Step |
Hyp |
Ref |
Expression |
1 |
|
elrspunidl.n |
⊢ 𝑁 = ( RSpan ‘ 𝑅 ) |
2 |
|
elrspunidl.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
3 |
|
elrspunidl.1 |
⊢ 0 = ( 0g ‘ 𝑅 ) |
4 |
|
elrspunidl.x |
⊢ · = ( .r ‘ 𝑅 ) |
5 |
|
elrspunidl.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
6 |
|
elrspunidl.i |
⊢ ( 𝜑 → 𝑆 ⊆ ( LIdeal ‘ 𝑅 ) ) |
7 |
6
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑆 ) → 𝑖 ∈ ( LIdeal ‘ 𝑅 ) ) |
8 |
|
eqid |
⊢ ( LIdeal ‘ 𝑅 ) = ( LIdeal ‘ 𝑅 ) |
9 |
2 8
|
lidlss |
⊢ ( 𝑖 ∈ ( LIdeal ‘ 𝑅 ) → 𝑖 ⊆ 𝐵 ) |
10 |
7 9
|
syl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑆 ) → 𝑖 ⊆ 𝐵 ) |
11 |
10
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑖 ∈ 𝑆 𝑖 ⊆ 𝐵 ) |
12 |
|
unissb |
⊢ ( ∪ 𝑆 ⊆ 𝐵 ↔ ∀ 𝑖 ∈ 𝑆 𝑖 ⊆ 𝐵 ) |
13 |
11 12
|
sylibr |
⊢ ( 𝜑 → ∪ 𝑆 ⊆ 𝐵 ) |
14 |
1 2 3 4 5 13
|
elrsp |
⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑁 ‘ ∪ 𝑆 ) ↔ ∃ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ( 𝑏 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ) ) |
15 |
|
fvexd |
⊢ ( 𝜑 → ( LIdeal ‘ 𝑅 ) ∈ V ) |
16 |
15 6
|
ssexd |
⊢ ( 𝜑 → 𝑆 ∈ V ) |
17 |
16
|
uniexd |
⊢ ( 𝜑 → ∪ 𝑆 ∈ V ) |
18 |
|
eluni2 |
⊢ ( 𝑗 ∈ ∪ 𝑆 ↔ ∃ 𝑖 ∈ 𝑆 𝑗 ∈ 𝑖 ) |
19 |
18
|
biimpi |
⊢ ( 𝑗 ∈ ∪ 𝑆 → ∃ 𝑖 ∈ 𝑆 𝑗 ∈ 𝑖 ) |
20 |
19
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ∪ 𝑆 ) → ∃ 𝑖 ∈ 𝑆 𝑗 ∈ 𝑖 ) |
21 |
20
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑗 ∈ ∪ 𝑆 ∃ 𝑖 ∈ 𝑆 𝑗 ∈ 𝑖 ) |
22 |
|
eleq2 |
⊢ ( 𝑖 = ( 𝑓 ‘ 𝑗 ) → ( 𝑗 ∈ 𝑖 ↔ 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ) |
23 |
22
|
ac6sg |
⊢ ( ∪ 𝑆 ∈ V → ( ∀ 𝑗 ∈ ∪ 𝑆 ∃ 𝑖 ∈ 𝑆 𝑗 ∈ 𝑖 → ∃ 𝑓 ( 𝑓 : ∪ 𝑆 ⟶ 𝑆 ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ) ) |
24 |
17 21 23
|
sylc |
⊢ ( 𝜑 → ∃ 𝑓 ( 𝑓 : ∪ 𝑆 ⟶ 𝑆 ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ) |
25 |
24
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) → ∃ 𝑓 ( 𝑓 : ∪ 𝑆 ⟶ 𝑆 ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ) |
26 |
|
simp-5l |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → 𝜑 ) |
27 |
26
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) → 𝜑 ) |
28 |
|
ringcmn |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ CMnd ) |
29 |
27 5 28
|
3syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) → 𝑅 ∈ CMnd ) |
30 |
|
vex |
⊢ 𝑓 ∈ V |
31 |
|
cnvexg |
⊢ ( 𝑓 ∈ V → ◡ 𝑓 ∈ V ) |
32 |
|
imaexg |
⊢ ( ◡ 𝑓 ∈ V → ( ◡ 𝑓 “ { 𝑖 } ) ∈ V ) |
33 |
30 31 32
|
mp2b |
⊢ ( ◡ 𝑓 “ { 𝑖 } ) ∈ V |
34 |
33
|
a1i |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) → ( ◡ 𝑓 “ { 𝑖 } ) ∈ V ) |
35 |
5
|
ad7antr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ) → 𝑅 ∈ Ring ) |
36 |
|
elmapi |
⊢ ( 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) → 𝑏 : ∪ 𝑆 ⟶ 𝐵 ) |
37 |
36
|
ad7antlr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ) → 𝑏 : ∪ 𝑆 ⟶ 𝐵 ) |
38 |
|
cnvimass |
⊢ ( ◡ 𝑓 “ { 𝑖 } ) ⊆ dom 𝑓 |
39 |
|
fdm |
⊢ ( 𝑓 : ∪ 𝑆 ⟶ 𝑆 → dom 𝑓 = ∪ 𝑆 ) |
40 |
38 39
|
sseqtrid |
⊢ ( 𝑓 : ∪ 𝑆 ⟶ 𝑆 → ( ◡ 𝑓 “ { 𝑖 } ) ⊆ ∪ 𝑆 ) |
41 |
40
|
ad3antlr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) → ( ◡ 𝑓 “ { 𝑖 } ) ⊆ ∪ 𝑆 ) |
42 |
41
|
sselda |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ) → 𝑙 ∈ ∪ 𝑆 ) |
43 |
37 42
|
ffvelrnd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ) → ( 𝑏 ‘ 𝑙 ) ∈ 𝐵 ) |
44 |
13
|
ad7antr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ) → ∪ 𝑆 ⊆ 𝐵 ) |
45 |
44 42
|
sseldd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ) → 𝑙 ∈ 𝐵 ) |
46 |
2 4
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑏 ‘ 𝑙 ) ∈ 𝐵 ∧ 𝑙 ∈ 𝐵 ) → ( ( 𝑏 ‘ 𝑙 ) · 𝑙 ) ∈ 𝐵 ) |
47 |
35 43 45 46
|
syl3anc |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ) → ( ( 𝑏 ‘ 𝑙 ) · 𝑙 ) ∈ 𝐵 ) |
48 |
|
fveq2 |
⊢ ( 𝑗 = 𝑙 → ( 𝑏 ‘ 𝑗 ) = ( 𝑏 ‘ 𝑙 ) ) |
49 |
|
id |
⊢ ( 𝑗 = 𝑙 → 𝑗 = 𝑙 ) |
50 |
48 49
|
oveq12d |
⊢ ( 𝑗 = 𝑙 → ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) = ( ( 𝑏 ‘ 𝑙 ) · 𝑙 ) ) |
51 |
50
|
cbvmptv |
⊢ ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) = ( 𝑙 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑙 ) · 𝑙 ) ) |
52 |
47 51
|
fmptd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) → ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) : ( ◡ 𝑓 “ { 𝑖 } ) ⟶ 𝐵 ) |
53 |
34
|
mptexd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) → ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ∈ V ) |
54 |
52
|
ffund |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) → Fun ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) |
55 |
|
simp-5r |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) → 𝑏 finSupp 0 ) |
56 |
|
nfv |
⊢ Ⅎ 𝑗 ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) |
57 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑅 |
58 |
|
nfcv |
⊢ Ⅎ 𝑗 Σg |
59 |
|
nfmpt1 |
⊢ Ⅎ 𝑗 ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) |
60 |
57 58 59
|
nfov |
⊢ Ⅎ 𝑗 ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) |
61 |
60
|
nfeq2 |
⊢ Ⅎ 𝑗 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) |
62 |
56 61
|
nfan |
⊢ Ⅎ 𝑗 ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) |
63 |
|
nfv |
⊢ Ⅎ 𝑗 𝑓 : ∪ 𝑆 ⟶ 𝑆 |
64 |
62 63
|
nfan |
⊢ Ⅎ 𝑗 ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) |
65 |
|
nfra1 |
⊢ Ⅎ 𝑗 ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) |
66 |
64 65
|
nfan |
⊢ Ⅎ 𝑗 ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) |
67 |
|
nfv |
⊢ Ⅎ 𝑗 𝑖 ∈ 𝑆 |
68 |
66 67
|
nfan |
⊢ Ⅎ 𝑗 ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) |
69 |
|
nfcv |
⊢ Ⅎ 𝑗 ( ◡ 𝑓 “ { 𝑖 } ) |
70 |
|
nfcv |
⊢ Ⅎ 𝑗 ( 𝑏 supp 0 ) |
71 |
36
|
ad7antlr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑖 } ) ∖ ( 𝑏 supp 0 ) ) ) → 𝑏 : ∪ 𝑆 ⟶ 𝐵 ) |
72 |
71
|
ffnd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑖 } ) ∖ ( 𝑏 supp 0 ) ) ) → 𝑏 Fn ∪ 𝑆 ) |
73 |
26 17
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ∪ 𝑆 ∈ V ) |
74 |
73
|
ad2antrr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑖 } ) ∖ ( 𝑏 supp 0 ) ) ) → ∪ 𝑆 ∈ V ) |
75 |
3
|
fvexi |
⊢ 0 ∈ V |
76 |
75
|
a1i |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑖 } ) ∖ ( 𝑏 supp 0 ) ) ) → 0 ∈ V ) |
77 |
41
|
ssdifd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) → ( ( ◡ 𝑓 “ { 𝑖 } ) ∖ ( 𝑏 supp 0 ) ) ⊆ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) |
78 |
77
|
sselda |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑖 } ) ∖ ( 𝑏 supp 0 ) ) ) → 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) |
79 |
72 74 76 78
|
fvdifsupp |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑖 } ) ∖ ( 𝑏 supp 0 ) ) ) → ( 𝑏 ‘ 𝑗 ) = 0 ) |
80 |
79
|
oveq1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑖 } ) ∖ ( 𝑏 supp 0 ) ) ) → ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) = ( 0 · 𝑗 ) ) |
81 |
5
|
ad7antr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑖 } ) ∖ ( 𝑏 supp 0 ) ) ) → 𝑅 ∈ Ring ) |
82 |
13
|
ad7antr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑖 } ) ∖ ( 𝑏 supp 0 ) ) ) → ∪ 𝑆 ⊆ 𝐵 ) |
83 |
78
|
eldifad |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑖 } ) ∖ ( 𝑏 supp 0 ) ) ) → 𝑗 ∈ ∪ 𝑆 ) |
84 |
82 83
|
sseldd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑖 } ) ∖ ( 𝑏 supp 0 ) ) ) → 𝑗 ∈ 𝐵 ) |
85 |
2 4 3
|
ringlz |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ 𝐵 ) → ( 0 · 𝑗 ) = 0 ) |
86 |
81 84 85
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑖 } ) ∖ ( 𝑏 supp 0 ) ) ) → ( 0 · 𝑗 ) = 0 ) |
87 |
80 86
|
eqtrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑖 } ) ∖ ( 𝑏 supp 0 ) ) ) → ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) = 0 ) |
88 |
68 69 70 87 34
|
suppss2f |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) → ( ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) supp 0 ) ⊆ ( 𝑏 supp 0 ) ) |
89 |
|
fsuppsssupp |
⊢ ( ( ( ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ∈ V ∧ Fun ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ∧ ( 𝑏 finSupp 0 ∧ ( ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) supp 0 ) ⊆ ( 𝑏 supp 0 ) ) ) → ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) finSupp 0 ) |
90 |
53 54 55 88 89
|
syl22anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) → ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) finSupp 0 ) |
91 |
2 3 29 34 52 90
|
gsumcl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) → ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ∈ 𝐵 ) |
92 |
91
|
fmpttd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) : 𝑆 ⟶ 𝐵 ) |
93 |
2
|
fvexi |
⊢ 𝐵 ∈ V |
94 |
93
|
a1i |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
95 |
94 16
|
elmapd |
⊢ ( 𝜑 → ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∈ ( 𝐵 ↑m 𝑆 ) ↔ ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) : 𝑆 ⟶ 𝐵 ) ) |
96 |
95
|
biimpar |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) : 𝑆 ⟶ 𝐵 ) → ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∈ ( 𝐵 ↑m 𝑆 ) ) |
97 |
26 92 96
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∈ ( 𝐵 ↑m 𝑆 ) ) |
98 |
|
breq1 |
⊢ ( 𝑎 = ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) → ( 𝑎 finSupp 0 ↔ ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) finSupp 0 ) ) |
99 |
|
oveq2 |
⊢ ( 𝑎 = ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) → ( 𝑅 Σg 𝑎 ) = ( 𝑅 Σg ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ) ) |
100 |
99
|
eqeq2d |
⊢ ( 𝑎 = ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) → ( 𝑋 = ( 𝑅 Σg 𝑎 ) ↔ 𝑋 = ( 𝑅 Σg ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ) ) ) |
101 |
|
fveq1 |
⊢ ( 𝑎 = ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) → ( 𝑎 ‘ 𝑘 ) = ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ‘ 𝑘 ) ) |
102 |
101
|
eleq1d |
⊢ ( 𝑎 = ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) → ( ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ↔ ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ‘ 𝑘 ) ∈ 𝑘 ) ) |
103 |
102
|
ralbidv |
⊢ ( 𝑎 = ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) → ( ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ↔ ∀ 𝑘 ∈ 𝑆 ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ‘ 𝑘 ) ∈ 𝑘 ) ) |
104 |
98 100 103
|
3anbi123d |
⊢ ( 𝑎 = ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) → ( ( 𝑎 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ↔ ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ‘ 𝑘 ) ∈ 𝑘 ) ) ) |
105 |
104
|
adantl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑎 = ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ) → ( ( 𝑎 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ↔ ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ‘ 𝑘 ) ∈ 𝑘 ) ) ) |
106 |
26 16
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → 𝑆 ∈ V ) |
107 |
106
|
mptexd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∈ V ) |
108 |
75
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → 0 ∈ V ) |
109 |
|
funmpt |
⊢ Fun ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) |
110 |
109
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → Fun ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ) |
111 |
|
simplr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) |
112 |
111
|
ffund |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → Fun 𝑓 ) |
113 |
|
simp-4r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → 𝑏 finSupp 0 ) |
114 |
113
|
fsuppimpd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( 𝑏 supp 0 ) ∈ Fin ) |
115 |
|
imafi |
⊢ ( ( Fun 𝑓 ∧ ( 𝑏 supp 0 ) ∈ Fin ) → ( 𝑓 “ ( 𝑏 supp 0 ) ) ∈ Fin ) |
116 |
112 114 115
|
syl2anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( 𝑓 “ ( 𝑏 supp 0 ) ) ∈ Fin ) |
117 |
|
nfv |
⊢ Ⅎ 𝑗 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) |
118 |
66 117
|
nfan |
⊢ Ⅎ 𝑗 ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) |
119 |
|
simpllr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) |
120 |
119
|
ffund |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → Fun 𝑓 ) |
121 |
|
snssi |
⊢ ( 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) → { 𝑖 } ⊆ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) |
122 |
121
|
adantl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → { 𝑖 } ⊆ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) |
123 |
|
sspreima |
⊢ ( ( Fun 𝑓 ∧ { 𝑖 } ⊆ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( ◡ 𝑓 “ { 𝑖 } ) ⊆ ( ◡ 𝑓 “ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) ) |
124 |
120 122 123
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( ◡ 𝑓 “ { 𝑖 } ) ⊆ ( ◡ 𝑓 “ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) ) |
125 |
|
difpreima |
⊢ ( Fun 𝑓 → ( ◡ 𝑓 “ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) = ( ( ◡ 𝑓 “ 𝑆 ) ∖ ( ◡ 𝑓 “ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) ) |
126 |
120 125
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( ◡ 𝑓 “ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) = ( ( ◡ 𝑓 “ 𝑆 ) ∖ ( ◡ 𝑓 “ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) ) |
127 |
124 126
|
sseqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( ◡ 𝑓 “ { 𝑖 } ) ⊆ ( ( ◡ 𝑓 “ 𝑆 ) ∖ ( ◡ 𝑓 “ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) ) |
128 |
|
suppssdm |
⊢ ( 𝑏 supp 0 ) ⊆ dom 𝑏 |
129 |
36
|
ad6antlr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → 𝑏 : ∪ 𝑆 ⟶ 𝐵 ) |
130 |
128 129
|
fssdm |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( 𝑏 supp 0 ) ⊆ ∪ 𝑆 ) |
131 |
119
|
fdmd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → dom 𝑓 = ∪ 𝑆 ) |
132 |
130 131
|
sseqtrrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( 𝑏 supp 0 ) ⊆ dom 𝑓 ) |
133 |
|
sseqin2 |
⊢ ( ( 𝑏 supp 0 ) ⊆ dom 𝑓 ↔ ( dom 𝑓 ∩ ( 𝑏 supp 0 ) ) = ( 𝑏 supp 0 ) ) |
134 |
133
|
biimpi |
⊢ ( ( 𝑏 supp 0 ) ⊆ dom 𝑓 → ( dom 𝑓 ∩ ( 𝑏 supp 0 ) ) = ( 𝑏 supp 0 ) ) |
135 |
|
dminss |
⊢ ( dom 𝑓 ∩ ( 𝑏 supp 0 ) ) ⊆ ( ◡ 𝑓 “ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) |
136 |
134 135
|
eqsstrrdi |
⊢ ( ( 𝑏 supp 0 ) ⊆ dom 𝑓 → ( 𝑏 supp 0 ) ⊆ ( ◡ 𝑓 “ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) |
137 |
132 136
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( 𝑏 supp 0 ) ⊆ ( ◡ 𝑓 “ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) |
138 |
137
|
sscond |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( ( ◡ 𝑓 “ 𝑆 ) ∖ ( ◡ 𝑓 “ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) ⊆ ( ( ◡ 𝑓 “ 𝑆 ) ∖ ( 𝑏 supp 0 ) ) ) |
139 |
127 138
|
sstrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( ◡ 𝑓 “ { 𝑖 } ) ⊆ ( ( ◡ 𝑓 “ 𝑆 ) ∖ ( 𝑏 supp 0 ) ) ) |
140 |
|
fimacnv |
⊢ ( 𝑓 : ∪ 𝑆 ⟶ 𝑆 → ( ◡ 𝑓 “ 𝑆 ) = ∪ 𝑆 ) |
141 |
119 140
|
syl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( ◡ 𝑓 “ 𝑆 ) = ∪ 𝑆 ) |
142 |
141
|
difeq1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( ( ◡ 𝑓 “ 𝑆 ) ∖ ( 𝑏 supp 0 ) ) = ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) |
143 |
139 142
|
sseqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( ◡ 𝑓 “ { 𝑖 } ) ⊆ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) |
144 |
143
|
sselda |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) ∧ 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ) → 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) |
145 |
|
ssidd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( 𝑏 supp 0 ) ⊆ ( 𝑏 supp 0 ) ) |
146 |
73
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ∪ 𝑆 ∈ V ) |
147 |
75
|
a1i |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → 0 ∈ V ) |
148 |
129 145 146 147
|
suppssr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) → ( 𝑏 ‘ 𝑗 ) = 0 ) |
149 |
144 148
|
syldan |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) ∧ 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ) → ( 𝑏 ‘ 𝑗 ) = 0 ) |
150 |
149
|
oveq1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) ∧ 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ) → ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) = ( 0 · 𝑗 ) ) |
151 |
5
|
ad7antr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) ∧ 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ) → 𝑅 ∈ Ring ) |
152 |
13
|
ad7antr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) ∧ 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ) → ∪ 𝑆 ⊆ 𝐵 ) |
153 |
40
|
ad3antlr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( ◡ 𝑓 “ { 𝑖 } ) ⊆ ∪ 𝑆 ) |
154 |
153
|
sselda |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) ∧ 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ) → 𝑗 ∈ ∪ 𝑆 ) |
155 |
152 154
|
sseldd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) ∧ 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ) → 𝑗 ∈ 𝐵 ) |
156 |
151 155 85
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) ∧ 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ) → ( 0 · 𝑗 ) = 0 ) |
157 |
150 156
|
eqtrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) ∧ 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ) → ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) = 0 ) |
158 |
118 157
|
mpteq2da |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) = ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ 0 ) ) |
159 |
158
|
oveq2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) = ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ 0 ) ) ) |
160 |
5 28
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ CMnd ) |
161 |
160
|
cmnmndd |
⊢ ( 𝜑 → 𝑅 ∈ Mnd ) |
162 |
161
|
ad6antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → 𝑅 ∈ Mnd ) |
163 |
3
|
gsumz |
⊢ ( ( 𝑅 ∈ Mnd ∧ ( ◡ 𝑓 “ { 𝑖 } ) ∈ V ) → ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ 0 ) ) = 0 ) |
164 |
162 33 163
|
sylancl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ 0 ) ) = 0 ) |
165 |
159 164
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ ( 𝑆 ∖ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) ) → ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) = 0 ) |
166 |
165 106
|
suppss2 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) supp 0 ) ⊆ ( 𝑓 “ ( 𝑏 supp 0 ) ) ) |
167 |
116 166
|
ssfid |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) supp 0 ) ∈ Fin ) |
168 |
|
isfsupp |
⊢ ( ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∈ V ∧ 0 ∈ V ) → ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) finSupp 0 ↔ ( Fun ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) supp 0 ) ∈ Fin ) ) ) |
169 |
168
|
biimpar |
⊢ ( ( ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∈ V ∧ 0 ∈ V ) ∧ ( Fun ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) supp 0 ) ∈ Fin ) ) → ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) finSupp 0 ) |
170 |
107 108 110 167 169
|
syl22anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) finSupp 0 ) |
171 |
|
simpllr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) |
172 |
26 160
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → 𝑅 ∈ CMnd ) |
173 |
5
|
ad6antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ∪ 𝑆 ) → 𝑅 ∈ Ring ) |
174 |
36
|
ad5antlr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → 𝑏 : ∪ 𝑆 ⟶ 𝐵 ) |
175 |
174
|
ffvelrnda |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ∪ 𝑆 ) → ( 𝑏 ‘ 𝑗 ) ∈ 𝐵 ) |
176 |
26 13
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ∪ 𝑆 ⊆ 𝐵 ) |
177 |
176
|
sselda |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ∪ 𝑆 ) → 𝑗 ∈ 𝐵 ) |
178 |
2 4
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑏 ‘ 𝑗 ) ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) → ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ∈ 𝐵 ) |
179 |
173 175 177 178
|
syl3anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ∪ 𝑆 ) → ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ∈ 𝐵 ) |
180 |
|
eqid |
⊢ ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) = ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) |
181 |
66 179 180
|
fmptdf |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) : ∪ 𝑆 ⟶ 𝐵 ) |
182 |
73
|
mptexd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ∈ V ) |
183 |
|
funmpt |
⊢ Fun ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) |
184 |
183
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → Fun ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) |
185 |
|
nfcv |
⊢ Ⅎ 𝑗 ∪ 𝑆 |
186 |
174
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) → 𝑏 : ∪ 𝑆 ⟶ 𝐵 ) |
187 |
186
|
ffnd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) → 𝑏 Fn ∪ 𝑆 ) |
188 |
73
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) → ∪ 𝑆 ∈ V ) |
189 |
75
|
a1i |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) → 0 ∈ V ) |
190 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) → 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) |
191 |
187 188 189 190
|
fvdifsupp |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) → ( 𝑏 ‘ 𝑗 ) = 0 ) |
192 |
191
|
oveq1d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) → ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) = ( 0 · 𝑗 ) ) |
193 |
5
|
ad6antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) → 𝑅 ∈ Ring ) |
194 |
176
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) → ∪ 𝑆 ⊆ 𝐵 ) |
195 |
190
|
eldifad |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) → 𝑗 ∈ ∪ 𝑆 ) |
196 |
194 195
|
sseldd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) → 𝑗 ∈ 𝐵 ) |
197 |
193 196 85
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) → ( 0 · 𝑗 ) = 0 ) |
198 |
192 197
|
eqtrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) → ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) = 0 ) |
199 |
66 185 70 198 73
|
suppss2f |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) supp 0 ) ⊆ ( 𝑏 supp 0 ) ) |
200 |
|
fsuppsssupp |
⊢ ( ( ( ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ∈ V ∧ Fun ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ∧ ( 𝑏 finSupp 0 ∧ ( ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) supp 0 ) ⊆ ( 𝑏 supp 0 ) ) ) → ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) finSupp 0 ) |
201 |
182 184 113 199 200
|
syl22anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) finSupp 0 ) |
202 |
|
sndisj |
⊢ Disj 𝑖 ∈ 𝑆 { 𝑖 } |
203 |
|
disjpreima |
⊢ ( ( Fun 𝑓 ∧ Disj 𝑖 ∈ 𝑆 { 𝑖 } ) → Disj 𝑖 ∈ 𝑆 ( ◡ 𝑓 “ { 𝑖 } ) ) |
204 |
112 202 203
|
sylancl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → Disj 𝑖 ∈ 𝑆 ( ◡ 𝑓 “ { 𝑖 } ) ) |
205 |
|
iunid |
⊢ ∪ 𝑖 ∈ 𝑆 { 𝑖 } = 𝑆 |
206 |
205
|
imaeq2i |
⊢ ( ◡ 𝑓 “ ∪ 𝑖 ∈ 𝑆 { 𝑖 } ) = ( ◡ 𝑓 “ 𝑆 ) |
207 |
|
iunpreima |
⊢ ( Fun 𝑓 → ( ◡ 𝑓 “ ∪ 𝑖 ∈ 𝑆 { 𝑖 } ) = ∪ 𝑖 ∈ 𝑆 ( ◡ 𝑓 “ { 𝑖 } ) ) |
208 |
112 207
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( ◡ 𝑓 “ ∪ 𝑖 ∈ 𝑆 { 𝑖 } ) = ∪ 𝑖 ∈ 𝑆 ( ◡ 𝑓 “ { 𝑖 } ) ) |
209 |
140
|
ad2antlr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( ◡ 𝑓 “ 𝑆 ) = ∪ 𝑆 ) |
210 |
206 208 209
|
3eqtr3a |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ∪ 𝑖 ∈ 𝑆 ( ◡ 𝑓 “ { 𝑖 } ) = ∪ 𝑆 ) |
211 |
2 3 172 73 106 181 201 204 210
|
gsumpart |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) = ( 𝑅 Σg ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ↾ ( ◡ 𝑓 “ { 𝑖 } ) ) ) ) ) ) |
212 |
41
|
resmptd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) → ( ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ↾ ( ◡ 𝑓 “ { 𝑖 } ) ) = ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) |
213 |
212
|
oveq2d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑖 ∈ 𝑆 ) → ( 𝑅 Σg ( ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ↾ ( ◡ 𝑓 “ { 𝑖 } ) ) ) = ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) |
214 |
213
|
mpteq2dva |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ↾ ( ◡ 𝑓 “ { 𝑖 } ) ) ) ) = ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ) |
215 |
214
|
oveq2d |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( 𝑅 Σg ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ↾ ( ◡ 𝑓 “ { 𝑖 } ) ) ) ) ) = ( 𝑅 Σg ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ) ) |
216 |
171 211 215
|
3eqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → 𝑋 = ( 𝑅 Σg ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ) ) |
217 |
|
eqid |
⊢ ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) = ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) |
218 |
|
simpr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑖 = 𝑘 ) → 𝑖 = 𝑘 ) |
219 |
218
|
sneqd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑖 = 𝑘 ) → { 𝑖 } = { 𝑘 } ) |
220 |
219
|
imaeq2d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑖 = 𝑘 ) → ( ◡ 𝑓 “ { 𝑖 } ) = ( ◡ 𝑓 “ { 𝑘 } ) ) |
221 |
220
|
mpteq1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑖 = 𝑘 ) → ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) = ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) |
222 |
221
|
oveq2d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑖 = 𝑘 ) → ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) = ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) |
223 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → 𝑘 ∈ 𝑆 ) |
224 |
|
ovexd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ∈ V ) |
225 |
217 222 223 224
|
fvmptd2 |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ‘ 𝑘 ) = ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) |
226 |
160
|
ad6antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → 𝑅 ∈ CMnd ) |
227 |
30
|
cnvex |
⊢ ◡ 𝑓 ∈ V |
228 |
227
|
imaex |
⊢ ( ◡ 𝑓 “ { 𝑘 } ) ∈ V |
229 |
228
|
a1i |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → ( ◡ 𝑓 “ { 𝑘 } ) ∈ V ) |
230 |
5
|
ad6antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → 𝑅 ∈ Ring ) |
231 |
26 6
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → 𝑆 ⊆ ( LIdeal ‘ 𝑅 ) ) |
232 |
231
|
sselda |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) |
233 |
8
|
lidlsubg |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) → 𝑘 ∈ ( SubGrp ‘ 𝑅 ) ) |
234 |
|
subgsubm |
⊢ ( 𝑘 ∈ ( SubGrp ‘ 𝑅 ) → 𝑘 ∈ ( SubMnd ‘ 𝑅 ) ) |
235 |
233 234
|
syl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) → 𝑘 ∈ ( SubMnd ‘ 𝑅 ) ) |
236 |
230 232 235
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → 𝑘 ∈ ( SubMnd ‘ 𝑅 ) ) |
237 |
230
|
adantr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ) → 𝑅 ∈ Ring ) |
238 |
232
|
adantr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ) → 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) |
239 |
36
|
ad7antlr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ) → 𝑏 : ∪ 𝑆 ⟶ 𝐵 ) |
240 |
|
cnvimass |
⊢ ( ◡ 𝑓 “ { 𝑘 } ) ⊆ dom 𝑓 |
241 |
240 39
|
sseqtrid |
⊢ ( 𝑓 : ∪ 𝑆 ⟶ 𝑆 → ( ◡ 𝑓 “ { 𝑘 } ) ⊆ ∪ 𝑆 ) |
242 |
241
|
ad3antlr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → ( ◡ 𝑓 “ { 𝑘 } ) ⊆ ∪ 𝑆 ) |
243 |
242
|
sselda |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ) → 𝑙 ∈ ∪ 𝑆 ) |
244 |
239 243
|
ffvelrnd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ) → ( 𝑏 ‘ 𝑙 ) ∈ 𝐵 ) |
245 |
|
fveq2 |
⊢ ( 𝑗 = 𝑙 → ( 𝑓 ‘ 𝑗 ) = ( 𝑓 ‘ 𝑙 ) ) |
246 |
49 245
|
eleq12d |
⊢ ( 𝑗 = 𝑙 → ( 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ↔ 𝑙 ∈ ( 𝑓 ‘ 𝑙 ) ) ) |
247 |
|
simpllr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ) → ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) |
248 |
246 247 243
|
rspcdva |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ) → 𝑙 ∈ ( 𝑓 ‘ 𝑙 ) ) |
249 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ) → 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) |
250 |
249
|
ffnd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ) → 𝑓 Fn ∪ 𝑆 ) |
251 |
|
elpreima |
⊢ ( 𝑓 Fn ∪ 𝑆 → ( 𝑙 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ↔ ( 𝑙 ∈ ∪ 𝑆 ∧ ( 𝑓 ‘ 𝑙 ) ∈ { 𝑘 } ) ) ) |
252 |
251
|
biimpa |
⊢ ( ( 𝑓 Fn ∪ 𝑆 ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ) → ( 𝑙 ∈ ∪ 𝑆 ∧ ( 𝑓 ‘ 𝑙 ) ∈ { 𝑘 } ) ) |
253 |
|
elsni |
⊢ ( ( 𝑓 ‘ 𝑙 ) ∈ { 𝑘 } → ( 𝑓 ‘ 𝑙 ) = 𝑘 ) |
254 |
252 253
|
simpl2im |
⊢ ( ( 𝑓 Fn ∪ 𝑆 ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ) → ( 𝑓 ‘ 𝑙 ) = 𝑘 ) |
255 |
250 254
|
sylancom |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ) → ( 𝑓 ‘ 𝑙 ) = 𝑘 ) |
256 |
248 255
|
eleqtrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ) → 𝑙 ∈ 𝑘 ) |
257 |
8 2 4
|
lidlmcl |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑘 ∈ ( LIdeal ‘ 𝑅 ) ) ∧ ( ( 𝑏 ‘ 𝑙 ) ∈ 𝐵 ∧ 𝑙 ∈ 𝑘 ) ) → ( ( 𝑏 ‘ 𝑙 ) · 𝑙 ) ∈ 𝑘 ) |
258 |
237 238 244 256 257
|
syl22anc |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑙 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ) → ( ( 𝑏 ‘ 𝑙 ) · 𝑙 ) ∈ 𝑘 ) |
259 |
50
|
cbvmptv |
⊢ ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) = ( 𝑙 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ↦ ( ( 𝑏 ‘ 𝑙 ) · 𝑙 ) ) |
260 |
258 259
|
fmptd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) : ( ◡ 𝑓 “ { 𝑘 } ) ⟶ 𝑘 ) |
261 |
229
|
mptexd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ∈ V ) |
262 |
260
|
ffund |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → Fun ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) |
263 |
|
simp-5r |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → 𝑏 finSupp 0 ) |
264 |
|
nfv |
⊢ Ⅎ 𝑗 𝑘 ∈ 𝑆 |
265 |
66 264
|
nfan |
⊢ Ⅎ 𝑗 ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) |
266 |
|
nfcv |
⊢ Ⅎ 𝑗 ( ◡ 𝑓 “ { 𝑘 } ) |
267 |
36
|
ad7antlr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑘 } ) ∖ ( 𝑏 supp 0 ) ) ) → 𝑏 : ∪ 𝑆 ⟶ 𝐵 ) |
268 |
267
|
ffnd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑘 } ) ∖ ( 𝑏 supp 0 ) ) ) → 𝑏 Fn ∪ 𝑆 ) |
269 |
73
|
ad2antrr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑘 } ) ∖ ( 𝑏 supp 0 ) ) ) → ∪ 𝑆 ∈ V ) |
270 |
75
|
a1i |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑘 } ) ∖ ( 𝑏 supp 0 ) ) ) → 0 ∈ V ) |
271 |
242
|
ssdifd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → ( ( ◡ 𝑓 “ { 𝑘 } ) ∖ ( 𝑏 supp 0 ) ) ⊆ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) |
272 |
271
|
sselda |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑘 } ) ∖ ( 𝑏 supp 0 ) ) ) → 𝑗 ∈ ( ∪ 𝑆 ∖ ( 𝑏 supp 0 ) ) ) |
273 |
268 269 270 272
|
fvdifsupp |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑘 } ) ∖ ( 𝑏 supp 0 ) ) ) → ( 𝑏 ‘ 𝑗 ) = 0 ) |
274 |
273
|
oveq1d |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑘 } ) ∖ ( 𝑏 supp 0 ) ) ) → ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) = ( 0 · 𝑗 ) ) |
275 |
13
|
ad7antr |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑘 } ) ∖ ( 𝑏 supp 0 ) ) ) → ∪ 𝑆 ⊆ 𝐵 ) |
276 |
272
|
eldifad |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑘 } ) ∖ ( 𝑏 supp 0 ) ) ) → 𝑗 ∈ ∪ 𝑆 ) |
277 |
275 276
|
sseldd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑘 } ) ∖ ( 𝑏 supp 0 ) ) ) → 𝑗 ∈ 𝐵 ) |
278 |
230 277 85
|
syl2an2r |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑘 } ) ∖ ( 𝑏 supp 0 ) ) ) → ( 0 · 𝑗 ) = 0 ) |
279 |
274 278
|
eqtrd |
⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) ∧ 𝑗 ∈ ( ( ◡ 𝑓 “ { 𝑘 } ) ∖ ( 𝑏 supp 0 ) ) ) → ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) = 0 ) |
280 |
265 266 70 279 229
|
suppss2f |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → ( ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) supp 0 ) ⊆ ( 𝑏 supp 0 ) ) |
281 |
|
fsuppsssupp |
⊢ ( ( ( ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ∈ V ∧ Fun ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ∧ ( 𝑏 finSupp 0 ∧ ( ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) supp 0 ) ⊆ ( 𝑏 supp 0 ) ) ) → ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) finSupp 0 ) |
282 |
261 262 263 280 281
|
syl22anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) finSupp 0 ) |
283 |
3 226 229 236 260 282
|
gsumsubmcl |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑘 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ∈ 𝑘 ) |
284 |
225 283
|
eqeltrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ∧ 𝑘 ∈ 𝑆 ) → ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ‘ 𝑘 ) ∈ 𝑘 ) |
285 |
284
|
ralrimiva |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ∀ 𝑘 ∈ 𝑆 ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ‘ 𝑘 ) ∈ 𝑘 ) |
286 |
170 216 285
|
3jca |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( ( 𝑖 ∈ 𝑆 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( ◡ 𝑓 “ { 𝑖 } ) ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ‘ 𝑘 ) ∈ 𝑘 ) ) |
287 |
97 105 286
|
rspcedvd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ 𝑓 : ∪ 𝑆 ⟶ 𝑆 ) ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) → ∃ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ( 𝑎 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ) |
288 |
287
|
anasss |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ∧ ( 𝑓 : ∪ 𝑆 ⟶ 𝑆 ∧ ∀ 𝑗 ∈ ∪ 𝑆 𝑗 ∈ ( 𝑓 ‘ 𝑗 ) ) ) → ∃ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ( 𝑎 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ) |
289 |
25 288
|
exlimddv |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ 𝑏 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) → ∃ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ( 𝑎 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ) |
290 |
289
|
anasss |
⊢ ( ( ( 𝜑 ∧ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) ∧ ( 𝑏 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ) → ∃ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ( 𝑎 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ) |
291 |
290
|
r19.29an |
⊢ ( ( 𝜑 ∧ ∃ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ( 𝑏 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ) → ∃ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ( 𝑎 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ) |
292 |
5
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → 𝑅 ∈ Ring ) |
293 |
292
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ) → 𝑅 ∈ Ring ) |
294 |
|
eqid |
⊢ ( ℤRHom ‘ 𝑅 ) = ( ℤRHom ‘ 𝑅 ) |
295 |
294
|
zrhrhm |
⊢ ( 𝑅 ∈ Ring → ( ℤRHom ‘ 𝑅 ) ∈ ( ℤring RingHom 𝑅 ) ) |
296 |
|
zringbas |
⊢ ℤ = ( Base ‘ ℤring ) |
297 |
296 2
|
rhmf |
⊢ ( ( ℤRHom ‘ 𝑅 ) ∈ ( ℤring RingHom 𝑅 ) → ( ℤRHom ‘ 𝑅 ) : ℤ ⟶ 𝐵 ) |
298 |
293 295 297
|
3syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ) → ( ℤRHom ‘ 𝑅 ) : ℤ ⟶ 𝐵 ) |
299 |
|
simp-5r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ) → 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) |
300 |
75
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ) → 0 ∈ V ) |
301 |
|
ssv |
⊢ ran 𝑎 ⊆ V |
302 |
|
ssdif |
⊢ ( ran 𝑎 ⊆ V → ( ran 𝑎 ∖ { 0 } ) ⊆ ( V ∖ { 0 } ) ) |
303 |
301 302
|
ax-mp |
⊢ ( ran 𝑎 ∖ { 0 } ) ⊆ ( V ∖ { 0 } ) |
304 |
303
|
sseli |
⊢ ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) → 𝑚 ∈ ( V ∖ { 0 } ) ) |
305 |
304
|
adantl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ) → 𝑚 ∈ ( V ∖ { 0 } ) ) |
306 |
|
simp-4r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ) → 𝑎 finSupp 0 ) |
307 |
299 300 305 306
|
fsuppinisegfi |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ) → ( ◡ 𝑎 “ { 𝑚 } ) ∈ Fin ) |
308 |
|
hashcl |
⊢ ( ( ◡ 𝑎 “ { 𝑚 } ) ∈ Fin → ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ∈ ℕ0 ) |
309 |
307 308
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ) → ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ∈ ℕ0 ) |
310 |
309
|
nn0zd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ) → ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ∈ ℤ ) |
311 |
298 310
|
ffvelrnd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ) → ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ∈ 𝐵 ) |
312 |
|
eqid |
⊢ ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) = ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) |
313 |
311 312
|
fmptd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) : ( ran 𝑎 ∖ { 0 } ) ⟶ 𝐵 ) |
314 |
2 3
|
ring0cl |
⊢ ( 𝑅 ∈ Ring → 0 ∈ 𝐵 ) |
315 |
|
fconst6g |
⊢ ( 0 ∈ 𝐵 → ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) : ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ⟶ 𝐵 ) |
316 |
292 314 315
|
3syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) : ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ⟶ 𝐵 ) |
317 |
|
disjdif |
⊢ ( ( ran 𝑎 ∖ { 0 } ) ∩ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) = ∅ |
318 |
317
|
a1i |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ( ran 𝑎 ∖ { 0 } ) ∩ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) = ∅ ) |
319 |
313 316 318
|
fun2d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) : ( ( ran 𝑎 ∖ { 0 } ) ∪ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) ⟶ 𝐵 ) |
320 |
|
simplll |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ) |
321 |
94 16
|
elmapd |
⊢ ( 𝜑 → ( 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ↔ 𝑎 : 𝑆 ⟶ 𝐵 ) ) |
322 |
321
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) → 𝑎 : 𝑆 ⟶ 𝐵 ) |
323 |
320 322
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → 𝑎 : 𝑆 ⟶ 𝐵 ) |
324 |
323
|
ffnd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → 𝑎 Fn 𝑆 ) |
325 |
|
elssuni |
⊢ ( 𝑘 ∈ 𝑆 → 𝑘 ⊆ ∪ 𝑆 ) |
326 |
325
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ 𝑘 ∈ 𝑆 ) → 𝑘 ⊆ ∪ 𝑆 ) |
327 |
326
|
sseld |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ 𝑘 ∈ 𝑆 ) → ( ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 → ( 𝑎 ‘ 𝑘 ) ∈ ∪ 𝑆 ) ) |
328 |
327
|
ralimdva |
⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) → ( ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 → ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ ∪ 𝑆 ) ) |
329 |
328
|
imp |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ ∪ 𝑆 ) |
330 |
|
fnfvrnss |
⊢ ( ( 𝑎 Fn 𝑆 ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ ∪ 𝑆 ) → ran 𝑎 ⊆ ∪ 𝑆 ) |
331 |
324 329 330
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ran 𝑎 ⊆ ∪ 𝑆 ) |
332 |
331
|
ssdifssd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ran 𝑎 ∖ { 0 } ) ⊆ ∪ 𝑆 ) |
333 |
|
undif |
⊢ ( ( ran 𝑎 ∖ { 0 } ) ⊆ ∪ 𝑆 ↔ ( ( ran 𝑎 ∖ { 0 } ) ∪ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) = ∪ 𝑆 ) |
334 |
332 333
|
sylib |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ( ran 𝑎 ∖ { 0 } ) ∪ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) = ∪ 𝑆 ) |
335 |
334
|
feq2d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) : ( ( ran 𝑎 ∖ { 0 } ) ∪ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) ⟶ 𝐵 ↔ ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) : ∪ 𝑆 ⟶ 𝐵 ) ) |
336 |
319 335
|
mpbid |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) : ∪ 𝑆 ⟶ 𝐵 ) |
337 |
93
|
a1i |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → 𝐵 ∈ V ) |
338 |
17
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ∪ 𝑆 ∈ V ) |
339 |
337 338
|
elmapd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ∈ ( 𝐵 ↑m ∪ 𝑆 ) ↔ ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) : ∪ 𝑆 ⟶ 𝐵 ) ) |
340 |
336 339
|
mpbird |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ∈ ( 𝐵 ↑m ∪ 𝑆 ) ) |
341 |
|
breq1 |
⊢ ( 𝑏 = ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) → ( 𝑏 finSupp 0 ↔ ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) finSupp 0 ) ) |
342 |
|
fveq1 |
⊢ ( 𝑏 = ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) → ( 𝑏 ‘ 𝑗 ) = ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) ) |
343 |
342
|
oveq1d |
⊢ ( 𝑏 = ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) → ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) = ( ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) · 𝑗 ) ) |
344 |
343
|
mpteq2dv |
⊢ ( 𝑏 = ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) → ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) = ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) · 𝑗 ) ) ) |
345 |
344
|
oveq2d |
⊢ ( 𝑏 = ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) → ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) · 𝑗 ) ) ) ) |
346 |
345
|
eqeq2d |
⊢ ( 𝑏 = ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) → ( 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ↔ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) · 𝑗 ) ) ) ) ) |
347 |
341 346
|
anbi12d |
⊢ ( 𝑏 = ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) → ( ( 𝑏 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ↔ ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) · 𝑗 ) ) ) ) ) ) |
348 |
347
|
adantl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑏 = ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ) → ( ( 𝑏 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ↔ ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) · 𝑗 ) ) ) ) ) ) |
349 |
319
|
ffund |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → Fun ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ) |
350 |
340
|
elexd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ∈ V ) |
351 |
75
|
a1i |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → 0 ∈ V ) |
352 |
323
|
ffund |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → Fun 𝑎 ) |
353 |
320
|
simprd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) |
354 |
|
simpllr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → 𝑎 finSupp 0 ) |
355 |
|
fsupprnfi |
⊢ ( ( ( Fun 𝑎 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ ( 0 ∈ V ∧ 𝑎 finSupp 0 ) ) → ran 𝑎 ∈ Fin ) |
356 |
|
diffi |
⊢ ( ran 𝑎 ∈ Fin → ( ran 𝑎 ∖ { 0 } ) ∈ Fin ) |
357 |
355 356
|
syl |
⊢ ( ( ( Fun 𝑎 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ ( 0 ∈ V ∧ 𝑎 finSupp 0 ) ) → ( ran 𝑎 ∖ { 0 } ) ∈ Fin ) |
358 |
352 353 351 354 357
|
syl22anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ran 𝑎 ∖ { 0 } ) ∈ Fin ) |
359 |
313 358 351
|
fdmfifsupp |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) finSupp 0 ) |
360 |
13
|
ssdifssd |
⊢ ( 𝜑 → ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ⊆ 𝐵 ) |
361 |
360
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ⊆ 𝐵 ) |
362 |
337 361
|
ssexd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ∈ V ) |
363 |
362 351
|
fczfsuppd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) finSupp 0 ) |
364 |
359 363
|
fsuppun |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) supp 0 ) ∈ Fin ) |
365 |
|
funisfsupp |
⊢ ( ( Fun ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ∧ ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ∈ V ∧ 0 ∈ V ) → ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) finSupp 0 ↔ ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) supp 0 ) ∈ Fin ) ) |
366 |
365
|
biimpar |
⊢ ( ( ( Fun ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ∧ ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ∈ V ∧ 0 ∈ V ) ∧ ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) supp 0 ) ∈ Fin ) → ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) finSupp 0 ) |
367 |
349 350 351 364 366
|
syl31anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) finSupp 0 ) |
368 |
|
fvex |
⊢ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ∈ V |
369 |
368 312
|
fnmpti |
⊢ ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) Fn ( ran 𝑎 ∖ { 0 } ) |
370 |
369
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) Fn ( ran 𝑎 ∖ { 0 } ) ) |
371 |
|
fnconstg |
⊢ ( 0 ∈ V → ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) Fn ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) |
372 |
75 371
|
ax-mp |
⊢ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) Fn ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) |
373 |
372
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) Fn ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) |
374 |
317
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → ( ( ran 𝑎 ∖ { 0 } ) ∩ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) = ∅ ) |
375 |
|
simpr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) |
376 |
370 373 374 375
|
fvun1d |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) = ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ‘ 𝑗 ) ) |
377 |
|
sneq |
⊢ ( 𝑚 = 𝑗 → { 𝑚 } = { 𝑗 } ) |
378 |
377
|
imaeq2d |
⊢ ( 𝑚 = 𝑗 → ( ◡ 𝑎 “ { 𝑚 } ) = ( ◡ 𝑎 “ { 𝑗 } ) ) |
379 |
378
|
fveq2d |
⊢ ( 𝑚 = 𝑗 → ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) = ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ) |
380 |
379
|
fveq2d |
⊢ ( 𝑚 = 𝑗 → ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) = ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ) ) |
381 |
|
fvexd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ) ∈ V ) |
382 |
312 380 375 381
|
fvmptd3 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ‘ 𝑗 ) = ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ) ) |
383 |
376 382
|
eqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) = ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ) ) |
384 |
383
|
oveq1d |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → ( ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) · 𝑗 ) = ( ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ) · 𝑗 ) ) |
385 |
384
|
mpteq2dva |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) · 𝑗 ) ) = ( 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ) · 𝑗 ) ) ) |
386 |
385
|
oveq2d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( 𝑅 Σg ( 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) · 𝑗 ) ) ) = ( 𝑅 Σg ( 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ) · 𝑗 ) ) ) ) |
387 |
292 28
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → 𝑅 ∈ CMnd ) |
388 |
317
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) → ( ( ran 𝑎 ∖ { 0 } ) ∩ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) = ∅ ) |
389 |
|
fvun2 |
⊢ ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) Fn ( ran 𝑎 ∖ { 0 } ) ∧ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) Fn ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ∧ ( ( ( ran 𝑎 ∖ { 0 } ) ∩ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) = ∅ ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) ) → ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) = ( ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ‘ 𝑗 ) ) |
390 |
369 372 389
|
mp3an12 |
⊢ ( ( ( ( ran 𝑎 ∖ { 0 } ) ∩ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) = ∅ ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) → ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) = ( ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ‘ 𝑗 ) ) |
391 |
388 390
|
sylancom |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) → ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) = ( ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ‘ 𝑗 ) ) |
392 |
75
|
fvconst2 |
⊢ ( 𝑗 ∈ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) → ( ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ‘ 𝑗 ) = 0 ) |
393 |
392
|
adantl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) → ( ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ‘ 𝑗 ) = 0 ) |
394 |
391 393
|
eqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) → ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) = 0 ) |
395 |
394
|
oveq1d |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) → ( ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) · 𝑗 ) = ( 0 · 𝑗 ) ) |
396 |
361
|
sselda |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) → 𝑗 ∈ 𝐵 ) |
397 |
292 396 85
|
syl2an2r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) → ( 0 · 𝑗 ) = 0 ) |
398 |
395 397
|
eqtrd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) ) → ( ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) · 𝑗 ) = 0 ) |
399 |
292
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ∪ 𝑆 ) → 𝑅 ∈ Ring ) |
400 |
336
|
ffvelrnda |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ∪ 𝑆 ) → ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) ∈ 𝐵 ) |
401 |
13
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ∪ 𝑆 ⊆ 𝐵 ) |
402 |
401
|
sselda |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ∪ 𝑆 ) → 𝑗 ∈ 𝐵 ) |
403 |
2 4
|
ringcl |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) → ( ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) · 𝑗 ) ∈ 𝐵 ) |
404 |
399 400 402 403
|
syl3anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ∪ 𝑆 ) → ( ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) · 𝑗 ) ∈ 𝐵 ) |
405 |
2 3 387 338 398 358 404 332
|
gsummptres2 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) · 𝑗 ) ) ) = ( 𝑅 Σg ( 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) · 𝑗 ) ) ) ) |
406 |
|
eqid |
⊢ ( .g ‘ 𝑅 ) = ( .g ‘ 𝑅 ) |
407 |
2 3 406 387 323 354
|
gsumhashmul |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( 𝑅 Σg 𝑎 ) = ( 𝑅 Σg ( 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ( .g ‘ 𝑅 ) 𝑗 ) ) ) ) |
408 |
|
simplr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → 𝑋 = ( 𝑅 Σg 𝑎 ) ) |
409 |
292
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → 𝑅 ∈ Ring ) |
410 |
353
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) |
411 |
75
|
a1i |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → 0 ∈ V ) |
412 |
303 375
|
sselid |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → 𝑗 ∈ ( V ∖ { 0 } ) ) |
413 |
354
|
adantr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → 𝑎 finSupp 0 ) |
414 |
410 411 412 413
|
fsuppinisegfi |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → ( ◡ 𝑎 “ { 𝑗 } ) ∈ Fin ) |
415 |
|
hashcl |
⊢ ( ( ◡ 𝑎 “ { 𝑗 } ) ∈ Fin → ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ∈ ℕ0 ) |
416 |
414 415
|
syl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ∈ ℕ0 ) |
417 |
416
|
nn0zd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ∈ ℤ ) |
418 |
332 401
|
sstrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ran 𝑎 ∖ { 0 } ) ⊆ 𝐵 ) |
419 |
418
|
sselda |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → 𝑗 ∈ 𝐵 ) |
420 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
421 |
294 406 420
|
zrhmulg |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ∈ ℤ ) → ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ) = ( ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ( .g ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) ) |
422 |
421
|
adantr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ∈ ℤ ) ∧ 𝑗 ∈ 𝐵 ) → ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ) = ( ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ( .g ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) ) |
423 |
422
|
oveq1d |
⊢ ( ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ∈ ℤ ) ∧ 𝑗 ∈ 𝐵 ) → ( ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ) · 𝑗 ) = ( ( ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ( .g ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) · 𝑗 ) ) |
424 |
|
simpll |
⊢ ( ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ∈ ℤ ) ∧ 𝑗 ∈ 𝐵 ) → 𝑅 ∈ Ring ) |
425 |
|
simplr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ∈ ℤ ) ∧ 𝑗 ∈ 𝐵 ) → ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ∈ ℤ ) |
426 |
2 420
|
ringidcl |
⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ 𝐵 ) |
427 |
426
|
ad2antrr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ∈ ℤ ) ∧ 𝑗 ∈ 𝐵 ) → ( 1r ‘ 𝑅 ) ∈ 𝐵 ) |
428 |
|
simpr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ∈ ℤ ) ∧ 𝑗 ∈ 𝐵 ) → 𝑗 ∈ 𝐵 ) |
429 |
2 406 4
|
mulgass2 |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ∈ ℤ ∧ ( 1r ‘ 𝑅 ) ∈ 𝐵 ∧ 𝑗 ∈ 𝐵 ) ) → ( ( ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ( .g ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) · 𝑗 ) = ( ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ( .g ‘ 𝑅 ) ( ( 1r ‘ 𝑅 ) · 𝑗 ) ) ) |
430 |
424 425 427 428 429
|
syl13anc |
⊢ ( ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ∈ ℤ ) ∧ 𝑗 ∈ 𝐵 ) → ( ( ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ( .g ‘ 𝑅 ) ( 1r ‘ 𝑅 ) ) · 𝑗 ) = ( ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ( .g ‘ 𝑅 ) ( ( 1r ‘ 𝑅 ) · 𝑗 ) ) ) |
431 |
2 4 420
|
ringlidm |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑗 ∈ 𝐵 ) → ( ( 1r ‘ 𝑅 ) · 𝑗 ) = 𝑗 ) |
432 |
424 431
|
sylancom |
⊢ ( ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ∈ ℤ ) ∧ 𝑗 ∈ 𝐵 ) → ( ( 1r ‘ 𝑅 ) · 𝑗 ) = 𝑗 ) |
433 |
432
|
oveq2d |
⊢ ( ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ∈ ℤ ) ∧ 𝑗 ∈ 𝐵 ) → ( ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ( .g ‘ 𝑅 ) ( ( 1r ‘ 𝑅 ) · 𝑗 ) ) = ( ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ( .g ‘ 𝑅 ) 𝑗 ) ) |
434 |
423 430 433
|
3eqtrd |
⊢ ( ( ( 𝑅 ∈ Ring ∧ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ∈ ℤ ) ∧ 𝑗 ∈ 𝐵 ) → ( ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ) · 𝑗 ) = ( ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ( .g ‘ 𝑅 ) 𝑗 ) ) |
435 |
409 417 419 434
|
syl21anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ∧ 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ) → ( ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ) · 𝑗 ) = ( ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ( .g ‘ 𝑅 ) 𝑗 ) ) |
436 |
435
|
mpteq2dva |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ) · 𝑗 ) ) = ( 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ( .g ‘ 𝑅 ) 𝑗 ) ) ) |
437 |
436
|
oveq2d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( 𝑅 Σg ( 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ) · 𝑗 ) ) ) = ( 𝑅 Σg ( 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ( .g ‘ 𝑅 ) 𝑗 ) ) ) ) |
438 |
407 408 437
|
3eqtr4d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑗 } ) ) ) · 𝑗 ) ) ) ) |
439 |
386 405 438
|
3eqtr4rd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) · 𝑗 ) ) ) ) |
440 |
367 439
|
jca |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( ( ( 𝑚 ∈ ( ran 𝑎 ∖ { 0 } ) ↦ ( ( ℤRHom ‘ 𝑅 ) ‘ ( ♯ ‘ ( ◡ 𝑎 “ { 𝑚 } ) ) ) ) ∪ ( ( ∪ 𝑆 ∖ ( ran 𝑎 ∖ { 0 } ) ) × { 0 } ) ) ‘ 𝑗 ) · 𝑗 ) ) ) ) ) |
441 |
340 348 440
|
rspcedvd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ 𝑎 finSupp 0 ) ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) → ∃ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ( 𝑏 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ) |
442 |
441
|
exp41 |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) → ( 𝑎 finSupp 0 → ( 𝑋 = ( 𝑅 Σg 𝑎 ) → ( ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 → ∃ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ( 𝑏 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ) ) ) ) |
443 |
442
|
3imp2 |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ) ∧ ( 𝑎 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ) → ∃ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ( 𝑏 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ) |
444 |
443
|
r19.29an |
⊢ ( ( 𝜑 ∧ ∃ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ( 𝑎 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ) → ∃ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ( 𝑏 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ) |
445 |
291 444
|
impbida |
⊢ ( 𝜑 → ( ∃ 𝑏 ∈ ( 𝐵 ↑m ∪ 𝑆 ) ( 𝑏 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg ( 𝑗 ∈ ∪ 𝑆 ↦ ( ( 𝑏 ‘ 𝑗 ) · 𝑗 ) ) ) ) ↔ ∃ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ( 𝑎 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ) ) |
446 |
14 445
|
bitrd |
⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑁 ‘ ∪ 𝑆 ) ↔ ∃ 𝑎 ∈ ( 𝐵 ↑m 𝑆 ) ( 𝑎 finSupp 0 ∧ 𝑋 = ( 𝑅 Σg 𝑎 ) ∧ ∀ 𝑘 ∈ 𝑆 ( 𝑎 ‘ 𝑘 ) ∈ 𝑘 ) ) ) |