Metamath Proof Explorer
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 |
⊢ + = ( +g ‘ 𝐺 ) |
|
|
gsumunsn.g |
⊢ ( 𝜑 → 𝐺 ∈ CMnd ) |
|
|
gsumunsn.a |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
|
|
gsumunsn.f |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 ) |
|
|
gsumunsn.m |
⊢ ( 𝜑 → 𝑀 ∈ 𝑉 ) |
|
|
gsumunsn.d |
⊢ ( 𝜑 → ¬ 𝑀 ∈ 𝐴 ) |
|
|
gsumunsn.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
|
|
gsumunsn.s |
⊢ ( 𝑘 = 𝑀 → 𝑋 = 𝑌 ) |
|
Assertion |
gsumunsn |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 ∪ { 𝑀 } ) ↦ 𝑋 ) ) = ( ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) + 𝑌 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
gsumunsn.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
gsumunsn.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
gsumunsn.g |
⊢ ( 𝜑 → 𝐺 ∈ CMnd ) |
4 |
|
gsumunsn.a |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
5 |
|
gsumunsn.f |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑋 ∈ 𝐵 ) |
6 |
|
gsumunsn.m |
⊢ ( 𝜑 → 𝑀 ∈ 𝑉 ) |
7 |
|
gsumunsn.d |
⊢ ( 𝜑 → ¬ 𝑀 ∈ 𝐴 ) |
8 |
|
gsumunsn.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
9 |
|
gsumunsn.s |
⊢ ( 𝑘 = 𝑀 → 𝑋 = 𝑌 ) |
10 |
9
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝑀 ) → 𝑋 = 𝑌 ) |
11 |
1 2 3 4 5 6 7 8 10
|
gsumunsnd |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 ∪ { 𝑀 } ) ↦ 𝑋 ) ) = ( ( 𝐺 Σg ( 𝑘 ∈ 𝐴 ↦ 𝑋 ) ) + 𝑌 ) ) |