Metamath Proof Explorer


Theorem mgpsumz

Description: If the group sum for the multiplicative group of a unital ring contains a summand/factor that is the zero of the ring, the group sum itself is zero. (Contributed by AV, 29-Dec-2018)

Ref Expression
Hypotheses mgpsumunsn.m 𝑀 = ( mulGrp ‘ 𝑅 )
mgpsumunsn.t · = ( .r𝑅 )
mgpsumunsn.r ( 𝜑𝑅 ∈ CRing )
mgpsumunsn.n ( 𝜑𝑁 ∈ Fin )
mgpsumunsn.i ( 𝜑𝐼𝑁 )
mgpsumunsn.a ( ( 𝜑𝑘𝑁 ) → 𝐴 ∈ ( Base ‘ 𝑅 ) )
mgpsumz.z 0 = ( 0g𝑅 )
mgpsumz.0 ( 𝑘 = 𝐼𝐴 = 0 )
Assertion mgpsumz ( 𝜑 → ( 𝑀 Σg ( 𝑘𝑁𝐴 ) ) = 0 )

Proof

Step Hyp Ref Expression
1 mgpsumunsn.m 𝑀 = ( mulGrp ‘ 𝑅 )
2 mgpsumunsn.t · = ( .r𝑅 )
3 mgpsumunsn.r ( 𝜑𝑅 ∈ CRing )
4 mgpsumunsn.n ( 𝜑𝑁 ∈ Fin )
5 mgpsumunsn.i ( 𝜑𝐼𝑁 )
6 mgpsumunsn.a ( ( 𝜑𝑘𝑁 ) → 𝐴 ∈ ( Base ‘ 𝑅 ) )
7 mgpsumz.z 0 = ( 0g𝑅 )
8 mgpsumz.0 ( 𝑘 = 𝐼𝐴 = 0 )
9 crngring ( 𝑅 ∈ CRing → 𝑅 ∈ Ring )
10 ringmnd ( 𝑅 ∈ Ring → 𝑅 ∈ Mnd )
11 9 10 syl ( 𝑅 ∈ CRing → 𝑅 ∈ Mnd )
12 3 11 syl ( 𝜑𝑅 ∈ Mnd )
13 eqid ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
14 13 7 mndidcl ( 𝑅 ∈ Mnd → 0 ∈ ( Base ‘ 𝑅 ) )
15 12 14 syl ( 𝜑0 ∈ ( Base ‘ 𝑅 ) )
16 1 2 3 4 5 6 15 8 mgpsumunsn ( 𝜑 → ( 𝑀 Σg ( 𝑘𝑁𝐴 ) ) = ( ( 𝑀 Σg ( 𝑘 ∈ ( 𝑁 ∖ { 𝐼 } ) ↦ 𝐴 ) ) · 0 ) )
17 3 9 syl ( 𝜑𝑅 ∈ Ring )
18 1 13 mgpbas ( Base ‘ 𝑅 ) = ( Base ‘ 𝑀 )
19 1 crngmgp ( 𝑅 ∈ CRing → 𝑀 ∈ CMnd )
20 3 19 syl ( 𝜑𝑀 ∈ CMnd )
21 diffi ( 𝑁 ∈ Fin → ( 𝑁 ∖ { 𝐼 } ) ∈ Fin )
22 4 21 syl ( 𝜑 → ( 𝑁 ∖ { 𝐼 } ) ∈ Fin )
23 eldifi ( 𝑘 ∈ ( 𝑁 ∖ { 𝐼 } ) → 𝑘𝑁 )
24 23 6 sylan2 ( ( 𝜑𝑘 ∈ ( 𝑁 ∖ { 𝐼 } ) ) → 𝐴 ∈ ( Base ‘ 𝑅 ) )
25 24 ralrimiva ( 𝜑 → ∀ 𝑘 ∈ ( 𝑁 ∖ { 𝐼 } ) 𝐴 ∈ ( Base ‘ 𝑅 ) )
26 18 20 22 25 gsummptcl ( 𝜑 → ( 𝑀 Σg ( 𝑘 ∈ ( 𝑁 ∖ { 𝐼 } ) ↦ 𝐴 ) ) ∈ ( Base ‘ 𝑅 ) )
27 13 2 7 ringrz ( ( 𝑅 ∈ Ring ∧ ( 𝑀 Σg ( 𝑘 ∈ ( 𝑁 ∖ { 𝐼 } ) ↦ 𝐴 ) ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑀 Σg ( 𝑘 ∈ ( 𝑁 ∖ { 𝐼 } ) ↦ 𝐴 ) ) · 0 ) = 0 )
28 17 26 27 syl2anc ( 𝜑 → ( ( 𝑀 Σg ( 𝑘 ∈ ( 𝑁 ∖ { 𝐼 } ) ↦ 𝐴 ) ) · 0 ) = 0 )
29 16 28 eqtrd ( 𝜑 → ( 𝑀 Σg ( 𝑘𝑁𝐴 ) ) = 0 )