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Metamath Proof Explorer

Theorem eqcoe1ply1eq

Description: Two polynomials over the same ring are equal if they have identical coefficients. (Contributed by AV, 7-Oct-2019)

Ref Expression
Hypotheses eqcoe1ply1eq.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
eqcoe1ply1eq.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
eqcoe1ply1eq.a ⊢ 𝐴 = ( coe1 ‘ 𝐾 )
eqcoe1ply1eq.c ⊢ 𝐶 = ( coe1 ‘ 𝐿 )
Assertion eqcoe1ply1eq ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵 ) → ( ∀ 𝑘 ∈ ℕ0 ( 𝐴 ‘ 𝑘 ) = ( 𝐶 ‘ 𝑘 ) → 𝐾 = 𝐿 ) )

Proof

Step Hyp Ref Expression
1 eqcoe1ply1eq.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
2 eqcoe1ply1eq.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
3 eqcoe1ply1eq.a ⊢ 𝐴 = ( coe1 ‘ 𝐾 )
4 eqcoe1ply1eq.c ⊢ 𝐶 = ( coe1 ‘ 𝐿 )
5 fveq2 ⊢ ( 𝑘 = 𝑛 → ( 𝐴 ‘ 𝑘 ) = ( 𝐴 ‘ 𝑛 ) )
6 fveq2 ⊢ ( 𝑘 = 𝑛 → ( 𝐶 ‘ 𝑘 ) = ( 𝐶 ‘ 𝑛 ) )
7 5 6 eqeq12d ⊢ ( 𝑘 = 𝑛 → ( ( 𝐴 ‘ 𝑘 ) = ( 𝐶 ‘ 𝑘 ) ↔ ( 𝐴 ‘ 𝑛 ) = ( 𝐶 ‘ 𝑛 ) ) )
8 7 rspccv ⊢ ( ∀ 𝑘 ∈ ℕ0 ( 𝐴 ‘ 𝑘 ) = ( 𝐶 ‘ 𝑘 ) → ( 𝑛 ∈ ℕ0 → ( 𝐴 ‘ 𝑛 ) = ( 𝐶 ‘ 𝑛 ) ) )
9 8 adantl ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵 ) ∧ ∀ 𝑘 ∈ ℕ0 ( 𝐴 ‘ 𝑘 ) = ( 𝐶 ‘ 𝑘 ) ) → ( 𝑛 ∈ ℕ0 → ( 𝐴 ‘ 𝑛 ) = ( 𝐶 ‘ 𝑛 ) ) )
10 9 imp ⊢ ( ( ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵 ) ∧ ∀ 𝑘 ∈ ℕ0 ( 𝐴 ‘ 𝑘 ) = ( 𝐶 ‘ 𝑘 ) ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝐴 ‘ 𝑛 ) = ( 𝐶 ‘ 𝑛 ) )
11 3 fveq1i ⊢ ( 𝐴 ‘ 𝑛 ) = ( ( coe1 ‘ 𝐾 ) ‘ 𝑛 )
12 4 fveq1i ⊢ ( 𝐶 ‘ 𝑛 ) = ( ( coe1 ‘ 𝐿 ) ‘ 𝑛 )
13 10 11 12 3eqtr3g ⊢ ( ( ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵 ) ∧ ∀ 𝑘 ∈ ℕ0 ( 𝐴 ‘ 𝑘 ) = ( 𝐶 ‘ 𝑘 ) ) ∧ 𝑛 ∈ ℕ0 ) → ( ( coe1 ‘ 𝐾 ) ‘ 𝑛 ) = ( ( coe1 ‘ 𝐿 ) ‘ 𝑛 ) )
14 13 oveq1d ⊢ ( ( ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵 ) ∧ ∀ 𝑘 ∈ ℕ0 ( 𝐴 ‘ 𝑘 ) = ( 𝐶 ‘ 𝑘 ) ) ∧ 𝑛 ∈ ℕ0 ) → ( ( ( coe1 ‘ 𝐾 ) ‘ 𝑛 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑛 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) = ( ( ( coe1 ‘ 𝐿 ) ‘ 𝑛 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑛 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) )
15 14 mpteq2dva ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵 ) ∧ ∀ 𝑘 ∈ ℕ0 ( 𝐴 ‘ 𝑘 ) = ( 𝐶 ‘ 𝑘 ) ) → ( 𝑛 ∈ ℕ0 ↦ ( ( ( coe1 ‘ 𝐾 ) ‘ 𝑛 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑛 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( coe1 ‘ 𝐿 ) ‘ 𝑛 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑛 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) )
16 15 oveq2d ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵 ) ∧ ∀ 𝑘 ∈ ℕ0 ( 𝐴 ‘ 𝑘 ) = ( 𝐶 ‘ 𝑘 ) ) → ( 𝑃 Σg ( 𝑛 ∈ ℕ0 ↦ ( ( ( coe1 ‘ 𝐾 ) ‘ 𝑛 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑛 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) ) = ( 𝑃 Σg ( 𝑛 ∈ ℕ0 ↦ ( ( ( coe1 ‘ 𝐿 ) ‘ 𝑛 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑛 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) ) )
17 eqid ⊢ ( var1 ‘ 𝑅 ) = ( var1 ‘ 𝑅 )
18 eqid ⊢ ( ·𝑠 ‘ 𝑃 ) = ( ·𝑠 ‘ 𝑃 )
19 eqid ⊢ ( mulGrp ‘ 𝑃 ) = ( mulGrp ‘ 𝑃 )
20 eqid ⊢ ( .g ‘ ( mulGrp ‘ 𝑃 ) ) = ( .g ‘ ( mulGrp ‘ 𝑃 ) )
21 eqid ⊢ ( coe1 ‘ 𝐾 ) = ( coe1 ‘ 𝐾 )
22 1 17 2 18 19 20 21 ply1coe ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ) → 𝐾 = ( 𝑃 Σg ( 𝑛 ∈ ℕ0 ↦ ( ( ( coe1 ‘ 𝐾 ) ‘ 𝑛 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑛 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) ) )
23 22 3adant3 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵 ) → 𝐾 = ( 𝑃 Σg ( 𝑛 ∈ ℕ0 ↦ ( ( ( coe1 ‘ 𝐾 ) ‘ 𝑛 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑛 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) ) )
24 eqid ⊢ ( coe1 ‘ 𝐿 ) = ( coe1 ‘ 𝐿 )
25 1 17 2 18 19 20 24 ply1coe ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐿 ∈ 𝐵 ) → 𝐿 = ( 𝑃 Σg ( 𝑛 ∈ ℕ0 ↦ ( ( ( coe1 ‘ 𝐿 ) ‘ 𝑛 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑛 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) ) )
26 25 3adant2 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵 ) → 𝐿 = ( 𝑃 Σg ( 𝑛 ∈ ℕ0 ↦ ( ( ( coe1 ‘ 𝐿 ) ‘ 𝑛 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑛 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) ) )
27 23 26 eqeq12d ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵 ) → ( 𝐾 = 𝐿 ↔ ( 𝑃 Σg ( 𝑛 ∈ ℕ0 ↦ ( ( ( coe1 ‘ 𝐾 ) ‘ 𝑛 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑛 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) ) = ( 𝑃 Σg ( 𝑛 ∈ ℕ0 ↦ ( ( ( coe1 ‘ 𝐿 ) ‘ 𝑛 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑛 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) ) ) )
28 27 adantr ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵 ) ∧ ∀ 𝑘 ∈ ℕ0 ( 𝐴 ‘ 𝑘 ) = ( 𝐶 ‘ 𝑘 ) ) → ( 𝐾 = 𝐿 ↔ ( 𝑃 Σg ( 𝑛 ∈ ℕ0 ↦ ( ( ( coe1 ‘ 𝐾 ) ‘ 𝑛 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑛 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) ) = ( 𝑃 Σg ( 𝑛 ∈ ℕ0 ↦ ( ( ( coe1 ‘ 𝐿 ) ‘ 𝑛 ) ( ·𝑠 ‘ 𝑃 ) ( 𝑛 ( .g ‘ ( mulGrp ‘ 𝑃 ) ) ( var1 ‘ 𝑅 ) ) ) ) ) ) )
29 16 28 mpbird ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵 ) ∧ ∀ 𝑘 ∈ ℕ0 ( 𝐴 ‘ 𝑘 ) = ( 𝐶 ‘ 𝑘 ) ) → 𝐾 = 𝐿 )
30 29 ex ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ∧ 𝐿 ∈ 𝐵 ) → ( ∀ 𝑘 ∈ ℕ0 ( 𝐴 ‘ 𝑘 ) = ( 𝐶 ‘ 𝑘 ) → 𝐾 = 𝐿 ) )