Description: The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014)
Ref | Expression | ||
---|---|---|---|
Hypotheses | psrplusg.s | ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) | |
psrplusg.b | ⊢ 𝐵 = ( Base ‘ 𝑆 ) | ||
psrplusg.a | ⊢ + = ( +g ‘ 𝑅 ) | ||
psrplusg.p | ⊢ ✚ = ( +g ‘ 𝑆 ) | ||
psradd.x | ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) | ||
psradd.y | ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) | ||
Assertion | psradd | ⊢ ( 𝜑 → ( 𝑋 ✚ 𝑌 ) = ( 𝑋 ∘f + 𝑌 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrplusg.s | ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) | |
2 | psrplusg.b | ⊢ 𝐵 = ( Base ‘ 𝑆 ) | |
3 | psrplusg.a | ⊢ + = ( +g ‘ 𝑅 ) | |
4 | psrplusg.p | ⊢ ✚ = ( +g ‘ 𝑆 ) | |
5 | psradd.x | ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) | |
6 | psradd.y | ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) | |
7 | 1 2 3 4 | psrplusg | ⊢ ✚ = ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) |
8 | 7 | oveqi | ⊢ ( 𝑋 ✚ 𝑌 ) = ( 𝑋 ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) 𝑌 ) |
9 | 5 6 | ofmresval | ⊢ ( 𝜑 → ( 𝑋 ( ∘f + ↾ ( 𝐵 × 𝐵 ) ) 𝑌 ) = ( 𝑋 ∘f + 𝑌 ) ) |
10 | 8 9 | syl5eq | ⊢ ( 𝜑 → ( 𝑋 ✚ 𝑌 ) = ( 𝑋 ∘f + 𝑌 ) ) |