| Step |
Hyp |
Ref |
Expression |
| 1 |
|
gsumfs2d.p |
⊢ Ⅎ 𝑥 𝜑 |
| 2 |
|
gsumfs2d.b |
⊢ 𝐵 = ( Base ‘ 𝑊 ) |
| 3 |
|
gsumfs2d.1 |
⊢ 0 = ( 0g ‘ 𝑊 ) |
| 4 |
|
gsumfs2d.r |
⊢ ( 𝜑 → Rel 𝐴 ) |
| 5 |
|
gsumfs2d.2 |
⊢ ( 𝜑 → 𝐹 finSupp 0 ) |
| 6 |
|
gsumfs2d.w |
⊢ ( 𝜑 → 𝑊 ∈ CMnd ) |
| 7 |
|
gsumfs2d.3 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
| 8 |
|
gsumfs2d.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑋 ) |
| 9 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) → 𝑊 ∈ CMnd ) |
| 10 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) → 𝐴 ∈ 𝑋 ) |
| 11 |
10
|
imaexd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) → ( 𝐴 “ { 𝑥 } ) ∈ V ) |
| 12 |
7
|
ffnd |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
| 13 |
12
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) ∧ 𝑦 ∈ ( ( 𝐴 “ { 𝑥 } ) ∖ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ) → 𝐹 Fn 𝐴 ) |
| 14 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) ∧ 𝑦 ∈ ( ( 𝐴 “ { 𝑥 } ) ∖ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ) → 𝐴 ∈ 𝑋 ) |
| 15 |
3
|
fvexi |
⊢ 0 ∈ V |
| 16 |
15
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) ∧ 𝑦 ∈ ( ( 𝐴 “ { 𝑥 } ) ∖ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ) → 0 ∈ V ) |
| 17 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) ∧ 𝑦 ∈ ( ( 𝐴 “ { 𝑥 } ) ∖ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ) → 𝑦 ∈ ( ( 𝐴 “ { 𝑥 } ) ∖ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ) |
| 18 |
17
|
eldifad |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) ∧ 𝑦 ∈ ( ( 𝐴 “ { 𝑥 } ) ∖ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ) → 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) |
| 19 |
|
vex |
⊢ 𝑥 ∈ V |
| 20 |
|
vex |
⊢ 𝑦 ∈ V |
| 21 |
19 20
|
elimasn |
⊢ ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↔ 〈 𝑥 , 𝑦 〉 ∈ 𝐴 ) |
| 22 |
21
|
biimpi |
⊢ ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) → 〈 𝑥 , 𝑦 〉 ∈ 𝐴 ) |
| 23 |
18 22
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) ∧ 𝑦 ∈ ( ( 𝐴 “ { 𝑥 } ) ∖ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ) → 〈 𝑥 , 𝑦 〉 ∈ 𝐴 ) |
| 24 |
17
|
eldifbd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) ∧ 𝑦 ∈ ( ( 𝐴 “ { 𝑥 } ) ∖ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ) → ¬ 𝑦 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) |
| 25 |
19 20
|
elimasn |
⊢ ( 𝑦 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ↔ 〈 𝑥 , 𝑦 〉 ∈ ( 𝐹 supp 0 ) ) |
| 26 |
25
|
biimpri |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐹 supp 0 ) → 𝑦 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) |
| 27 |
24 26
|
nsyl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) ∧ 𝑦 ∈ ( ( 𝐴 “ { 𝑥 } ) ∖ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ) → ¬ 〈 𝑥 , 𝑦 〉 ∈ ( 𝐹 supp 0 ) ) |
| 28 |
23 27
|
eldifd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) ∧ 𝑦 ∈ ( ( 𝐴 “ { 𝑥 } ) ∖ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ) → 〈 𝑥 , 𝑦 〉 ∈ ( 𝐴 ∖ ( 𝐹 supp 0 ) ) ) |
| 29 |
13 14 16 28
|
fvdifsupp |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) ∧ 𝑦 ∈ ( ( 𝐴 “ { 𝑥 } ) ∖ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ) → ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) = 0 ) |
| 30 |
5
|
fsuppimpd |
⊢ ( 𝜑 → ( 𝐹 supp 0 ) ∈ Fin ) |
| 31 |
30
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) → ( 𝐹 supp 0 ) ∈ Fin ) |
| 32 |
|
imafi2 |
⊢ ( ( 𝐹 supp 0 ) ∈ Fin → ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ∈ Fin ) |
| 33 |
31 32
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) → ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ∈ Fin ) |
| 34 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
| 35 |
22
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) → 〈 𝑥 , 𝑦 〉 ∈ 𝐴 ) |
| 36 |
34 35
|
ffvelcdmd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) → ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ∈ 𝐵 ) |
| 37 |
|
suppssdm |
⊢ ( 𝐹 supp 0 ) ⊆ dom 𝐹 |
| 38 |
37 7
|
fssdm |
⊢ ( 𝜑 → ( 𝐹 supp 0 ) ⊆ 𝐴 ) |
| 39 |
38
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) → ( 𝐹 supp 0 ) ⊆ 𝐴 ) |
| 40 |
|
imass1 |
⊢ ( ( 𝐹 supp 0 ) ⊆ 𝐴 → ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ⊆ ( 𝐴 “ { 𝑥 } ) ) |
| 41 |
39 40
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) → ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ⊆ ( 𝐴 “ { 𝑥 } ) ) |
| 42 |
2 3 9 11 29 33 36 41
|
gsummptres2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) → ( 𝑊 Σg ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ) = ( 𝑊 Σg ( 𝑦 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ) ) |
| 43 |
42
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ dom ( 𝐹 supp 0 ) ↦ ( 𝑊 Σg ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ) ) = ( 𝑥 ∈ dom ( 𝐹 supp 0 ) ↦ ( 𝑊 Σg ( 𝑦 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ) ) ) |
| 44 |
43
|
oveq2d |
⊢ ( 𝜑 → ( 𝑊 Σg ( 𝑥 ∈ dom ( 𝐹 supp 0 ) ↦ ( 𝑊 Σg ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ) ) ) = ( 𝑊 Σg ( 𝑥 ∈ dom ( 𝐹 supp 0 ) ↦ ( 𝑊 Σg ( 𝑦 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ) ) ) ) |
| 45 |
8
|
dmexd |
⊢ ( 𝜑 → dom 𝐴 ∈ V ) |
| 46 |
12
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( dom 𝐴 ∖ dom ( 𝐹 supp 0 ) ) ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) → 𝐹 Fn 𝐴 ) |
| 47 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( dom 𝐴 ∖ dom ( 𝐹 supp 0 ) ) ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) → 𝐴 ∈ 𝑋 ) |
| 48 |
15
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( dom 𝐴 ∖ dom ( 𝐹 supp 0 ) ) ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) → 0 ∈ V ) |
| 49 |
22
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( dom 𝐴 ∖ dom ( 𝐹 supp 0 ) ) ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) → 〈 𝑥 , 𝑦 〉 ∈ 𝐴 ) |
| 50 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( dom 𝐴 ∖ dom ( 𝐹 supp 0 ) ) ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) → 𝑥 ∈ ( dom 𝐴 ∖ dom ( 𝐹 supp 0 ) ) ) |
| 51 |
50
|
eldifbd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( dom 𝐴 ∖ dom ( 𝐹 supp 0 ) ) ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) → ¬ 𝑥 ∈ dom ( 𝐹 supp 0 ) ) |
| 52 |
19 20
|
opeldm |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐹 supp 0 ) → 𝑥 ∈ dom ( 𝐹 supp 0 ) ) |
| 53 |
51 52
|
nsyl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( dom 𝐴 ∖ dom ( 𝐹 supp 0 ) ) ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) → ¬ 〈 𝑥 , 𝑦 〉 ∈ ( 𝐹 supp 0 ) ) |
| 54 |
49 53
|
eldifd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( dom 𝐴 ∖ dom ( 𝐹 supp 0 ) ) ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) → 〈 𝑥 , 𝑦 〉 ∈ ( 𝐴 ∖ ( 𝐹 supp 0 ) ) ) |
| 55 |
46 47 48 54
|
fvdifsupp |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( dom 𝐴 ∖ dom ( 𝐹 supp 0 ) ) ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) → ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) = 0 ) |
| 56 |
55
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( dom 𝐴 ∖ dom ( 𝐹 supp 0 ) ) ) → ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) = ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ 0 ) ) |
| 57 |
56
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( dom 𝐴 ∖ dom ( 𝐹 supp 0 ) ) ) → ( 𝑊 Σg ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ) = ( 𝑊 Σg ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ 0 ) ) ) |
| 58 |
6
|
cmnmndd |
⊢ ( 𝜑 → 𝑊 ∈ Mnd ) |
| 59 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( dom 𝐴 ∖ dom ( 𝐹 supp 0 ) ) ) → 𝐴 ∈ 𝑋 ) |
| 60 |
59
|
imaexd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( dom 𝐴 ∖ dom ( 𝐹 supp 0 ) ) ) → ( 𝐴 “ { 𝑥 } ) ∈ V ) |
| 61 |
3
|
gsumz |
⊢ ( ( 𝑊 ∈ Mnd ∧ ( 𝐴 “ { 𝑥 } ) ∈ V ) → ( 𝑊 Σg ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ 0 ) ) = 0 ) |
| 62 |
58 60 61
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( dom 𝐴 ∖ dom ( 𝐹 supp 0 ) ) ) → ( 𝑊 Σg ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ 0 ) ) = 0 ) |
| 63 |
57 62
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( dom 𝐴 ∖ dom ( 𝐹 supp 0 ) ) ) → ( 𝑊 Σg ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ) = 0 ) |
| 64 |
|
dmfi |
⊢ ( ( 𝐹 supp 0 ) ∈ Fin → dom ( 𝐹 supp 0 ) ∈ Fin ) |
| 65 |
30 64
|
syl |
⊢ ( 𝜑 → dom ( 𝐹 supp 0 ) ∈ Fin ) |
| 66 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) → 𝑊 ∈ CMnd ) |
| 67 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) → 𝐴 ∈ 𝑋 ) |
| 68 |
67
|
imaexd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) → ( 𝐴 “ { 𝑥 } ) ∈ V ) |
| 69 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
| 70 |
22
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) → 〈 𝑥 , 𝑦 〉 ∈ 𝐴 ) |
| 71 |
69 70
|
ffvelcdmd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) → ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ∈ 𝐵 ) |
| 72 |
71
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) → ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) : ( 𝐴 “ { 𝑥 } ) ⟶ 𝐵 ) |
| 73 |
68
|
mptexd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) → ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ∈ V ) |
| 74 |
72
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) → ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) Fn ( 𝐴 “ { 𝑥 } ) ) |
| 75 |
15
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) → 0 ∈ V ) |
| 76 |
30
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) → ( 𝐹 supp 0 ) ∈ Fin ) |
| 77 |
76 32
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) → ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ∈ Fin ) |
| 78 |
|
eqid |
⊢ ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) = ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) |
| 79 |
|
simp-4l |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑡 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑡 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ∧ 𝑦 = 𝑡 ) → 𝜑 ) |
| 80 |
|
simp-4r |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑡 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑡 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ∧ 𝑦 = 𝑡 ) → 𝑥 ∈ dom 𝐴 ) |
| 81 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑡 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑡 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ∧ 𝑦 = 𝑡 ) → 𝑦 = 𝑡 ) |
| 82 |
|
simpllr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑡 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑡 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ∧ 𝑦 = 𝑡 ) → 𝑡 ∈ ( 𝐴 “ { 𝑥 } ) ) |
| 83 |
81 82
|
eqeltrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑡 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑡 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ∧ 𝑦 = 𝑡 ) → 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) |
| 84 |
|
simplr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑡 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑡 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ∧ 𝑦 = 𝑡 ) → ¬ 𝑡 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) |
| 85 |
81 84
|
eqneltrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑡 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑡 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ∧ 𝑦 = 𝑡 ) → ¬ 𝑦 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) |
| 86 |
12
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑦 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) → 𝐹 Fn 𝐴 ) |
| 87 |
8
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑦 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) → 𝐴 ∈ 𝑋 ) |
| 88 |
15
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑦 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) → 0 ∈ V ) |
| 89 |
70
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑦 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) → 〈 𝑥 , 𝑦 〉 ∈ 𝐴 ) |
| 90 |
26
|
con3i |
⊢ ( ¬ 𝑦 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) → ¬ 〈 𝑥 , 𝑦 〉 ∈ ( 𝐹 supp 0 ) ) |
| 91 |
90
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑦 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) → ¬ 〈 𝑥 , 𝑦 〉 ∈ ( 𝐹 supp 0 ) ) |
| 92 |
89 91
|
eldifd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑦 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) → 〈 𝑥 , 𝑦 〉 ∈ ( 𝐴 ∖ ( 𝐹 supp 0 ) ) ) |
| 93 |
86 87 88 92
|
fvdifsupp |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑦 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) → ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) = 0 ) |
| 94 |
79 80 83 85 93
|
syl1111anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑡 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑡 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) ∧ 𝑦 = 𝑡 ) → ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) = 0 ) |
| 95 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑡 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑡 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) → 𝑡 ∈ ( 𝐴 “ { 𝑥 } ) ) |
| 96 |
15
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑡 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑡 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) → 0 ∈ V ) |
| 97 |
78 94 95 96
|
fvmptd2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑡 ∈ ( 𝐴 “ { 𝑥 } ) ) ∧ ¬ 𝑡 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ) → ( ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ‘ 𝑡 ) = 0 ) |
| 98 |
97
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑡 ∈ ( 𝐴 “ { 𝑥 } ) ) → ( ¬ 𝑡 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) → ( ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ‘ 𝑡 ) = 0 ) ) |
| 99 |
98
|
orrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) ∧ 𝑡 ∈ ( 𝐴 “ { 𝑥 } ) ) → ( 𝑡 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ∨ ( ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ‘ 𝑡 ) = 0 ) ) |
| 100 |
73 74 75 77 99
|
finnzfsuppd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) → ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) finSupp 0 ) |
| 101 |
2 3 66 68 72 100
|
gsumcl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ dom 𝐴 ) → ( 𝑊 Σg ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ) ∈ 𝐵 ) |
| 102 |
|
dmss |
⊢ ( ( 𝐹 supp 0 ) ⊆ 𝐴 → dom ( 𝐹 supp 0 ) ⊆ dom 𝐴 ) |
| 103 |
38 102
|
syl |
⊢ ( 𝜑 → dom ( 𝐹 supp 0 ) ⊆ dom 𝐴 ) |
| 104 |
2 3 6 45 63 65 101 103
|
gsummptres2 |
⊢ ( 𝜑 → ( 𝑊 Σg ( 𝑥 ∈ dom 𝐴 ↦ ( 𝑊 Σg ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ) ) ) = ( 𝑊 Σg ( 𝑥 ∈ dom ( 𝐹 supp 0 ) ↦ ( 𝑊 Σg ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ) ) ) ) |
| 105 |
7 38
|
feqresmpt |
⊢ ( 𝜑 → ( 𝐹 ↾ ( 𝐹 supp 0 ) ) = ( 𝑧 ∈ ( 𝐹 supp 0 ) ↦ ( 𝐹 ‘ 𝑧 ) ) ) |
| 106 |
105
|
oveq2d |
⊢ ( 𝜑 → ( 𝑊 Σg ( 𝐹 ↾ ( 𝐹 supp 0 ) ) ) = ( 𝑊 Σg ( 𝑧 ∈ ( 𝐹 supp 0 ) ↦ ( 𝐹 ‘ 𝑧 ) ) ) ) |
| 107 |
|
ssidd |
⊢ ( 𝜑 → ( 𝐹 supp 0 ) ⊆ ( 𝐹 supp 0 ) ) |
| 108 |
2 3 6 8 7 107 5
|
gsumres |
⊢ ( 𝜑 → ( 𝑊 Σg ( 𝐹 ↾ ( 𝐹 supp 0 ) ) ) = ( 𝑊 Σg 𝐹 ) ) |
| 109 |
|
nfcv |
⊢ Ⅎ 𝑦 ( 𝐹 ‘ 𝑧 ) |
| 110 |
|
fveq2 |
⊢ ( 𝑧 = 〈 𝑥 , 𝑦 〉 → ( 𝐹 ‘ 𝑧 ) = ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) |
| 111 |
|
relss |
⊢ ( ( 𝐹 supp 0 ) ⊆ 𝐴 → ( Rel 𝐴 → Rel ( 𝐹 supp 0 ) ) ) |
| 112 |
38 4 111
|
sylc |
⊢ ( 𝜑 → Rel ( 𝐹 supp 0 ) ) |
| 113 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐹 supp 0 ) ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
| 114 |
38
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐹 supp 0 ) ) → 𝑧 ∈ 𝐴 ) |
| 115 |
113 114
|
ffvelcdmd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐹 supp 0 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝐵 ) |
| 116 |
109 1 2 110 112 30 6 115
|
gsummpt2d |
⊢ ( 𝜑 → ( 𝑊 Σg ( 𝑧 ∈ ( 𝐹 supp 0 ) ↦ ( 𝐹 ‘ 𝑧 ) ) ) = ( 𝑊 Σg ( 𝑥 ∈ dom ( 𝐹 supp 0 ) ↦ ( 𝑊 Σg ( 𝑦 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ) ) ) ) |
| 117 |
106 108 116
|
3eqtr3d |
⊢ ( 𝜑 → ( 𝑊 Σg 𝐹 ) = ( 𝑊 Σg ( 𝑥 ∈ dom ( 𝐹 supp 0 ) ↦ ( 𝑊 Σg ( 𝑦 ∈ ( ( 𝐹 supp 0 ) “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ) ) ) ) |
| 118 |
44 104 117
|
3eqtr4rd |
⊢ ( 𝜑 → ( 𝑊 Σg 𝐹 ) = ( 𝑊 Σg ( 𝑥 ∈ dom 𝐴 ↦ ( 𝑊 Σg ( 𝑦 ∈ ( 𝐴 “ { 𝑥 } ) ↦ ( 𝐹 ‘ 〈 𝑥 , 𝑦 〉 ) ) ) ) ) ) |