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 ) → ⨣ = + ) |