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Metamath Proof Explorer

Theorem mplneg

Description: The negative function on multivariate polynomials. (Contributed by SN, 25-May-2024)

Ref Expression
Hypotheses mplneg.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
mplneg.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
mplneg.n ⊢ 𝑁 = ( invg ‘ 𝑅 )
mplneg.m ⊢ 𝑀 = ( invg ‘ 𝑃 )
mplneg.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
mplneg.r ⊢ ( 𝜑 → 𝑅 ∈ Grp )
mplneg.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion mplneg ( 𝜑 → ( 𝑀 ‘ 𝑋 ) = ( 𝑁 ∘ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 mplneg.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
2 mplneg.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
3 mplneg.n ⊢ 𝑁 = ( invg ‘ 𝑅 )
4 mplneg.m ⊢ 𝑀 = ( invg ‘ 𝑃 )
5 mplneg.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
6 mplneg.r ⊢ ( 𝜑 → 𝑅 ∈ Grp )
7 mplneg.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
8 eqid ⊢ ( 𝐼 mPwSer 𝑅 ) = ( 𝐼 mPwSer 𝑅 )
9 8 1 2 5 6 mplsubg ⊢ ( 𝜑 → 𝐵 ∈ ( SubGrp ‘ ( 𝐼 mPwSer 𝑅 ) ) )
10 1 8 2 mplval2 ⊢ 𝑃 = ( ( 𝐼 mPwSer 𝑅 ) ↾s 𝐵 )
11 eqid ⊢ ( invg ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( invg ‘ ( 𝐼 mPwSer 𝑅 ) )
12 10 11 4 subginv ⊢ ( ( 𝐵 ∈ ( SubGrp ‘ ( 𝐼 mPwSer 𝑅 ) ) ∧ 𝑋 ∈ 𝐵 ) → ( ( invg ‘ ( 𝐼 mPwSer 𝑅 ) ) ‘ 𝑋 ) = ( 𝑀 ‘ 𝑋 ) )
13 9 7 12 syl2anc ⊢ ( 𝜑 → ( ( invg ‘ ( 𝐼 mPwSer 𝑅 ) ) ‘ 𝑋 ) = ( 𝑀 ‘ 𝑋 ) )
14 eqid ⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
15 eqid ⊢ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Base ‘ ( 𝐼 mPwSer 𝑅 ) )
16 1 8 2 15 mplbasss ⊢ 𝐵 ⊆ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) )
17 16 sseli ⊢ ( 𝑋 ∈ 𝐵 → 𝑋 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) )
18 7 17 syl ⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ ( 𝐼 mPwSer 𝑅 ) ) )
19 8 5 6 14 3 15 11 18 psrneg ⊢ ( 𝜑 → ( ( invg ‘ ( 𝐼 mPwSer 𝑅 ) ) ‘ 𝑋 ) = ( 𝑁 ∘ 𝑋 ) )
20 13 19 eqtr3d ⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) = ( 𝑁 ∘ 𝑋 ) )