Metamath Proof Explorer


Theorem prodf1f

Description: A one-valued infinite product is equal to the constant one function. (Contributed by Scott Fenton, 5-Dec-2017)

Ref Expression
Hypothesis prodf1.1 𝑍 = ( ℤ𝑀 )
Assertion prodf1f ( 𝑀 ∈ ℤ → seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) = ( 𝑍 × { 1 } ) )

Proof

Step Hyp Ref Expression
1 prodf1.1 𝑍 = ( ℤ𝑀 )
2 1 prodf1 ( 𝑘𝑍 → ( seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) ‘ 𝑘 ) = 1 )
3 1ex 1 ∈ V
4 3 fvconst2 ( 𝑘𝑍 → ( ( 𝑍 × { 1 } ) ‘ 𝑘 ) = 1 )
5 2 4 eqtr4d ( 𝑘𝑍 → ( seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) ‘ 𝑘 ) = ( ( 𝑍 × { 1 } ) ‘ 𝑘 ) )
6 5 rgen 𝑘𝑍 ( seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) ‘ 𝑘 ) = ( ( 𝑍 × { 1 } ) ‘ 𝑘 )
7 seqfn ( 𝑀 ∈ ℤ → seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) Fn ( ℤ𝑀 ) )
8 1 fneq2i ( seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) Fn 𝑍 ↔ seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) Fn ( ℤ𝑀 ) )
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 } ) ↔ ∀ 𝑘𝑍 ( seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) ‘ 𝑘 ) = ( ( 𝑍 × { 1 } ) ‘ 𝑘 ) ) )
14 9 12 13 sylancl ( 𝑀 ∈ ℤ → ( seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) = ( 𝑍 × { 1 } ) ↔ ∀ 𝑘𝑍 ( seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) ‘ 𝑘 ) = ( ( 𝑍 × { 1 } ) ‘ 𝑘 ) ) )
15 6 14 mpbiri ( 𝑀 ∈ ℤ → seq 𝑀 ( · , ( 𝑍 × { 1 } ) ) = ( 𝑍 × { 1 } ) )