Step |
Hyp |
Ref |
Expression |
1 |
|
gsum2d.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
gsum2d.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
gsum2d.g |
⊢ ( 𝜑 → 𝐺 ∈ CMnd ) |
4 |
|
gsum2d.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
5 |
|
gsum2d.r |
⊢ ( 𝜑 → Rel 𝐴 ) |
6 |
|
gsum2d.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑊 ) |
7 |
|
gsum2d.s |
⊢ ( 𝜑 → dom 𝐴 ⊆ 𝐷 ) |
8 |
|
gsum2d.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
9 |
|
gsum2d.w |
⊢ ( 𝜑 → 𝐹 finSupp 0 ) |
10 |
9
|
fsuppimpd |
⊢ ( 𝜑 → ( 𝐹 supp 0 ) ∈ Fin ) |
11 |
|
dmfi |
⊢ ( ( 𝐹 supp 0 ) ∈ Fin → dom ( 𝐹 supp 0 ) ∈ Fin ) |
12 |
10 11
|
syl |
⊢ ( 𝜑 → dom ( 𝐹 supp 0 ) ∈ Fin ) |
13 |
|
reseq2 |
⊢ ( 𝑥 = ∅ → ( 𝐴 ↾ 𝑥 ) = ( 𝐴 ↾ ∅ ) ) |
14 |
|
res0 |
⊢ ( 𝐴 ↾ ∅ ) = ∅ |
15 |
13 14
|
eqtrdi |
⊢ ( 𝑥 = ∅ → ( 𝐴 ↾ 𝑥 ) = ∅ ) |
16 |
15
|
reseq2d |
⊢ ( 𝑥 = ∅ → ( 𝐹 ↾ ( 𝐴 ↾ 𝑥 ) ) = ( 𝐹 ↾ ∅ ) ) |
17 |
|
res0 |
⊢ ( 𝐹 ↾ ∅ ) = ∅ |
18 |
16 17
|
eqtrdi |
⊢ ( 𝑥 = ∅ → ( 𝐹 ↾ ( 𝐴 ↾ 𝑥 ) ) = ∅ ) |
19 |
18
|
oveq2d |
⊢ ( 𝑥 = ∅ → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑥 ) ) ) = ( 𝐺 Σg ∅ ) ) |
20 |
|
mpteq1 |
⊢ ( 𝑥 = ∅ → ( 𝑗 ∈ 𝑥 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) = ( 𝑗 ∈ ∅ ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) |
21 |
|
mpt0 |
⊢ ( 𝑗 ∈ ∅ ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) = ∅ |
22 |
20 21
|
eqtrdi |
⊢ ( 𝑥 = ∅ → ( 𝑗 ∈ 𝑥 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) = ∅ ) |
23 |
22
|
oveq2d |
⊢ ( 𝑥 = ∅ → ( 𝐺 Σg ( 𝑗 ∈ 𝑥 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) = ( 𝐺 Σg ∅ ) ) |
24 |
19 23
|
eqeq12d |
⊢ ( 𝑥 = ∅ → ( ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑥 ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝑥 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ↔ ( 𝐺 Σg ∅ ) = ( 𝐺 Σg ∅ ) ) ) |
25 |
24
|
imbi2d |
⊢ ( 𝑥 = ∅ → ( ( 𝜑 → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑥 ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝑥 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) ↔ ( 𝜑 → ( 𝐺 Σg ∅ ) = ( 𝐺 Σg ∅ ) ) ) ) |
26 |
|
reseq2 |
⊢ ( 𝑥 = 𝑦 → ( 𝐴 ↾ 𝑥 ) = ( 𝐴 ↾ 𝑦 ) ) |
27 |
26
|
reseq2d |
⊢ ( 𝑥 = 𝑦 → ( 𝐹 ↾ ( 𝐴 ↾ 𝑥 ) ) = ( 𝐹 ↾ ( 𝐴 ↾ 𝑦 ) ) ) |
28 |
27
|
oveq2d |
⊢ ( 𝑥 = 𝑦 → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑥 ) ) ) = ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑦 ) ) ) ) |
29 |
|
mpteq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑗 ∈ 𝑥 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) = ( 𝑗 ∈ 𝑦 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) |
30 |
29
|
oveq2d |
⊢ ( 𝑥 = 𝑦 → ( 𝐺 Σg ( 𝑗 ∈ 𝑥 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝑦 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) |
31 |
28 30
|
eqeq12d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑥 ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝑥 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ↔ ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑦 ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝑦 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) ) |
32 |
31
|
imbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑥 ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝑥 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) ↔ ( 𝜑 → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑦 ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝑦 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) ) ) |
33 |
|
reseq2 |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝐴 ↾ 𝑥 ) = ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) |
34 |
33
|
reseq2d |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝐹 ↾ ( 𝐴 ↾ 𝑥 ) ) = ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ) |
35 |
34
|
oveq2d |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑥 ) ) ) = ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ) ) |
36 |
|
mpteq1 |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝑗 ∈ 𝑥 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) = ( 𝑗 ∈ ( 𝑦 ∪ { 𝑧 } ) ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) |
37 |
36
|
oveq2d |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝐺 Σg ( 𝑗 ∈ 𝑥 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ ( 𝑦 ∪ { 𝑧 } ) ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) |
38 |
35 37
|
eqeq12d |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑥 ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝑥 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ↔ ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ ( 𝑦 ∪ { 𝑧 } ) ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) ) |
39 |
38
|
imbi2d |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ( 𝜑 → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑥 ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝑥 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) ↔ ( 𝜑 → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ ( 𝑦 ∪ { 𝑧 } ) ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) ) ) |
40 |
|
reseq2 |
⊢ ( 𝑥 = dom ( 𝐹 supp 0 ) → ( 𝐴 ↾ 𝑥 ) = ( 𝐴 ↾ dom ( 𝐹 supp 0 ) ) ) |
41 |
40
|
reseq2d |
⊢ ( 𝑥 = dom ( 𝐹 supp 0 ) → ( 𝐹 ↾ ( 𝐴 ↾ 𝑥 ) ) = ( 𝐹 ↾ ( 𝐴 ↾ dom ( 𝐹 supp 0 ) ) ) ) |
42 |
41
|
oveq2d |
⊢ ( 𝑥 = dom ( 𝐹 supp 0 ) → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑥 ) ) ) = ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ dom ( 𝐹 supp 0 ) ) ) ) ) |
43 |
|
mpteq1 |
⊢ ( 𝑥 = dom ( 𝐹 supp 0 ) → ( 𝑗 ∈ 𝑥 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) = ( 𝑗 ∈ dom ( 𝐹 supp 0 ) ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) |
44 |
43
|
oveq2d |
⊢ ( 𝑥 = dom ( 𝐹 supp 0 ) → ( 𝐺 Σg ( 𝑗 ∈ 𝑥 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ dom ( 𝐹 supp 0 ) ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) |
45 |
42 44
|
eqeq12d |
⊢ ( 𝑥 = dom ( 𝐹 supp 0 ) → ( ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑥 ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝑥 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ↔ ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ dom ( 𝐹 supp 0 ) ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ dom ( 𝐹 supp 0 ) ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) ) |
46 |
45
|
imbi2d |
⊢ ( 𝑥 = dom ( 𝐹 supp 0 ) → ( ( 𝜑 → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑥 ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝑥 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) ↔ ( 𝜑 → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ dom ( 𝐹 supp 0 ) ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ dom ( 𝐹 supp 0 ) ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) ) ) |
47 |
|
eqidd |
⊢ ( 𝜑 → ( 𝐺 Σg ∅ ) = ( 𝐺 Σg ∅ ) ) |
48 |
|
oveq1 |
⊢ ( ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑦 ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝑦 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) → ( ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑦 ) ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) ) = ( ( 𝐺 Σg ( 𝑗 ∈ 𝑦 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) ) ) |
49 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
50 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → 𝐺 ∈ CMnd ) |
51 |
4
|
resexd |
⊢ ( 𝜑 → ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ∈ V ) |
52 |
51
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ∈ V ) |
53 |
|
resss |
⊢ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ⊆ 𝐴 |
54 |
|
fssres |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ⊆ 𝐴 ) → ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) : ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ⟶ 𝐵 ) |
55 |
8 53 54
|
sylancl |
⊢ ( 𝜑 → ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) : ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ⟶ 𝐵 ) |
56 |
55
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) : ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ⟶ 𝐵 ) |
57 |
8
|
ffund |
⊢ ( 𝜑 → Fun 𝐹 ) |
58 |
57
|
funresd |
⊢ ( 𝜑 → Fun ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ) |
59 |
58
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → Fun ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ) |
60 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( 𝐹 supp 0 ) ∈ Fin ) |
61 |
8 4
|
fexd |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
62 |
2
|
fvexi |
⊢ 0 ∈ V |
63 |
|
ressuppss |
⊢ ( ( 𝐹 ∈ V ∧ 0 ∈ V ) → ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) supp 0 ) ⊆ ( 𝐹 supp 0 ) ) |
64 |
61 62 63
|
sylancl |
⊢ ( 𝜑 → ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) supp 0 ) ⊆ ( 𝐹 supp 0 ) ) |
65 |
64
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) supp 0 ) ⊆ ( 𝐹 supp 0 ) ) |
66 |
60 65
|
ssfid |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) supp 0 ) ∈ Fin ) |
67 |
61
|
resexd |
⊢ ( 𝜑 → ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ∈ V ) |
68 |
|
isfsupp |
⊢ ( ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ∈ V ∧ 0 ∈ V ) → ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) finSupp 0 ↔ ( Fun ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) supp 0 ) ∈ Fin ) ) ) |
69 |
67 62 68
|
sylancl |
⊢ ( 𝜑 → ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) finSupp 0 ↔ ( Fun ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) supp 0 ) ∈ Fin ) ) ) |
70 |
69
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) finSupp 0 ↔ ( Fun ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ∧ ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) supp 0 ) ∈ Fin ) ) ) |
71 |
59 66 70
|
mpbir2and |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) finSupp 0 ) |
72 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ¬ 𝑧 ∈ 𝑦 ) |
73 |
|
disjsn |
⊢ ( ( 𝑦 ∩ { 𝑧 } ) = ∅ ↔ ¬ 𝑧 ∈ 𝑦 ) |
74 |
72 73
|
sylibr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( 𝑦 ∩ { 𝑧 } ) = ∅ ) |
75 |
74
|
reseq2d |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( 𝐴 ↾ ( 𝑦 ∩ { 𝑧 } ) ) = ( 𝐴 ↾ ∅ ) ) |
76 |
|
resindi |
⊢ ( 𝐴 ↾ ( 𝑦 ∩ { 𝑧 } ) ) = ( ( 𝐴 ↾ 𝑦 ) ∩ ( 𝐴 ↾ { 𝑧 } ) ) |
77 |
75 76 14
|
3eqtr3g |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( ( 𝐴 ↾ 𝑦 ) ∩ ( 𝐴 ↾ { 𝑧 } ) ) = ∅ ) |
78 |
|
resundi |
⊢ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) = ( ( 𝐴 ↾ 𝑦 ) ∪ ( 𝐴 ↾ { 𝑧 } ) ) |
79 |
78
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) = ( ( 𝐴 ↾ 𝑦 ) ∪ ( 𝐴 ↾ { 𝑧 } ) ) ) |
80 |
1 2 49 50 52 56 71 77 79
|
gsumsplit |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ) = ( ( 𝐺 Σg ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ↾ ( 𝐴 ↾ 𝑦 ) ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) ) ) |
81 |
|
ssun1 |
⊢ 𝑦 ⊆ ( 𝑦 ∪ { 𝑧 } ) |
82 |
|
ssres2 |
⊢ ( 𝑦 ⊆ ( 𝑦 ∪ { 𝑧 } ) → ( 𝐴 ↾ 𝑦 ) ⊆ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) |
83 |
|
resabs1 |
⊢ ( ( 𝐴 ↾ 𝑦 ) ⊆ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) → ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ↾ ( 𝐴 ↾ 𝑦 ) ) = ( 𝐹 ↾ ( 𝐴 ↾ 𝑦 ) ) ) |
84 |
81 82 83
|
mp2b |
⊢ ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ↾ ( 𝐴 ↾ 𝑦 ) ) = ( 𝐹 ↾ ( 𝐴 ↾ 𝑦 ) ) |
85 |
84
|
oveq2i |
⊢ ( 𝐺 Σg ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ↾ ( 𝐴 ↾ 𝑦 ) ) ) = ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑦 ) ) ) |
86 |
|
ssun2 |
⊢ { 𝑧 } ⊆ ( 𝑦 ∪ { 𝑧 } ) |
87 |
|
ssres2 |
⊢ ( { 𝑧 } ⊆ ( 𝑦 ∪ { 𝑧 } ) → ( 𝐴 ↾ { 𝑧 } ) ⊆ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) |
88 |
|
resabs1 |
⊢ ( ( 𝐴 ↾ { 𝑧 } ) ⊆ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) → ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ↾ ( 𝐴 ↾ { 𝑧 } ) ) = ( 𝐹 ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) |
89 |
86 87 88
|
mp2b |
⊢ ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ↾ ( 𝐴 ↾ { 𝑧 } ) ) = ( 𝐹 ↾ ( 𝐴 ↾ { 𝑧 } ) ) |
90 |
89
|
oveq2i |
⊢ ( 𝐺 Σg ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) = ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) |
91 |
85 90
|
oveq12i |
⊢ ( ( 𝐺 Σg ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ↾ ( 𝐴 ↾ 𝑦 ) ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) ) = ( ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑦 ) ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) ) |
92 |
80 91
|
eqtrdi |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ) = ( ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑦 ) ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) ) ) |
93 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → 𝑦 ∈ Fin ) |
94 |
1 2 3 4 5 6 7 8 9
|
gsum2dlem1 |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ∈ 𝐵 ) |
95 |
94
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) ∧ 𝑗 ∈ 𝑦 ) → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ∈ 𝐵 ) |
96 |
|
vex |
⊢ 𝑧 ∈ V |
97 |
96
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → 𝑧 ∈ V ) |
98 |
|
sneq |
⊢ ( 𝑗 = 𝑧 → { 𝑗 } = { 𝑧 } ) |
99 |
98
|
imaeq2d |
⊢ ( 𝑗 = 𝑧 → ( 𝐴 “ { 𝑗 } ) = ( 𝐴 “ { 𝑧 } ) ) |
100 |
|
oveq1 |
⊢ ( 𝑗 = 𝑧 → ( 𝑗 𝐹 𝑘 ) = ( 𝑧 𝐹 𝑘 ) ) |
101 |
99 100
|
mpteq12dv |
⊢ ( 𝑗 = 𝑧 → ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) = ( 𝑘 ∈ ( 𝐴 “ { 𝑧 } ) ↦ ( 𝑧 𝐹 𝑘 ) ) ) |
102 |
101
|
oveq2d |
⊢ ( 𝑗 = 𝑧 → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) = ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑧 } ) ↦ ( 𝑧 𝐹 𝑘 ) ) ) ) |
103 |
102
|
eleq1d |
⊢ ( 𝑗 = 𝑧 → ( ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ∈ 𝐵 ↔ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑧 } ) ↦ ( 𝑧 𝐹 𝑘 ) ) ) ∈ 𝐵 ) ) |
104 |
103
|
imbi2d |
⊢ ( 𝑗 = 𝑧 → ( ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ∈ 𝐵 ) ↔ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑧 } ) ↦ ( 𝑧 𝐹 𝑘 ) ) ) ∈ 𝐵 ) ) ) |
105 |
104 94
|
chvarvv |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑧 } ) ↦ ( 𝑧 𝐹 𝑘 ) ) ) ∈ 𝐵 ) |
106 |
105
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑧 } ) ↦ ( 𝑧 𝐹 𝑘 ) ) ) ∈ 𝐵 ) |
107 |
1 49 50 93 95 97 72 106 102
|
gsumunsn |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( 𝐺 Σg ( 𝑗 ∈ ( 𝑦 ∪ { 𝑧 } ) ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) = ( ( 𝐺 Σg ( 𝑗 ∈ 𝑦 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑧 } ) ↦ ( 𝑧 𝐹 𝑘 ) ) ) ) ) |
108 |
98
|
reseq2d |
⊢ ( 𝑗 = 𝑧 → ( 𝐴 ↾ { 𝑗 } ) = ( 𝐴 ↾ { 𝑧 } ) ) |
109 |
108
|
reseq2d |
⊢ ( 𝑗 = 𝑧 → ( 𝐹 ↾ ( 𝐴 ↾ { 𝑗 } ) ) = ( 𝐹 ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) |
110 |
109
|
oveq2d |
⊢ ( 𝑗 = 𝑧 → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑗 } ) ) ) = ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) ) |
111 |
102 110
|
eqeq12d |
⊢ ( 𝑗 = 𝑧 → ( ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) = ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑗 } ) ) ) ↔ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑧 } ) ↦ ( 𝑧 𝐹 𝑘 ) ) ) = ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) ) ) |
112 |
111
|
imbi2d |
⊢ ( 𝑗 = 𝑧 → ( ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) = ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑗 } ) ) ) ) ↔ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑧 } ) ↦ ( 𝑧 𝐹 𝑘 ) ) ) = ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) ) ) ) |
113 |
|
imaexg |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 “ { 𝑗 } ) ∈ V ) |
114 |
4 113
|
syl |
⊢ ( 𝜑 → ( 𝐴 “ { 𝑗 } ) ∈ V ) |
115 |
|
vex |
⊢ 𝑗 ∈ V |
116 |
|
vex |
⊢ 𝑘 ∈ V |
117 |
115 116
|
elimasn |
⊢ ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↔ 〈 𝑗 , 𝑘 〉 ∈ 𝐴 ) |
118 |
|
df-ov |
⊢ ( 𝑗 𝐹 𝑘 ) = ( 𝐹 ‘ 〈 𝑗 , 𝑘 〉 ) |
119 |
8
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 〈 𝑗 , 𝑘 〉 ∈ 𝐴 ) → ( 𝐹 ‘ 〈 𝑗 , 𝑘 〉 ) ∈ 𝐵 ) |
120 |
118 119
|
eqeltrid |
⊢ ( ( 𝜑 ∧ 〈 𝑗 , 𝑘 〉 ∈ 𝐴 ) → ( 𝑗 𝐹 𝑘 ) ∈ 𝐵 ) |
121 |
117 120
|
sylan2b |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ) → ( 𝑗 𝐹 𝑘 ) ∈ 𝐵 ) |
122 |
121
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) : ( 𝐴 “ { 𝑗 } ) ⟶ 𝐵 ) |
123 |
|
funmpt |
⊢ Fun ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) |
124 |
123
|
a1i |
⊢ ( 𝜑 → Fun ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) |
125 |
|
rnfi |
⊢ ( ( 𝐹 supp 0 ) ∈ Fin → ran ( 𝐹 supp 0 ) ∈ Fin ) |
126 |
10 125
|
syl |
⊢ ( 𝜑 → ran ( 𝐹 supp 0 ) ∈ Fin ) |
127 |
117
|
biimpi |
⊢ ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) → 〈 𝑗 , 𝑘 〉 ∈ 𝐴 ) |
128 |
115 116
|
opelrn |
⊢ ( 〈 𝑗 , 𝑘 〉 ∈ ( 𝐹 supp 0 ) → 𝑘 ∈ ran ( 𝐹 supp 0 ) ) |
129 |
128
|
con3i |
⊢ ( ¬ 𝑘 ∈ ran ( 𝐹 supp 0 ) → ¬ 〈 𝑗 , 𝑘 〉 ∈ ( 𝐹 supp 0 ) ) |
130 |
127 129
|
anim12i |
⊢ ( ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ∧ ¬ 𝑘 ∈ ran ( 𝐹 supp 0 ) ) → ( 〈 𝑗 , 𝑘 〉 ∈ 𝐴 ∧ ¬ 〈 𝑗 , 𝑘 〉 ∈ ( 𝐹 supp 0 ) ) ) |
131 |
|
eldif |
⊢ ( 𝑘 ∈ ( ( 𝐴 “ { 𝑗 } ) ∖ ran ( 𝐹 supp 0 ) ) ↔ ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ∧ ¬ 𝑘 ∈ ran ( 𝐹 supp 0 ) ) ) |
132 |
|
eldif |
⊢ ( 〈 𝑗 , 𝑘 〉 ∈ ( 𝐴 ∖ ( 𝐹 supp 0 ) ) ↔ ( 〈 𝑗 , 𝑘 〉 ∈ 𝐴 ∧ ¬ 〈 𝑗 , 𝑘 〉 ∈ ( 𝐹 supp 0 ) ) ) |
133 |
130 131 132
|
3imtr4i |
⊢ ( 𝑘 ∈ ( ( 𝐴 “ { 𝑗 } ) ∖ ran ( 𝐹 supp 0 ) ) → 〈 𝑗 , 𝑘 〉 ∈ ( 𝐴 ∖ ( 𝐹 supp 0 ) ) ) |
134 |
|
ssidd |
⊢ ( 𝜑 → ( 𝐹 supp 0 ) ⊆ ( 𝐹 supp 0 ) ) |
135 |
62
|
a1i |
⊢ ( 𝜑 → 0 ∈ V ) |
136 |
8 134 4 135
|
suppssr |
⊢ ( ( 𝜑 ∧ 〈 𝑗 , 𝑘 〉 ∈ ( 𝐴 ∖ ( 𝐹 supp 0 ) ) ) → ( 𝐹 ‘ 〈 𝑗 , 𝑘 〉 ) = 0 ) |
137 |
118 136
|
syl5eq |
⊢ ( ( 𝜑 ∧ 〈 𝑗 , 𝑘 〉 ∈ ( 𝐴 ∖ ( 𝐹 supp 0 ) ) ) → ( 𝑗 𝐹 𝑘 ) = 0 ) |
138 |
133 137
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐴 “ { 𝑗 } ) ∖ ran ( 𝐹 supp 0 ) ) ) → ( 𝑗 𝐹 𝑘 ) = 0 ) |
139 |
138 114
|
suppss2 |
⊢ ( 𝜑 → ( ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) supp 0 ) ⊆ ran ( 𝐹 supp 0 ) ) |
140 |
126 139
|
ssfid |
⊢ ( 𝜑 → ( ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) supp 0 ) ∈ Fin ) |
141 |
114
|
mptexd |
⊢ ( 𝜑 → ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ∈ V ) |
142 |
|
isfsupp |
⊢ ( ( ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ∈ V ∧ 0 ∈ V ) → ( ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) finSupp 0 ↔ ( Fun ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ∧ ( ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) supp 0 ) ∈ Fin ) ) ) |
143 |
141 62 142
|
sylancl |
⊢ ( 𝜑 → ( ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) finSupp 0 ↔ ( Fun ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ∧ ( ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) supp 0 ) ∈ Fin ) ) ) |
144 |
124 140 143
|
mpbir2and |
⊢ ( 𝜑 → ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) finSupp 0 ) |
145 |
|
2ndconst |
⊢ ( 𝑗 ∈ V → ( 2nd ↾ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ) : ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) –1-1-onto→ ( 𝐴 “ { 𝑗 } ) ) |
146 |
115 145
|
mp1i |
⊢ ( 𝜑 → ( 2nd ↾ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ) : ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) –1-1-onto→ ( 𝐴 “ { 𝑗 } ) ) |
147 |
1 2 3 114 122 144 146
|
gsumf1o |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) = ( 𝐺 Σg ( ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ∘ ( 2nd ↾ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ) ) ) ) |
148 |
|
1st2nd2 |
⊢ ( 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) → 𝑥 = 〈 ( 1st ‘ 𝑥 ) , ( 2nd ‘ 𝑥 ) 〉 ) |
149 |
|
xp1st |
⊢ ( 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) → ( 1st ‘ 𝑥 ) ∈ { 𝑗 } ) |
150 |
|
elsni |
⊢ ( ( 1st ‘ 𝑥 ) ∈ { 𝑗 } → ( 1st ‘ 𝑥 ) = 𝑗 ) |
151 |
149 150
|
syl |
⊢ ( 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) → ( 1st ‘ 𝑥 ) = 𝑗 ) |
152 |
151
|
opeq1d |
⊢ ( 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) → 〈 ( 1st ‘ 𝑥 ) , ( 2nd ‘ 𝑥 ) 〉 = 〈 𝑗 , ( 2nd ‘ 𝑥 ) 〉 ) |
153 |
148 152
|
eqtrd |
⊢ ( 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) → 𝑥 = 〈 𝑗 , ( 2nd ‘ 𝑥 ) 〉 ) |
154 |
153
|
fveq2d |
⊢ ( 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 〈 𝑗 , ( 2nd ‘ 𝑥 ) 〉 ) ) |
155 |
|
df-ov |
⊢ ( 𝑗 𝐹 ( 2nd ‘ 𝑥 ) ) = ( 𝐹 ‘ 〈 𝑗 , ( 2nd ‘ 𝑥 ) 〉 ) |
156 |
154 155
|
eqtr4di |
⊢ ( 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) → ( 𝐹 ‘ 𝑥 ) = ( 𝑗 𝐹 ( 2nd ‘ 𝑥 ) ) ) |
157 |
156
|
mpteq2ia |
⊢ ( 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) = ( 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ↦ ( 𝑗 𝐹 ( 2nd ‘ 𝑥 ) ) ) |
158 |
8
|
feqmptd |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
159 |
158
|
reseq1d |
⊢ ( 𝜑 → ( 𝐹 ↾ ( 𝐴 ↾ { 𝑗 } ) ) = ( ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) ↾ ( 𝐴 ↾ { 𝑗 } ) ) ) |
160 |
|
resss |
⊢ ( 𝐴 ↾ { 𝑗 } ) ⊆ 𝐴 |
161 |
|
resmpt |
⊢ ( ( 𝐴 ↾ { 𝑗 } ) ⊆ 𝐴 → ( ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) ↾ ( 𝐴 ↾ { 𝑗 } ) ) = ( 𝑥 ∈ ( 𝐴 ↾ { 𝑗 } ) ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
162 |
160 161
|
ax-mp |
⊢ ( ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) ↾ ( 𝐴 ↾ { 𝑗 } ) ) = ( 𝑥 ∈ ( 𝐴 ↾ { 𝑗 } ) ↦ ( 𝐹 ‘ 𝑥 ) ) |
163 |
|
ressn |
⊢ ( 𝐴 ↾ { 𝑗 } ) = ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) |
164 |
163
|
mpteq1i |
⊢ ( 𝑥 ∈ ( 𝐴 ↾ { 𝑗 } ) ↦ ( 𝐹 ‘ 𝑥 ) ) = ( 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) |
165 |
162 164
|
eqtri |
⊢ ( ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) ↾ ( 𝐴 ↾ { 𝑗 } ) ) = ( 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) |
166 |
159 165
|
eqtrdi |
⊢ ( 𝜑 → ( 𝐹 ↾ ( 𝐴 ↾ { 𝑗 } ) ) = ( 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
167 |
|
xp2nd |
⊢ ( 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) → ( 2nd ‘ 𝑥 ) ∈ ( 𝐴 “ { 𝑗 } ) ) |
168 |
167
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ) → ( 2nd ‘ 𝑥 ) ∈ ( 𝐴 “ { 𝑗 } ) ) |
169 |
|
fo2nd |
⊢ 2nd : V –onto→ V |
170 |
|
fof |
⊢ ( 2nd : V –onto→ V → 2nd : V ⟶ V ) |
171 |
169 170
|
mp1i |
⊢ ( 𝜑 → 2nd : V ⟶ V ) |
172 |
171
|
feqmptd |
⊢ ( 𝜑 → 2nd = ( 𝑥 ∈ V ↦ ( 2nd ‘ 𝑥 ) ) ) |
173 |
172
|
reseq1d |
⊢ ( 𝜑 → ( 2nd ↾ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ) = ( ( 𝑥 ∈ V ↦ ( 2nd ‘ 𝑥 ) ) ↾ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ) ) |
174 |
|
ssv |
⊢ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ⊆ V |
175 |
|
resmpt |
⊢ ( ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ⊆ V → ( ( 𝑥 ∈ V ↦ ( 2nd ‘ 𝑥 ) ) ↾ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ) = ( 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ↦ ( 2nd ‘ 𝑥 ) ) ) |
176 |
174 175
|
ax-mp |
⊢ ( ( 𝑥 ∈ V ↦ ( 2nd ‘ 𝑥 ) ) ↾ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ) = ( 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ↦ ( 2nd ‘ 𝑥 ) ) |
177 |
173 176
|
eqtrdi |
⊢ ( 𝜑 → ( 2nd ↾ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ) = ( 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ↦ ( 2nd ‘ 𝑥 ) ) ) |
178 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) = ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) |
179 |
|
oveq2 |
⊢ ( 𝑘 = ( 2nd ‘ 𝑥 ) → ( 𝑗 𝐹 𝑘 ) = ( 𝑗 𝐹 ( 2nd ‘ 𝑥 ) ) ) |
180 |
168 177 178 179
|
fmptco |
⊢ ( 𝜑 → ( ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ∘ ( 2nd ↾ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ) ) = ( 𝑥 ∈ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ↦ ( 𝑗 𝐹 ( 2nd ‘ 𝑥 ) ) ) ) |
181 |
157 166 180
|
3eqtr4a |
⊢ ( 𝜑 → ( 𝐹 ↾ ( 𝐴 ↾ { 𝑗 } ) ) = ( ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ∘ ( 2nd ↾ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ) ) ) |
182 |
181
|
oveq2d |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑗 } ) ) ) = ( 𝐺 Σg ( ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ∘ ( 2nd ↾ ( { 𝑗 } × ( 𝐴 “ { 𝑗 } ) ) ) ) ) ) |
183 |
147 182
|
eqtr4d |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) = ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑗 } ) ) ) ) |
184 |
112 183
|
chvarvv |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑧 } ) ↦ ( 𝑧 𝐹 𝑘 ) ) ) = ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) ) |
185 |
184
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑧 } ) ↦ ( 𝑧 𝐹 𝑘 ) ) ) = ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) ) |
186 |
185
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( ( 𝐺 Σg ( 𝑗 ∈ 𝑦 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑧 } ) ↦ ( 𝑧 𝐹 𝑘 ) ) ) ) = ( ( 𝐺 Σg ( 𝑗 ∈ 𝑦 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) ) ) |
187 |
107 186
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( 𝐺 Σg ( 𝑗 ∈ ( 𝑦 ∪ { 𝑧 } ) ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) = ( ( 𝐺 Σg ( 𝑗 ∈ 𝑦 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) ) ) |
188 |
92 187
|
eqeq12d |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ ( 𝑦 ∪ { 𝑧 } ) ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ↔ ( ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑦 ) ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) ) = ( ( 𝐺 Σg ( 𝑗 ∈ 𝑦 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ( +g ‘ 𝐺 ) ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ { 𝑧 } ) ) ) ) ) ) |
189 |
48 188
|
syl5ibr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ( ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑦 ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝑦 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ ( 𝑦 ∪ { 𝑧 } ) ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) ) |
190 |
189
|
expcom |
⊢ ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) → ( 𝜑 → ( ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑦 ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝑦 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ ( 𝑦 ∪ { 𝑧 } ) ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) ) ) |
191 |
190
|
a2d |
⊢ ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) → ( ( 𝜑 → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ 𝑦 ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ 𝑦 ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) → ( 𝜑 → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ ( 𝑦 ∪ { 𝑧 } ) ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ ( 𝑦 ∪ { 𝑧 } ) ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) ) ) |
192 |
25 32 39 46 47 191
|
findcard2s |
⊢ ( dom ( 𝐹 supp 0 ) ∈ Fin → ( 𝜑 → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ dom ( 𝐹 supp 0 ) ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ dom ( 𝐹 supp 0 ) ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) ) |
193 |
12 192
|
mpcom |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝐹 ↾ ( 𝐴 ↾ dom ( 𝐹 supp 0 ) ) ) ) = ( 𝐺 Σg ( 𝑗 ∈ dom ( 𝐹 supp 0 ) ↦ ( 𝐺 Σg ( 𝑘 ∈ ( 𝐴 “ { 𝑗 } ) ↦ ( 𝑗 𝐹 𝑘 ) ) ) ) ) ) |