Metamath Proof Explorer


Theorem gsumunsn

Description: Append an element to a finite group sum. (Contributed by Mario Carneiro, 19-Dec-2014) (Proof shortened by AV, 8-Mar-2019)

Ref Expression
Hypotheses gsumunsn.b 𝐵 = ( Base ‘ 𝐺 )
gsumunsn.p + = ( +g𝐺 )
gsumunsn.g ( 𝜑𝐺 ∈ CMnd )
gsumunsn.a ( 𝜑𝐴 ∈ Fin )
gsumunsn.f ( ( 𝜑𝑘𝐴 ) → 𝑋𝐵 )
gsumunsn.m ( 𝜑𝑀𝑉 )
gsumunsn.d ( 𝜑 → ¬ 𝑀𝐴 )
gsumunsn.y ( 𝜑𝑌𝐵 )
gsumunsn.s ( 𝑘 = 𝑀𝑋 = 𝑌 )
Assertion gsumunsn ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 ∪ { 𝑀 } ) ↦ 𝑋 ) ) = ( ( 𝐺 Σg ( 𝑘𝐴𝑋 ) ) + 𝑌 ) )

Proof

Step Hyp Ref Expression
1 gsumunsn.b 𝐵 = ( Base ‘ 𝐺 )
2 gsumunsn.p + = ( +g𝐺 )
3 gsumunsn.g ( 𝜑𝐺 ∈ CMnd )
4 gsumunsn.a ( 𝜑𝐴 ∈ Fin )
5 gsumunsn.f ( ( 𝜑𝑘𝐴 ) → 𝑋𝐵 )
6 gsumunsn.m ( 𝜑𝑀𝑉 )
7 gsumunsn.d ( 𝜑 → ¬ 𝑀𝐴 )
8 gsumunsn.y ( 𝜑𝑌𝐵 )
9 gsumunsn.s ( 𝑘 = 𝑀𝑋 = 𝑌 )
10 9 adantl ( ( 𝜑𝑘 = 𝑀 ) → 𝑋 = 𝑌 )
11 1 2 3 4 5 6 7 8 10 gsumunsnd ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 ∪ { 𝑀 } ) ↦ 𝑋 ) ) = ( ( 𝐺 Σg ( 𝑘𝐴𝑋 ) ) + 𝑌 ) )