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Metamath Proof Explorer

Theorem prdsplusgsgrpcl

Description: Structure product pointwise sums are closed when the factors are semigroups. (Contributed by AV, 21-Feb-2025)

Ref Expression
Hypotheses prdsplusgsgrpcl.y ⊢ 𝑌 = ( 𝑆 Xs 𝑅 )
prdsplusgsgrpcl.b ⊢ 𝐵 = ( Base ‘ 𝑌 )
prdsplusgsgrpcl.p ⊢ + = ( +g ‘ 𝑌 )
prdsplusgsgrpcl.s ⊢ ( 𝜑 → 𝑆 ∈ 𝑉 )
prdsplusgsgrpcl.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑊 )
prdsplusgsgrpcl.r ⊢ ( 𝜑 → 𝑅 : 𝐼 ⟶ Smgrp )
prdsplusgsgrpcl.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
prdsplusgsgrpcl.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐵 )
Assertion prdsplusgsgrpcl ( 𝜑 → ( 𝐹 + 𝐺 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 prdsplusgsgrpcl.y ⊢ 𝑌 = ( 𝑆 Xs 𝑅 )
2 prdsplusgsgrpcl.b ⊢ 𝐵 = ( Base ‘ 𝑌 )
3 prdsplusgsgrpcl.p ⊢ + = ( +g ‘ 𝑌 )
4 prdsplusgsgrpcl.s ⊢ ( 𝜑 → 𝑆 ∈ 𝑉 )
5 prdsplusgsgrpcl.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑊 )
6 prdsplusgsgrpcl.r ⊢ ( 𝜑 → 𝑅 : 𝐼 ⟶ Smgrp )
7 prdsplusgsgrpcl.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
8 prdsplusgsgrpcl.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐵 )
9 6 ffnd ⊢ ( 𝜑 → 𝑅 Fn 𝐼 )
10 1 2 4 5 9 7 8 3 prdsplusgval ⊢ ( 𝜑 → ( 𝐹 + 𝐺 ) = ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) )
11 6 ffvelcdmda ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑅 ‘ 𝑥 ) ∈ Smgrp )
12 4 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝑆 ∈ 𝑉 )
13 5 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝐼 ∈ 𝑊 )
14 9 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝑅 Fn 𝐼 )
15 7 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝐹 ∈ 𝐵 )
16 simpr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝑥 ∈ 𝐼 )
17 1 2 12 13 14 15 16 prdsbasprj ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) )
18 8 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝐺 ∈ 𝐵 )
19 1 2 12 13 14 18 16 prdsbasprj ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) )
20 eqid ⊢ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) = ( Base ‘ ( 𝑅 ‘ 𝑥 ) )
21 eqid ⊢ ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) = ( +g ‘ ( 𝑅 ‘ 𝑥 ) )
22 20 21 sgrpcl ⊢ ( ( ( 𝑅 ‘ 𝑥 ) ∈ Smgrp ∧ ( 𝐹 ‘ 𝑥 ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ∧ ( 𝐺 ‘ 𝑥 ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) )
23 11 17 19 22 syl3anc ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) )
24 23 ralrimiva ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) )
25 1 2 4 5 9 prdsbasmpt ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐼 ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) )
26 24 25 mpbird ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ∈ 𝐵 )
27 10 26 eqeltrd ⊢ ( 𝜑 → ( 𝐹 + 𝐺 ) ∈ 𝐵 )