Metamath Proof Explorer


Theorem prdsvscacl

Description: Pointwise scalar multiplication is closed in products of modules. (Contributed by Stefan O'Rear, 10-Jan-2015)

Ref Expression
Hypotheses prdsvscacl.y 𝑌 = ( 𝑆 Xs 𝑅 )
prdsvscacl.b 𝐵 = ( Base ‘ 𝑌 )
prdsvscacl.t · = ( ·𝑠𝑌 )
prdsvscacl.k 𝐾 = ( Base ‘ 𝑆 )
prdsvscacl.s ( 𝜑𝑆 ∈ Ring )
prdsvscacl.i ( 𝜑𝐼𝑊 )
prdsvscacl.r ( 𝜑𝑅 : 𝐼 ⟶ LMod )
prdsvscacl.f ( 𝜑𝐹𝐾 )
prdsvscacl.g ( 𝜑𝐺𝐵 )
prdsvscacl.sr ( ( 𝜑𝑥𝐼 ) → ( Scalar ‘ ( 𝑅𝑥 ) ) = 𝑆 )
Assertion prdsvscacl ( 𝜑 → ( 𝐹 · 𝐺 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 prdsvscacl.y 𝑌 = ( 𝑆 Xs 𝑅 )
2 prdsvscacl.b 𝐵 = ( Base ‘ 𝑌 )
3 prdsvscacl.t · = ( ·𝑠𝑌 )
4 prdsvscacl.k 𝐾 = ( Base ‘ 𝑆 )
5 prdsvscacl.s ( 𝜑𝑆 ∈ Ring )
6 prdsvscacl.i ( 𝜑𝐼𝑊 )
7 prdsvscacl.r ( 𝜑𝑅 : 𝐼 ⟶ LMod )
8 prdsvscacl.f ( 𝜑𝐹𝐾 )
9 prdsvscacl.g ( 𝜑𝐺𝐵 )
10 prdsvscacl.sr ( ( 𝜑𝑥𝐼 ) → ( Scalar ‘ ( 𝑅𝑥 ) ) = 𝑆 )
11 7 ffnd ( 𝜑𝑅 Fn 𝐼 )
12 1 2 3 4 5 6 11 8 9 prdsvscaval ( 𝜑 → ( 𝐹 · 𝐺 ) = ( 𝑥𝐼 ↦ ( 𝐹 ( ·𝑠 ‘ ( 𝑅𝑥 ) ) ( 𝐺𝑥 ) ) ) )
13 7 ffvelrnda ( ( 𝜑𝑥𝐼 ) → ( 𝑅𝑥 ) ∈ LMod )
14 8 adantr ( ( 𝜑𝑥𝐼 ) → 𝐹𝐾 )
15 10 fveq2d ( ( 𝜑𝑥𝐼 ) → ( Base ‘ ( Scalar ‘ ( 𝑅𝑥 ) ) ) = ( Base ‘ 𝑆 ) )
16 15 4 eqtr4di ( ( 𝜑𝑥𝐼 ) → ( Base ‘ ( Scalar ‘ ( 𝑅𝑥 ) ) ) = 𝐾 )
17 14 16 eleqtrrd ( ( 𝜑𝑥𝐼 ) → 𝐹 ∈ ( Base ‘ ( Scalar ‘ ( 𝑅𝑥 ) ) ) )
18 5 adantr ( ( 𝜑𝑥𝐼 ) → 𝑆 ∈ Ring )
19 6 adantr ( ( 𝜑𝑥𝐼 ) → 𝐼𝑊 )
20 11 adantr ( ( 𝜑𝑥𝐼 ) → 𝑅 Fn 𝐼 )
21 9 adantr ( ( 𝜑𝑥𝐼 ) → 𝐺𝐵 )
22 simpr ( ( 𝜑𝑥𝐼 ) → 𝑥𝐼 )
23 1 2 18 19 20 21 22 prdsbasprj ( ( 𝜑𝑥𝐼 ) → ( 𝐺𝑥 ) ∈ ( Base ‘ ( 𝑅𝑥 ) ) )
24 eqid ( Base ‘ ( 𝑅𝑥 ) ) = ( Base ‘ ( 𝑅𝑥 ) )
25 eqid ( Scalar ‘ ( 𝑅𝑥 ) ) = ( Scalar ‘ ( 𝑅𝑥 ) )
26 eqid ( ·𝑠 ‘ ( 𝑅𝑥 ) ) = ( ·𝑠 ‘ ( 𝑅𝑥 ) )
27 eqid ( Base ‘ ( Scalar ‘ ( 𝑅𝑥 ) ) ) = ( Base ‘ ( Scalar ‘ ( 𝑅𝑥 ) ) )
28 24 25 26 27 lmodvscl ( ( ( 𝑅𝑥 ) ∈ LMod ∧ 𝐹 ∈ ( Base ‘ ( Scalar ‘ ( 𝑅𝑥 ) ) ) ∧ ( 𝐺𝑥 ) ∈ ( Base ‘ ( 𝑅𝑥 ) ) ) → ( 𝐹 ( ·𝑠 ‘ ( 𝑅𝑥 ) ) ( 𝐺𝑥 ) ) ∈ ( Base ‘ ( 𝑅𝑥 ) ) )
29 13 17 23 28 syl3anc ( ( 𝜑𝑥𝐼 ) → ( 𝐹 ( ·𝑠 ‘ ( 𝑅𝑥 ) ) ( 𝐺𝑥 ) ) ∈ ( Base ‘ ( 𝑅𝑥 ) ) )
30 29 ralrimiva ( 𝜑 → ∀ 𝑥𝐼 ( 𝐹 ( ·𝑠 ‘ ( 𝑅𝑥 ) ) ( 𝐺𝑥 ) ) ∈ ( Base ‘ ( 𝑅𝑥 ) ) )
31 1 2 5 6 11 prdsbasmpt ( 𝜑 → ( ( 𝑥𝐼 ↦ ( 𝐹 ( ·𝑠 ‘ ( 𝑅𝑥 ) ) ( 𝐺𝑥 ) ) ) ∈ 𝐵 ↔ ∀ 𝑥𝐼 ( 𝐹 ( ·𝑠 ‘ ( 𝑅𝑥 ) ) ( 𝐺𝑥 ) ) ∈ ( Base ‘ ( 𝑅𝑥 ) ) ) )
32 30 31 mpbird ( 𝜑 → ( 𝑥𝐼 ↦ ( 𝐹 ( ·𝑠 ‘ ( 𝑅𝑥 ) ) ( 𝐺𝑥 ) ) ) ∈ 𝐵 )
33 12 32 eqeltrd ( 𝜑 → ( 𝐹 · 𝐺 ) ∈ 𝐵 )