Description: If the addition operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015)
Ref | Expression | ||
---|---|---|---|
Hypotheses | plusffval.1 | ⊢ 𝐵 = ( Base ‘ 𝐺 ) | |
plusffval.2 | ⊢ + = ( +g ‘ 𝐺 ) | ||
plusffval.3 | ⊢ ⨣ = ( +𝑓 ‘ 𝐺 ) | ||
Assertion | plusfeq | ⊢ ( + Fn ( 𝐵 × 𝐵 ) → ⨣ = + ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plusffval.1 | ⊢ 𝐵 = ( Base ‘ 𝐺 ) | |
2 | plusffval.2 | ⊢ + = ( +g ‘ 𝐺 ) | |
3 | plusffval.3 | ⊢ ⨣ = ( +𝑓 ‘ 𝐺 ) | |
4 | 1 2 3 | plusffval | ⊢ ⨣ = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + 𝑦 ) ) |
5 | fnov | ⊢ ( + Fn ( 𝐵 × 𝐵 ) ↔ + = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + 𝑦 ) ) ) | |
6 | 5 | biimpi | ⊢ ( + Fn ( 𝐵 × 𝐵 ) → + = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( 𝑥 + 𝑦 ) ) ) |
7 | 4 6 | eqtr4id | ⊢ ( + Fn ( 𝐵 × 𝐵 ) → ⨣ = + ) |