Step |
Hyp |
Ref |
Expression |
1 |
|
prodf1.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
1
|
prodf1 |
⊢ ( 𝑘 ∈ 𝑍 → ( seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) ‘ 𝑘 ) = 1 ) |
3 |
|
1ex |
⊢ 1 ∈ V |
4 |
3
|
fvconst2 |
⊢ ( 𝑘 ∈ 𝑍 → ( ( 𝑍 × { 1 } ) ‘ 𝑘 ) = 1 ) |
5 |
2 4
|
eqtr4d |
⊢ ( 𝑘 ∈ 𝑍 → ( seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) ‘ 𝑘 ) = ( ( 𝑍 × { 1 } ) ‘ 𝑘 ) ) |
6 |
5
|
rgen |
⊢ ∀ 𝑘 ∈ 𝑍 ( seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) ‘ 𝑘 ) = ( ( 𝑍 × { 1 } ) ‘ 𝑘 ) |
7 |
|
seqfn |
⊢ ( 𝑀 ∈ ℤ → seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) Fn ( ℤ≥ ‘ 𝑀 ) ) |
8 |
1
|
fneq2i |
⊢ ( seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) Fn 𝑍 ↔ seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) Fn ( ℤ≥ ‘ 𝑀 ) ) |
9 |
7 8
|
sylibr |
⊢ ( 𝑀 ∈ ℤ → seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) Fn 𝑍 ) |
10 |
3
|
fconst |
⊢ ( 𝑍 × { 1 } ) : 𝑍 ⟶ { 1 } |
11 |
|
ffn |
⊢ ( ( 𝑍 × { 1 } ) : 𝑍 ⟶ { 1 } → ( 𝑍 × { 1 } ) Fn 𝑍 ) |
12 |
10 11
|
ax-mp |
⊢ ( 𝑍 × { 1 } ) Fn 𝑍 |
13 |
|
eqfnfv |
⊢ ( ( seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) Fn 𝑍 ∧ ( 𝑍 × { 1 } ) Fn 𝑍 ) → ( seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) = ( 𝑍 × { 1 } ) ↔ ∀ 𝑘 ∈ 𝑍 ( seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) ‘ 𝑘 ) = ( ( 𝑍 × { 1 } ) ‘ 𝑘 ) ) ) |
14 |
9 12 13
|
sylancl |
⊢ ( 𝑀 ∈ ℤ → ( seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) = ( 𝑍 × { 1 } ) ↔ ∀ 𝑘 ∈ 𝑍 ( seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) ‘ 𝑘 ) = ( ( 𝑍 × { 1 } ) ‘ 𝑘 ) ) ) |
15 |
6 14
|
mpbiri |
⊢ ( 𝑀 ∈ ℤ → seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) = ( 𝑍 × { 1 } ) ) |