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Metamath Proof Explorer

Theorem selvply1rhmlem5

Description: Lemma for selvply1rhm . (Contributed by Thierry Arnoux, 4-May-2026)

Ref Expression
Hypotheses selvply1rhm.1 ⊢ 𝐵 = ( Base ‘ 𝑃 )
selvply1rhm.2 ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
selvply1rhm.3 ⊢ 𝑈 = ( ( 𝐼 ∖ { 𝑋 } ) mPoly 𝑅 )
selvply1rhm.4 ⊢ 𝑄 = ( Poly1 ‘ 𝑈 )
selvply1rhm.5 ⊢ 𝐻 = ( 𝑓 ∈ 𝐵 ↦ ( 𝑛 ∈ ( ℕ0 ↑m 1o ) ↦ ( ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝑓 ) ‘ { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } ) ) )
selvply1rhm.6 ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
selvply1rhm.7 ⊢ ( 𝜑 → 𝑋 ∈ 𝐼 )
selvply1rhm.8 ⊢ ( 𝜑 → 𝑅 ∈ CRing )
selvply1rhmlem5.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
selvply1rhmlem5.m ⊢ 𝑀 = ( 𝑞 ∈ ( Base ‘ ( { 𝑋 } mPoly 𝑈 ) ) ↦ ( 𝑠 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑞 ‘ { ⟨ 𝑋 , ( 𝑠 ‘ ∅ ) ⟩ } ) ) )
Assertion selvply1rhmlem5 ( 𝜑 → ( 𝐻 ‘ 𝐹 ) = ( 𝑀 ‘ ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝐹 ) ) )

Proof

Step Hyp Ref Expression
1 selvply1rhm.1 ⊢ 𝐵 = ( Base ‘ 𝑃 )
2 selvply1rhm.2 ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
3 selvply1rhm.3 ⊢ 𝑈 = ( ( 𝐼 ∖ { 𝑋 } ) mPoly 𝑅 )
4 selvply1rhm.4 ⊢ 𝑄 = ( Poly1 ‘ 𝑈 )
5 selvply1rhm.5 ⊢ 𝐻 = ( 𝑓 ∈ 𝐵 ↦ ( 𝑛 ∈ ( ℕ0 ↑m 1o ) ↦ ( ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝑓 ) ‘ { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } ) ) )
6 selvply1rhm.6 ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
7 selvply1rhm.7 ⊢ ( 𝜑 → 𝑋 ∈ 𝐼 )
8 selvply1rhm.8 ⊢ ( 𝜑 → 𝑅 ∈ CRing )
9 selvply1rhmlem5.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
10 selvply1rhmlem5.m ⊢ 𝑀 = ( 𝑞 ∈ ( Base ‘ ( { 𝑋 } mPoly 𝑈 ) ) ↦ ( 𝑠 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑞 ‘ { ⟨ 𝑋 , ( 𝑠 ‘ ∅ ) ⟩ } ) ) )
11 fveq2 ⊢ ( 𝑓 = 𝐹 → ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝑓 ) = ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝐹 ) )
12 11 fveq1d ⊢ ( 𝑓 = 𝐹 → ( ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝑓 ) ‘ { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } ) = ( ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝐹 ) ‘ { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } ) )
13 12 mpteq2dv ⊢ ( 𝑓 = 𝐹 → ( 𝑛 ∈ ( ℕ0 ↑m 1o ) ↦ ( ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝑓 ) ‘ { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } ) ) = ( 𝑛 ∈ ( ℕ0 ↑m 1o ) ↦ ( ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝐹 ) ‘ { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } ) ) )
14 ovexd ⊢ ( 𝜑 → ( ℕ0 ↑m 1o ) ∈ V )
15 14 mptexd ⊢ ( 𝜑 → ( 𝑛 ∈ ( ℕ0 ↑m 1o ) ↦ ( ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝐹 ) ‘ { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } ) ) ∈ V )
16 5 13 9 15 fvmptd3 ⊢ ( 𝜑 → ( 𝐻 ‘ 𝐹 ) = ( 𝑛 ∈ ( ℕ0 ↑m 1o ) ↦ ( ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝐹 ) ‘ { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } ) ) )
17 fveq1 ⊢ ( 𝑠 = 𝑛 → ( 𝑠 ‘ ∅ ) = ( 𝑛 ‘ ∅ ) )
18 17 opeq2d ⊢ ( 𝑠 = 𝑛 → ⟨ 𝑋 , ( 𝑠 ‘ ∅ ) ⟩ = ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ )
19 18 sneqd ⊢ ( 𝑠 = 𝑛 → { ⟨ 𝑋 , ( 𝑠 ‘ ∅ ) ⟩ } = { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } )
20 19 fveq2d ⊢ ( 𝑠 = 𝑛 → ( 𝑞 ‘ { ⟨ 𝑋 , ( 𝑠 ‘ ∅ ) ⟩ } ) = ( 𝑞 ‘ { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } ) )
21 20 cbvmptv ⊢ ( 𝑠 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑞 ‘ { ⟨ 𝑋 , ( 𝑠 ‘ ∅ ) ⟩ } ) ) = ( 𝑛 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑞 ‘ { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } ) )
22 21 mpteq2i ⊢ ( 𝑞 ∈ ( Base ‘ ( { 𝑋 } mPoly 𝑈 ) ) ↦ ( 𝑠 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑞 ‘ { ⟨ 𝑋 , ( 𝑠 ‘ ∅ ) ⟩ } ) ) ) = ( 𝑞 ∈ ( Base ‘ ( { 𝑋 } mPoly 𝑈 ) ) ↦ ( 𝑛 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑞 ‘ { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } ) ) )
23 10 22 eqtri ⊢ 𝑀 = ( 𝑞 ∈ ( Base ‘ ( { 𝑋 } mPoly 𝑈 ) ) ↦ ( 𝑛 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑞 ‘ { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } ) ) )
24 fveq1 ⊢ ( 𝑞 = ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝐹 ) → ( 𝑞 ‘ { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } ) = ( ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝐹 ) ‘ { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } ) )
25 24 mpteq2dv ⊢ ( 𝑞 = ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝐹 ) → ( 𝑛 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑞 ‘ { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } ) ) = ( 𝑛 ∈ ( ℕ0 ↑m 1o ) ↦ ( ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝐹 ) ‘ { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } ) ) )
26 eqid ⊢ ( { 𝑋 } mPoly 𝑈 ) = ( { 𝑋 } mPoly 𝑈 )
27 eqid ⊢ ( Base ‘ ( { 𝑋 } mPoly 𝑈 ) ) = ( Base ‘ ( { 𝑋 } mPoly 𝑈 ) )
28 7 snssd ⊢ ( 𝜑 → { 𝑋 } ⊆ 𝐼 )
29 2 1 3 26 27 8 28 9 selvcl ⊢ ( 𝜑 → ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝐹 ) ∈ ( Base ‘ ( { 𝑋 } mPoly 𝑈 ) ) )
30 23 25 29 15 fvmptd3 ⊢ ( 𝜑 → ( 𝑀 ‘ ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝐹 ) ) = ( 𝑛 ∈ ( ℕ0 ↑m 1o ) ↦ ( ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝐹 ) ‘ { ⟨ 𝑋 , ( 𝑛 ‘ ∅ ) ⟩ } ) ) )
31 16 30 eqtr4d ⊢ ( 𝜑 → ( 𝐻 ‘ 𝐹 ) = ( 𝑀 ‘ ( ( ( 𝐼 selectVars 𝑅 ) ‘ { 𝑋 } ) ‘ 𝐹 ) ) )