1)))) |
| 28 | 26, 27 | eqeq12d 2753 |
. . . . . . 7
⊢ (𝑗 = (𝑛 + 1) → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))))) |
| 29 | 28 | imbi2d 340 |
. . . . . 6
⊢ (𝑗 = (𝑛 + 1) → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1)))))) |
| 30 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑗 = 𝑘 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (seq𝑀( · , (exp ∘ 𝐹))‘𝑘)) |
| 31 | | 2fveq3 6911 |
. . . . . . . 8
⊢ (𝑗 = 𝑘 → (exp‘(seq𝑀( + , 𝐹)‘𝑗)) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))) |
| 32 | 30, 31 | eqeq12d 2753 |
. . . . . . 7
⊢ (𝑗 = 𝑘 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗)) ↔ (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))) |
| 33 | 32 | imbi2d 340 |
. . . . . 6
⊢ (𝑗 = 𝑘 → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑗) = (exp‘(seq𝑀( + , 𝐹)‘𝑗))) ↔ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))))) |
| 34 | | uzid 12893 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
(ℤ≥‘𝑀)) |
| 35 | 2, 34 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
| 36 | 35, 1 | eleqtrrdi 2852 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ 𝑍) |
| 37 | | fvco3 7008 |
. . . . . . . . 9
⊢ ((𝐹:𝑍⟶ℂ ∧ 𝑀 ∈ 𝑍) → ((exp ∘ 𝐹)‘𝑀) = (exp‘(𝐹‘𝑀))) |
| 38 | 3, 36, 37 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → ((exp ∘ 𝐹)‘𝑀) = (exp‘(𝐹‘𝑀))) |
| 39 | | seq1 14055 |
. . . . . . . . 9
⊢ (𝑀 ∈ ℤ → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = ((exp ∘ 𝐹)‘𝑀)) |
| 40 | 2, 39 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = ((exp ∘ 𝐹)‘𝑀)) |
| 41 | | seq1 14055 |
. . . . . . . . . 10
⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) |
| 42 | 2, 41 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) |
| 43 | 42 | fveq2d 6910 |
. . . . . . . 8
⊢ (𝜑 → (exp‘(seq𝑀( + , 𝐹)‘𝑀)) = (exp‘(𝐹‘𝑀))) |
| 44 | 38, 40, 43 | 3eqtr4d 2787 |
. . . . . . 7
⊢ (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀))) |
| 45 | 44 | a1i 11 |
. . . . . 6
⊢ (𝑀 ∈ ℤ → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑀) = (exp‘(seq𝑀( + , 𝐹)‘𝑀)))) |
| 46 | | oveq1 7438 |
. . . . . . . . . . 11
⊢
((seq𝑀( · ,
(exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)) → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1)))) |
| 47 | 46 | 3ad2ant3 1136 |
. . . . . . . . . 10
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1)))) |
| 48 | 3 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ 𝜑) → 𝐹:𝑍⟶ℂ) |
| 49 | | peano2uz 12943 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → (𝑛 + 1) ∈
(ℤ≥‘𝑀)) |
| 50 | 49, 1 | eleqtrrdi 2852 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → (𝑛 + 1) ∈ 𝑍) |
| 51 | 50 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ 𝜑) → (𝑛 + 1) ∈ 𝑍) |
| 52 | | fvco3 7008 |
. . . . . . . . . . . . . 14
⊢ ((𝐹:𝑍⟶ℂ ∧ (𝑛 + 1) ∈ 𝑍) → ((exp ∘ 𝐹)‘(𝑛 + 1)) = (exp‘(𝐹‘(𝑛 + 1)))) |
| 53 | 48, 51, 52 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ 𝜑) → ((exp ∘ 𝐹)‘(𝑛 + 1)) = (exp‘(𝐹‘(𝑛 + 1)))) |
| 54 | 53 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1))))) |
| 55 | 15 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ) |
| 56 | 55 | expcom 413 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ 𝑍 → (𝜑 → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)) |
| 57 | 1 | eqcomi 2746 |
. . . . . . . . . . . . . . 15
⊢
(ℤ≥‘𝑀) = 𝑍 |
| 58 | 56, 57 | eleq2s 2859 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)) |
| 59 | 58 | imp 406 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ 𝜑) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ) |
| 60 | 48, 51 | ffvelcdmd 7105 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ 𝜑) → (𝐹‘(𝑛 + 1)) ∈ ℂ) |
| 61 | | efadd 16130 |
. . . . . . . . . . . . 13
⊢
(((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (𝐹‘(𝑛 + 1)) ∈ ℂ) →
(exp‘((seq𝑀( + ,
𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1))))) |
| 62 | 59, 60, 61 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ 𝜑) → (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) = ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · (exp‘(𝐹‘(𝑛 + 1))))) |
| 63 | 54, 62 | eqtr4d 2780 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ 𝜑) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))) |
| 64 | 63 | 3adant3 1133 |
. . . . . . . . . 10
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → ((exp‘(seq𝑀( + , 𝐹)‘𝑛)) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))) |
| 65 | 47, 64 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))) |
| 66 | | seqp1 14057 |
. . . . . . . . . . 11
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1)))) |
| 67 | 66 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ 𝜑) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1)))) |
| 68 | 67 | 3adant3 1133 |
. . . . . . . . 9
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) · ((exp ∘ 𝐹)‘(𝑛 + 1)))) |
| 69 | | seqp1 14057 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) |
| 70 | 69 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ 𝜑) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))) |
| 71 | 70 | fveq2d 6910 |
. . . . . . . . . 10
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ 𝜑) → (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))) |
| 72 | 71 | 3adant3 1133 |
. . . . . . . . 9
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1))) = (exp‘((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))) |
| 73 | 65, 68, 72 | 3eqtr4d 2787 |
. . . . . . . 8
⊢ ((𝑛 ∈
(ℤ≥‘𝑀) ∧ 𝜑 ∧ (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1)))) |
| 74 | 73 | 3exp 1120 |
. . . . . . 7
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → (𝜑 → ((seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛)) → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1)))))) |
| 75 | 74 | a2d 29 |
. . . . . 6
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → ((𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑛) = (exp‘(seq𝑀( + , 𝐹)‘𝑛))) → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘(𝑛 + 1)) = (exp‘(seq𝑀( + , 𝐹)‘(𝑛 + 1)))))) |
| 76 | 21, 25, 29, 33, 45, 75 | uzind4 12948 |
. . . . 5
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))) |
| 77 | 76, 1 | eleq2s 2859 |
. . . 4
⊢ (𝑘 ∈ 𝑍 → (𝜑 → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘)))) |
| 78 | 77 | impcom 407 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))) |
| 79 | | fvco3 7008 |
. . . 4
⊢
((seq𝑀( + , 𝐹):𝑍⟶ℂ ∧ 𝑘 ∈ 𝑍) → ((exp ∘ seq𝑀( + , 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))) |
| 80 | 15, 79 | sylan 580 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((exp ∘ seq𝑀( + , 𝐹))‘𝑘) = (exp‘(seq𝑀( + , 𝐹)‘𝑘))) |
| 81 | 78, 80 | eqtr4d 2780 |
. 2
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (seq𝑀( · , (exp ∘ 𝐹))‘𝑘) = ((exp ∘ seq𝑀( + , 𝐹))‘𝑘)) |
| 82 | 11, 17, 81 | eqfnfvd 7054 |
1
⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹))) |