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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 ⊢ ( 𝜑 → 𝑇 ⊆ ( 𝐼 × 𝐼 ) )
Assertion opsrmulr ( 𝜑 → ( .r ‘ 𝑆 ) = ( .r ‘ 𝑂 ) )

Proof

Step Hyp Ref Expression
1 opsrbas.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2 opsrbas.o ⊢ 𝑂 = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 )
3 opsrbas.t ⊢ ( 𝜑 → 𝑇 ⊆ ( 𝐼 × 𝐼 ) )
4 mulridx ⊢ .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 → ( .r ‘ 𝑆 ) = ( .r ‘ 𝑂 ) )