Metamath Proof Explorer


Theorem frlmplusgvalb

Description: Addition in a free module at the coordinates. (Contributed by AV, 16-Jan-2023)

Ref Expression
Hypotheses frlmplusgvalb.f 𝐹 = ( 𝑅 freeLMod 𝐼 )
frlmplusgvalb.b 𝐵 = ( Base ‘ 𝐹 )
frlmplusgvalb.i ( 𝜑𝐼𝑊 )
frlmplusgvalb.x ( 𝜑𝑋𝐵 )
frlmplusgvalb.z ( 𝜑𝑍𝐵 )
frlmplusgvalb.r ( 𝜑𝑅 ∈ Ring )
frlmplusgvalb.y ( 𝜑𝑌𝐵 )
frlmplusgvalb.a + = ( +g𝑅 )
frlmplusgvalb.p = ( +g𝐹 )
Assertion frlmplusgvalb ( 𝜑 → ( 𝑍 = ( 𝑋 𝑌 ) ↔ ∀ 𝑖𝐼 ( 𝑍𝑖 ) = ( ( 𝑋𝑖 ) + ( 𝑌𝑖 ) ) ) )

Proof

Step Hyp Ref Expression
1 frlmplusgvalb.f 𝐹 = ( 𝑅 freeLMod 𝐼 )
2 frlmplusgvalb.b 𝐵 = ( Base ‘ 𝐹 )
3 frlmplusgvalb.i ( 𝜑𝐼𝑊 )
4 frlmplusgvalb.x ( 𝜑𝑋𝐵 )
5 frlmplusgvalb.z ( 𝜑𝑍𝐵 )
6 frlmplusgvalb.r ( 𝜑𝑅 ∈ Ring )
7 frlmplusgvalb.y ( 𝜑𝑌𝐵 )
8 frlmplusgvalb.a + = ( +g𝑅 )
9 frlmplusgvalb.p = ( +g𝐹 )
10 eqid ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
11 1 10 2 frlmbasmap ( ( 𝐼𝑊𝑍𝐵 ) → 𝑍 ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) )
12 3 5 11 syl2anc ( 𝜑𝑍 ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) )
13 fvexd ( 𝜑 → ( Base ‘ 𝑅 ) ∈ V )
14 13 3 elmapd ( 𝜑 → ( 𝑍 ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) ↔ 𝑍 : 𝐼 ⟶ ( Base ‘ 𝑅 ) ) )
15 12 14 mpbid ( 𝜑𝑍 : 𝐼 ⟶ ( Base ‘ 𝑅 ) )
16 15 ffnd ( 𝜑𝑍 Fn 𝐼 )
17 1 frlmlmod ( ( 𝑅 ∈ Ring ∧ 𝐼𝑊 ) → 𝐹 ∈ LMod )
18 6 3 17 syl2anc ( 𝜑𝐹 ∈ LMod )
19 lmodgrp ( 𝐹 ∈ LMod → 𝐹 ∈ Grp )
20 18 19 syl ( 𝜑𝐹 ∈ Grp )
21 2 9 grpcl ( ( 𝐹 ∈ Grp ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 𝑌 ) ∈ 𝐵 )
22 20 4 7 21 syl3anc ( 𝜑 → ( 𝑋 𝑌 ) ∈ 𝐵 )
23 1 10 2 frlmbasmap ( ( 𝐼𝑊 ∧ ( 𝑋 𝑌 ) ∈ 𝐵 ) → ( 𝑋 𝑌 ) ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) )
24 3 22 23 syl2anc ( 𝜑 → ( 𝑋 𝑌 ) ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) )
25 13 3 elmapd ( 𝜑 → ( ( 𝑋 𝑌 ) ∈ ( ( Base ‘ 𝑅 ) ↑m 𝐼 ) ↔ ( 𝑋 𝑌 ) : 𝐼 ⟶ ( Base ‘ 𝑅 ) ) )
26 24 25 mpbid ( 𝜑 → ( 𝑋 𝑌 ) : 𝐼 ⟶ ( Base ‘ 𝑅 ) )
27 26 ffnd ( 𝜑 → ( 𝑋 𝑌 ) Fn 𝐼 )
28 eqfnfv ( ( 𝑍 Fn 𝐼 ∧ ( 𝑋 𝑌 ) Fn 𝐼 ) → ( 𝑍 = ( 𝑋 𝑌 ) ↔ ∀ 𝑖𝐼 ( 𝑍𝑖 ) = ( ( 𝑋 𝑌 ) ‘ 𝑖 ) ) )
29 16 27 28 syl2anc ( 𝜑 → ( 𝑍 = ( 𝑋 𝑌 ) ↔ ∀ 𝑖𝐼 ( 𝑍𝑖 ) = ( ( 𝑋 𝑌 ) ‘ 𝑖 ) ) )
30 6 adantr ( ( 𝜑𝑖𝐼 ) → 𝑅 ∈ Ring )
31 3 adantr ( ( 𝜑𝑖𝐼 ) → 𝐼𝑊 )
32 4 adantr ( ( 𝜑𝑖𝐼 ) → 𝑋𝐵 )
33 7 adantr ( ( 𝜑𝑖𝐼 ) → 𝑌𝐵 )
34 simpr ( ( 𝜑𝑖𝐼 ) → 𝑖𝐼 )
35 1 2 30 31 32 33 34 8 9 frlmvplusgvalc ( ( 𝜑𝑖𝐼 ) → ( ( 𝑋 𝑌 ) ‘ 𝑖 ) = ( ( 𝑋𝑖 ) + ( 𝑌𝑖 ) ) )
36 35 eqeq2d ( ( 𝜑𝑖𝐼 ) → ( ( 𝑍𝑖 ) = ( ( 𝑋 𝑌 ) ‘ 𝑖 ) ↔ ( 𝑍𝑖 ) = ( ( 𝑋𝑖 ) + ( 𝑌𝑖 ) ) ) )
37 36 ralbidva ( 𝜑 → ( ∀ 𝑖𝐼 ( 𝑍𝑖 ) = ( ( 𝑋 𝑌 ) ‘ 𝑖 ) ↔ ∀ 𝑖𝐼 ( 𝑍𝑖 ) = ( ( 𝑋𝑖 ) + ( 𝑌𝑖 ) ) ) )
38 29 37 bitrd ( 𝜑 → ( 𝑍 = ( 𝑋 𝑌 ) ↔ ∀ 𝑖𝐼 ( 𝑍𝑖 ) = ( ( 𝑋𝑖 ) + ( 𝑌𝑖 ) ) ) )