Step |
Hyp |
Ref |
Expression |
1 |
|
gsumcom3fi.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
gsumcom3fi.g |
⊢ ( 𝜑 → 𝐺 ∈ CMnd ) |
3 |
|
gsumcom3fi.a |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
4 |
|
gsumcom3fi.r |
⊢ ( 𝜑 → 𝐶 ∈ Fin ) |
5 |
|
gsumcom3fi.f |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ) → 𝑋 ∈ 𝐵 ) |
6 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
7 |
|
xpfi |
⊢ ( ( 𝐴 ∈ Fin ∧ 𝐶 ∈ Fin ) → ( 𝐴 × 𝐶 ) ∈ Fin ) |
8 |
3 4 7
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 × 𝐶 ) ∈ Fin ) |
9 |
|
brxp |
⊢ ( 𝑗 ( 𝐴 × 𝐶 ) 𝑘 ↔ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ) |
10 |
9
|
biimpri |
⊢ ( ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) → 𝑗 ( 𝐴 × 𝐶 ) 𝑘 ) |
11 |
10
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ) → 𝑗 ( 𝐴 × 𝐶 ) 𝑘 ) |
12 |
11
|
pm2.24d |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ) → ( ¬ 𝑗 ( 𝐴 × 𝐶 ) 𝑘 → 𝑋 = ( 0g ‘ 𝐺 ) ) ) |
13 |
12
|
impr |
⊢ ( ( 𝜑 ∧ ( ( 𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐶 ) ∧ ¬ 𝑗 ( 𝐴 × 𝐶 ) 𝑘 ) ) → 𝑋 = ( 0g ‘ 𝐺 ) ) |
14 |
1 6 2 3 4 5 8 13
|
gsumcom3 |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑗 ∈ 𝐴 ↦ ( 𝐺 Σg ( 𝑘 ∈ 𝐶 ↦ 𝑋 ) ) ) ) = ( 𝐺 Σg ( 𝑘 ∈ 𝐶 ↦ ( 𝐺 Σg ( 𝑗 ∈ 𝐴 ↦ 𝑋 ) ) ) ) ) |