Metamath Proof Explorer


Theorem opsrmulr

Description: The multiplication operation of the ordered power series structure. (Contributed by Mario Carneiro, 8-Feb-2015) (Revised by Mario Carneiro, 30-Aug-2015) (Revised by AV, 1-Nov-2024)

Ref Expression
Hypotheses opsrbas.s 𝑆 = ( 𝐼 mPwSer 𝑅 )
opsrbas.o 𝑂 = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 )
opsrbas.t ( 𝜑𝑇 ⊆ ( 𝐼 × 𝐼 ) )
Assertion opsrmulr ( 𝜑 → ( .r𝑆 ) = ( .r𝑂 ) )

Proof

Step Hyp Ref Expression
1 opsrbas.s 𝑆 = ( 𝐼 mPwSer 𝑅 )
2 opsrbas.o 𝑂 = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 )
3 opsrbas.t ( 𝜑𝑇 ⊆ ( 𝐼 × 𝐼 ) )
4 mulrid .r = Slot ( .r ‘ ndx )
5 plendxnmulrndx ( le ‘ ndx ) ≠ ( .r ‘ ndx )
6 5 necomi ( .r ‘ ndx ) ≠ ( le ‘ ndx )
7 1 2 3 4 6 opsrbaslem ( 𝜑 → ( .r𝑆 ) = ( .r𝑂 ) )