Metamath Proof Explorer


Theorem asclmulg

Description: Apply group multiplication to the algebra scalars. (Contributed by Thierry Arnoux, 24-Jul-2024)

Ref Expression
Hypotheses asclmulg.a 𝐴 = ( algSc ‘ 𝑊 )
asclmulg.f 𝐹 = ( Scalar ‘ 𝑊 )
asclmulg.k 𝐾 = ( Base ‘ 𝐹 )
asclmulg.m = ( .g𝑊 )
asclmulg.t = ( .g𝐹 )
Assertion asclmulg ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0𝑋𝐾 ) → ( 𝐴 ‘ ( 𝑁 𝑋 ) ) = ( 𝑁 ( 𝐴𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 asclmulg.a 𝐴 = ( algSc ‘ 𝑊 )
2 asclmulg.f 𝐹 = ( Scalar ‘ 𝑊 )
3 asclmulg.k 𝐾 = ( Base ‘ 𝐹 )
4 asclmulg.m = ( .g𝑊 )
5 asclmulg.t = ( .g𝐹 )
6 assalmod ( 𝑊 ∈ AssAlg → 𝑊 ∈ LMod )
7 6 3ad2ant1 ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0𝑋𝐾 ) → 𝑊 ∈ LMod )
8 simp3 ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0𝑋𝐾 ) → 𝑋𝐾 )
9 simp2 ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0𝑋𝐾 ) → 𝑁 ∈ ℕ0 )
10 assaring ( 𝑊 ∈ AssAlg → 𝑊 ∈ Ring )
11 eqid ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
12 eqid ( 1r𝑊 ) = ( 1r𝑊 )
13 11 12 ringidcl ( 𝑊 ∈ Ring → ( 1r𝑊 ) ∈ ( Base ‘ 𝑊 ) )
14 10 13 syl ( 𝑊 ∈ AssAlg → ( 1r𝑊 ) ∈ ( Base ‘ 𝑊 ) )
15 14 3ad2ant1 ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0𝑋𝐾 ) → ( 1r𝑊 ) ∈ ( Base ‘ 𝑊 ) )
16 eqid ( ·𝑠𝑊 ) = ( ·𝑠𝑊 )
17 11 2 16 3 4 5 lmodvsmmulgdi ( ( 𝑊 ∈ LMod ∧ ( 𝑋𝐾𝑁 ∈ ℕ0 ∧ ( 1r𝑊 ) ∈ ( Base ‘ 𝑊 ) ) ) → ( 𝑁 ( 𝑋 ( ·𝑠𝑊 ) ( 1r𝑊 ) ) ) = ( ( 𝑁 𝑋 ) ( ·𝑠𝑊 ) ( 1r𝑊 ) ) )
18 7 8 9 15 17 syl13anc ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0𝑋𝐾 ) → ( 𝑁 ( 𝑋 ( ·𝑠𝑊 ) ( 1r𝑊 ) ) ) = ( ( 𝑁 𝑋 ) ( ·𝑠𝑊 ) ( 1r𝑊 ) ) )
19 1 2 3 16 12 asclval ( 𝑋𝐾 → ( 𝐴𝑋 ) = ( 𝑋 ( ·𝑠𝑊 ) ( 1r𝑊 ) ) )
20 8 19 syl ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0𝑋𝐾 ) → ( 𝐴𝑋 ) = ( 𝑋 ( ·𝑠𝑊 ) ( 1r𝑊 ) ) )
21 20 oveq2d ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0𝑋𝐾 ) → ( 𝑁 ( 𝐴𝑋 ) ) = ( 𝑁 ( 𝑋 ( ·𝑠𝑊 ) ( 1r𝑊 ) ) ) )
22 2 assasca ( 𝑊 ∈ AssAlg → 𝐹 ∈ CRing )
23 22 3ad2ant1 ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0𝑋𝐾 ) → 𝐹 ∈ CRing )
24 23 crnggrpd ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0𝑋𝐾 ) → 𝐹 ∈ Grp )
25 9 nn0zd ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0𝑋𝐾 ) → 𝑁 ∈ ℤ )
26 3 5 24 25 8 mulgcld ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0𝑋𝐾 ) → ( 𝑁 𝑋 ) ∈ 𝐾 )
27 1 2 3 16 12 asclval ( ( 𝑁 𝑋 ) ∈ 𝐾 → ( 𝐴 ‘ ( 𝑁 𝑋 ) ) = ( ( 𝑁 𝑋 ) ( ·𝑠𝑊 ) ( 1r𝑊 ) ) )
28 26 27 syl ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0𝑋𝐾 ) → ( 𝐴 ‘ ( 𝑁 𝑋 ) ) = ( ( 𝑁 𝑋 ) ( ·𝑠𝑊 ) ( 1r𝑊 ) ) )
29 18 21 28 3eqtr4rd ( ( 𝑊 ∈ AssAlg ∧ 𝑁 ∈ ℕ0𝑋𝐾 ) → ( 𝐴 ‘ ( 𝑁 𝑋 ) ) = ( 𝑁 ( 𝐴𝑋 ) ) )