Metamath Proof Explorer


Theorem psradd

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 + = ( +g𝑅 )
psrplusg.p = ( +g𝑆 )
psradd.x ( 𝜑𝑋𝐵 )
psradd.y ( 𝜑𝑌𝐵 )
Assertion psradd ( 𝜑 → ( 𝑋 𝑌 ) = ( 𝑋f + 𝑌 ) )

Proof

Step Hyp Ref Expression
1 psrplusg.s 𝑆 = ( 𝐼 mPwSer 𝑅 )
2 psrplusg.b 𝐵 = ( Base ‘ 𝑆 )
3 psrplusg.a + = ( +g𝑅 )
4 psrplusg.p = ( +g𝑆 )
5 psradd.x ( 𝜑𝑋𝐵 )
6 psradd.y ( 𝜑𝑌𝐵 )
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 ) = ( 𝑋f + 𝑌 ) )
10 8 9 syl5eq ( 𝜑 → ( 𝑋 𝑌 ) = ( 𝑋f + 𝑌 ) )