Step |
Hyp |
Ref |
Expression |
1 |
|
gsumzcl.b |
|- B = ( Base ` G ) |
2 |
|
gsumzcl.0 |
|- .0. = ( 0g ` G ) |
3 |
|
gsumzcl.z |
|- Z = ( Cntz ` G ) |
4 |
|
gsumzcl.g |
|- ( ph -> G e. Mnd ) |
5 |
|
gsumzcl.a |
|- ( ph -> A e. V ) |
6 |
|
gsumzcl.f |
|- ( ph -> F : A --> B ) |
7 |
|
gsumzcl.c |
|- ( ph -> ran F C_ ( Z ` ran F ) ) |
8 |
|
gsumzcl2.w |
|- ( ph -> ( F supp .0. ) e. Fin ) |
9 |
2
|
fvexi |
|- .0. e. _V |
10 |
9
|
a1i |
|- ( ph -> .0. e. _V ) |
11 |
|
ssidd |
|- ( ph -> ( F supp .0. ) C_ ( F supp .0. ) ) |
12 |
6 5 10 11
|
gsumcllem |
|- ( ( ph /\ ( F supp .0. ) = (/) ) -> F = ( k e. A |-> .0. ) ) |
13 |
12
|
oveq2d |
|- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsum F ) = ( G gsum ( k e. A |-> .0. ) ) ) |
14 |
2
|
gsumz |
|- ( ( G e. Mnd /\ A e. V ) -> ( G gsum ( k e. A |-> .0. ) ) = .0. ) |
15 |
4 5 14
|
syl2anc |
|- ( ph -> ( G gsum ( k e. A |-> .0. ) ) = .0. ) |
16 |
15
|
adantr |
|- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsum ( k e. A |-> .0. ) ) = .0. ) |
17 |
13 16
|
eqtrd |
|- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsum F ) = .0. ) |
18 |
1 2
|
mndidcl |
|- ( G e. Mnd -> .0. e. B ) |
19 |
4 18
|
syl |
|- ( ph -> .0. e. B ) |
20 |
19
|
adantr |
|- ( ( ph /\ ( F supp .0. ) = (/) ) -> .0. e. B ) |
21 |
17 20
|
eqeltrd |
|- ( ( ph /\ ( F supp .0. ) = (/) ) -> ( G gsum F ) e. B ) |
22 |
21
|
ex |
|- ( ph -> ( ( F supp .0. ) = (/) -> ( G gsum F ) e. B ) ) |
23 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
24 |
4
|
adantr |
|- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> G e. Mnd ) |
25 |
5
|
adantr |
|- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> A e. V ) |
26 |
6
|
adantr |
|- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> F : A --> B ) |
27 |
7
|
adantr |
|- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ran F C_ ( Z ` ran F ) ) |
28 |
|
simprl |
|- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( # ` ( F supp .0. ) ) e. NN ) |
29 |
|
f1of1 |
|- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> ( F supp .0. ) ) |
30 |
29
|
ad2antll |
|- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> ( F supp .0. ) ) |
31 |
|
suppssdm |
|- ( F supp .0. ) C_ dom F |
32 |
31 6
|
fssdm |
|- ( ph -> ( F supp .0. ) C_ A ) |
33 |
32
|
adantr |
|- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( F supp .0. ) C_ A ) |
34 |
|
f1ss |
|- ( ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> ( F supp .0. ) /\ ( F supp .0. ) C_ A ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> A ) |
35 |
30 33 34
|
syl2anc |
|- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> A ) |
36 |
|
ssid |
|- ( F supp .0. ) C_ ( F supp .0. ) |
37 |
|
f1ofo |
|- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) -onto-> ( F supp .0. ) ) |
38 |
|
forn |
|- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -onto-> ( F supp .0. ) -> ran f = ( F supp .0. ) ) |
39 |
37 38
|
syl |
|- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> ran f = ( F supp .0. ) ) |
40 |
39
|
ad2antll |
|- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ran f = ( F supp .0. ) ) |
41 |
36 40
|
sseqtrrid |
|- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( F supp .0. ) C_ ran f ) |
42 |
|
eqid |
|- ( ( F o. f ) supp .0. ) = ( ( F o. f ) supp .0. ) |
43 |
1 2 23 3 24 25 26 27 28 35 41 42
|
gsumval3 |
|- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( G gsum F ) = ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( F supp .0. ) ) ) ) |
44 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
45 |
28 44
|
eleqtrdi |
|- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( # ` ( F supp .0. ) ) e. ( ZZ>= ` 1 ) ) |
46 |
|
f1f |
|- ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-> A -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) --> A ) |
47 |
35 46
|
syl |
|- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> f : ( 1 ... ( # ` ( F supp .0. ) ) ) --> A ) |
48 |
|
fco |
|- ( ( F : A --> B /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) --> A ) -> ( F o. f ) : ( 1 ... ( # ` ( F supp .0. ) ) ) --> B ) |
49 |
26 47 48
|
syl2anc |
|- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( F o. f ) : ( 1 ... ( # ` ( F supp .0. ) ) ) --> B ) |
50 |
49
|
ffvelrnda |
|- ( ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) /\ k e. ( 1 ... ( # ` ( F supp .0. ) ) ) ) -> ( ( F o. f ) ` k ) e. B ) |
51 |
1 23
|
mndcl |
|- ( ( G e. Mnd /\ k e. B /\ x e. B ) -> ( k ( +g ` G ) x ) e. B ) |
52 |
51
|
3expb |
|- ( ( G e. Mnd /\ ( k e. B /\ x e. B ) ) -> ( k ( +g ` G ) x ) e. B ) |
53 |
24 52
|
sylan |
|- ( ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) /\ ( k e. B /\ x e. B ) ) -> ( k ( +g ` G ) x ) e. B ) |
54 |
45 50 53
|
seqcl |
|- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( seq 1 ( ( +g ` G ) , ( F o. f ) ) ` ( # ` ( F supp .0. ) ) ) e. B ) |
55 |
43 54
|
eqeltrd |
|- ( ( ph /\ ( ( # ` ( F supp .0. ) ) e. NN /\ f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) -> ( G gsum F ) e. B ) |
56 |
55
|
expr |
|- ( ( ph /\ ( # ` ( F supp .0. ) ) e. NN ) -> ( f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> ( G gsum F ) e. B ) ) |
57 |
56
|
exlimdv |
|- ( ( ph /\ ( # ` ( F supp .0. ) ) e. NN ) -> ( E. f f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) -> ( G gsum F ) e. B ) ) |
58 |
57
|
expimpd |
|- ( ph -> ( ( ( # ` ( F supp .0. ) ) e. NN /\ E. f f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) -> ( G gsum F ) e. B ) ) |
59 |
|
fz1f1o |
|- ( ( F supp .0. ) e. Fin -> ( ( F supp .0. ) = (/) \/ ( ( # ` ( F supp .0. ) ) e. NN /\ E. f f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) ) |
60 |
8 59
|
syl |
|- ( ph -> ( ( F supp .0. ) = (/) \/ ( ( # ` ( F supp .0. ) ) e. NN /\ E. f f : ( 1 ... ( # ` ( F supp .0. ) ) ) -1-1-onto-> ( F supp .0. ) ) ) ) |
61 |
22 58 60
|
mpjaod |
|- ( ph -> ( G gsum F ) e. B ) |