Step |
Hyp |
Ref |
Expression |
1 |
|
gsumval1.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
gsumval1.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
gsumval1.p |
⊢ + = ( +g ‘ 𝐺 ) |
4 |
|
gsumval1.o |
⊢ 𝑂 = { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 + 𝑦 ) = 𝑦 ∧ ( 𝑦 + 𝑥 ) = 𝑦 ) } |
5 |
|
gsumval1.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝑉 ) |
6 |
|
gsumval1.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑊 ) |
7 |
|
gsumval1.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑂 ) |
8 |
|
eqidd |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ( V ∖ 𝑂 ) ) = ( ◡ 𝐹 “ ( V ∖ 𝑂 ) ) ) |
9 |
4
|
ssrab3 |
⊢ 𝑂 ⊆ 𝐵 |
10 |
|
fss |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝑂 ∧ 𝑂 ⊆ 𝐵 ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
11 |
7 9 10
|
sylancl |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
12 |
1 2 3 4 8 5 6 11
|
gsumval |
⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) = if ( ran 𝐹 ⊆ 𝑂 , 0 , if ( 𝐴 ∈ ran ... , ( ℩ 𝑧 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑧 = ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) ) ) , ( ℩ 𝑧 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ 𝑂 ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ 𝑂 ) ) ∧ 𝑧 = ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ 𝑂 ) ) ) ) ) ) ) ) ) |
13 |
|
frn |
⊢ ( 𝐹 : 𝐴 ⟶ 𝑂 → ran 𝐹 ⊆ 𝑂 ) |
14 |
|
iftrue |
⊢ ( ran 𝐹 ⊆ 𝑂 → if ( ran 𝐹 ⊆ 𝑂 , 0 , if ( 𝐴 ∈ ran ... , ( ℩ 𝑧 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑧 = ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) ) ) , ( ℩ 𝑧 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ 𝑂 ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ 𝑂 ) ) ∧ 𝑧 = ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ 𝑂 ) ) ) ) ) ) ) ) = 0 ) |
15 |
7 13 14
|
3syl |
⊢ ( 𝜑 → if ( ran 𝐹 ⊆ 𝑂 , 0 , if ( 𝐴 ∈ ran ... , ( ℩ 𝑧 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( 𝐴 = ( 𝑚 ... 𝑛 ) ∧ 𝑧 = ( seq 𝑚 ( + , 𝐹 ) ‘ 𝑛 ) ) ) , ( ℩ 𝑧 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ 𝑂 ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ 𝑂 ) ) ∧ 𝑧 = ( seq 1 ( + , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ 𝑂 ) ) ) ) ) ) ) ) = 0 ) |
16 |
12 15
|
eqtrd |
⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) = 0 ) |