Step |
Hyp |
Ref |
Expression |
1 |
|
gsumbagdiag.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
2 |
|
gsumbagdiag.s |
⊢ 𝑆 = { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹 } |
3 |
|
gsumbagdiag.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐷 ) |
4 |
|
gsumbagdiag.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
5 |
|
gsumbagdiag.g |
⊢ ( 𝜑 → 𝐺 ∈ CMnd ) |
6 |
|
gsumbagdiag.x |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑆 ∧ 𝑘 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑗 ) } ) ) → 𝑋 ∈ 𝐵 ) |
7 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
8 |
1
|
psrbaglefi |
⊢ ( 𝐹 ∈ 𝐷 → { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹 } ∈ Fin ) |
9 |
3 8
|
syl |
⊢ ( 𝜑 → { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹 } ∈ Fin ) |
10 |
2 9
|
eqeltrid |
⊢ ( 𝜑 → 𝑆 ∈ Fin ) |
11 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
12 |
1 11
|
rab2ex |
⊢ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑗 ) } ∈ V |
13 |
12
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑆 ) → { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑗 ) } ∈ V ) |
14 |
|
xpfi |
⊢ ( ( 𝑆 ∈ Fin ∧ 𝑆 ∈ Fin ) → ( 𝑆 × 𝑆 ) ∈ Fin ) |
15 |
10 10 14
|
syl2anc |
⊢ ( 𝜑 → ( 𝑆 × 𝑆 ) ∈ Fin ) |
16 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑆 ∧ 𝑘 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑗 ) } ) ) → 𝑗 ∈ 𝑆 ) |
17 |
1 2 3
|
gsumbagdiaglem |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑆 ∧ 𝑘 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑗 ) } ) ) → ( 𝑘 ∈ 𝑆 ∧ 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑘 ) } ) ) |
18 |
17
|
simpld |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑆 ∧ 𝑘 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑗 ) } ) ) → 𝑘 ∈ 𝑆 ) |
19 |
|
brxp |
⊢ ( 𝑗 ( 𝑆 × 𝑆 ) 𝑘 ↔ ( 𝑗 ∈ 𝑆 ∧ 𝑘 ∈ 𝑆 ) ) |
20 |
16 18 19
|
sylanbrc |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑆 ∧ 𝑘 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑗 ) } ) ) → 𝑗 ( 𝑆 × 𝑆 ) 𝑘 ) |
21 |
20
|
pm2.24d |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ 𝑆 ∧ 𝑘 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑗 ) } ) ) → ( ¬ 𝑗 ( 𝑆 × 𝑆 ) 𝑘 → 𝑋 = ( 0g ‘ 𝐺 ) ) ) |
22 |
21
|
impr |
⊢ ( ( 𝜑 ∧ ( ( 𝑗 ∈ 𝑆 ∧ 𝑘 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑗 ) } ) ∧ ¬ 𝑗 ( 𝑆 × 𝑆 ) 𝑘 ) ) → 𝑋 = ( 0g ‘ 𝐺 ) ) |
23 |
1 2 3
|
gsumbagdiaglem |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝑆 ∧ 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑘 ) } ) ) → ( 𝑗 ∈ 𝑆 ∧ 𝑘 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑗 ) } ) ) |
24 |
17 23
|
impbida |
⊢ ( 𝜑 → ( ( 𝑗 ∈ 𝑆 ∧ 𝑘 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑗 ) } ) ↔ ( 𝑘 ∈ 𝑆 ∧ 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑘 ) } ) ) ) |
25 |
4 7 5 10 13 6 15 22 10 24
|
gsumcom2 |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑗 ∈ 𝑆 , 𝑘 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑗 ) } ↦ 𝑋 ) ) = ( 𝐺 Σg ( 𝑘 ∈ 𝑆 , 𝑗 ∈ { 𝑥 ∈ 𝐷 ∣ 𝑥 ∘r ≤ ( 𝐹 ∘f − 𝑘 ) } ↦ 𝑋 ) ) ) |