Step |
Hyp |
Ref |
Expression |
1 |
|
gsumzadd.b |
|- B = ( Base ` G ) |
2 |
|
gsumzadd.0 |
|- .0. = ( 0g ` G ) |
3 |
|
gsumzadd.p |
|- .+ = ( +g ` G ) |
4 |
|
gsumzadd.z |
|- Z = ( Cntz ` G ) |
5 |
|
gsumzadd.g |
|- ( ph -> G e. Mnd ) |
6 |
|
gsumzadd.a |
|- ( ph -> A e. V ) |
7 |
|
gsumzadd.fn |
|- ( ph -> F finSupp .0. ) |
8 |
|
gsumzadd.hn |
|- ( ph -> H finSupp .0. ) |
9 |
|
gsumzaddlem.w |
|- W = ( ( F u. H ) supp .0. ) |
10 |
|
gsumzaddlem.f |
|- ( ph -> F : A --> B ) |
11 |
|
gsumzaddlem.h |
|- ( ph -> H : A --> B ) |
12 |
|
gsumzaddlem.1 |
|- ( ph -> ran F C_ ( Z ` ran F ) ) |
13 |
|
gsumzaddlem.2 |
|- ( ph -> ran H C_ ( Z ` ran H ) ) |
14 |
|
gsumzaddlem.3 |
|- ( ph -> ran ( F oF .+ H ) C_ ( Z ` ran ( F oF .+ H ) ) ) |
15 |
|
gsumzaddlem.4 |
|- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> ( F ` k ) e. ( Z ` { ( G gsum ( H |` x ) ) } ) ) |
16 |
1 2
|
mndidcl |
|- ( G e. Mnd -> .0. e. B ) |
17 |
5 16
|
syl |
|- ( ph -> .0. e. B ) |
18 |
1 3 2
|
mndlid |
|- ( ( G e. Mnd /\ .0. e. B ) -> ( .0. .+ .0. ) = .0. ) |
19 |
5 17 18
|
syl2anc |
|- ( ph -> ( .0. .+ .0. ) = .0. ) |
20 |
19
|
adantr |
|- ( ( ph /\ W = (/) ) -> ( .0. .+ .0. ) = .0. ) |
21 |
2
|
fvexi |
|- .0. e. _V |
22 |
21
|
a1i |
|- ( ph -> .0. e. _V ) |
23 |
11 6
|
fexd |
|- ( ph -> H e. _V ) |
24 |
23
|
suppun |
|- ( ph -> ( F supp .0. ) C_ ( ( F u. H ) supp .0. ) ) |
25 |
24 9
|
sseqtrrdi |
|- ( ph -> ( F supp .0. ) C_ W ) |
26 |
10 6 22 25
|
gsumcllem |
|- ( ( ph /\ W = (/) ) -> F = ( x e. A |-> .0. ) ) |
27 |
26
|
oveq2d |
|- ( ( ph /\ W = (/) ) -> ( G gsum F ) = ( G gsum ( x e. A |-> .0. ) ) ) |
28 |
2
|
gsumz |
|- ( ( G e. Mnd /\ A e. V ) -> ( G gsum ( x e. A |-> .0. ) ) = .0. ) |
29 |
5 6 28
|
syl2anc |
|- ( ph -> ( G gsum ( x e. A |-> .0. ) ) = .0. ) |
30 |
29
|
adantr |
|- ( ( ph /\ W = (/) ) -> ( G gsum ( x e. A |-> .0. ) ) = .0. ) |
31 |
27 30
|
eqtrd |
|- ( ( ph /\ W = (/) ) -> ( G gsum F ) = .0. ) |
32 |
10 6
|
fexd |
|- ( ph -> F e. _V ) |
33 |
32
|
suppun |
|- ( ph -> ( H supp .0. ) C_ ( ( H u. F ) supp .0. ) ) |
34 |
|
uncom |
|- ( F u. H ) = ( H u. F ) |
35 |
34
|
oveq1i |
|- ( ( F u. H ) supp .0. ) = ( ( H u. F ) supp .0. ) |
36 |
33 35
|
sseqtrrdi |
|- ( ph -> ( H supp .0. ) C_ ( ( F u. H ) supp .0. ) ) |
37 |
36 9
|
sseqtrrdi |
|- ( ph -> ( H supp .0. ) C_ W ) |
38 |
11 6 22 37
|
gsumcllem |
|- ( ( ph /\ W = (/) ) -> H = ( x e. A |-> .0. ) ) |
39 |
38
|
oveq2d |
|- ( ( ph /\ W = (/) ) -> ( G gsum H ) = ( G gsum ( x e. A |-> .0. ) ) ) |
40 |
39 30
|
eqtrd |
|- ( ( ph /\ W = (/) ) -> ( G gsum H ) = .0. ) |
41 |
31 40
|
oveq12d |
|- ( ( ph /\ W = (/) ) -> ( ( G gsum F ) .+ ( G gsum H ) ) = ( .0. .+ .0. ) ) |
42 |
6
|
adantr |
|- ( ( ph /\ W = (/) ) -> A e. V ) |
43 |
17
|
ad2antrr |
|- ( ( ( ph /\ W = (/) ) /\ x e. A ) -> .0. e. B ) |
44 |
42 43 43 26 38
|
offval2 |
|- ( ( ph /\ W = (/) ) -> ( F oF .+ H ) = ( x e. A |-> ( .0. .+ .0. ) ) ) |
45 |
20
|
mpteq2dv |
|- ( ( ph /\ W = (/) ) -> ( x e. A |-> ( .0. .+ .0. ) ) = ( x e. A |-> .0. ) ) |
46 |
44 45
|
eqtrd |
|- ( ( ph /\ W = (/) ) -> ( F oF .+ H ) = ( x e. A |-> .0. ) ) |
47 |
46
|
oveq2d |
|- ( ( ph /\ W = (/) ) -> ( G gsum ( F oF .+ H ) ) = ( G gsum ( x e. A |-> .0. ) ) ) |
48 |
47 30
|
eqtrd |
|- ( ( ph /\ W = (/) ) -> ( G gsum ( F oF .+ H ) ) = .0. ) |
49 |
20 41 48
|
3eqtr4rd |
|- ( ( ph /\ W = (/) ) -> ( G gsum ( F oF .+ H ) ) = ( ( G gsum F ) .+ ( G gsum H ) ) ) |
50 |
49
|
ex |
|- ( ph -> ( W = (/) -> ( G gsum ( F oF .+ H ) ) = ( ( G gsum F ) .+ ( G gsum H ) ) ) ) |
51 |
5
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> G e. Mnd ) |
52 |
1 3
|
mndcl |
|- ( ( G e. Mnd /\ z e. B /\ w e. B ) -> ( z .+ w ) e. B ) |
53 |
52
|
3expb |
|- ( ( G e. Mnd /\ ( z e. B /\ w e. B ) ) -> ( z .+ w ) e. B ) |
54 |
51 53
|
sylan |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ ( z e. B /\ w e. B ) ) -> ( z .+ w ) e. B ) |
55 |
54
|
caovclg |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ ( x e. B /\ y e. B ) ) -> ( x .+ y ) e. B ) |
56 |
|
simprl |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( # ` W ) e. NN ) |
57 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
58 |
56 57
|
eleqtrdi |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( # ` W ) e. ( ZZ>= ` 1 ) ) |
59 |
10
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> F : A --> B ) |
60 |
|
f1of1 |
|- ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W -> f : ( 1 ... ( # ` W ) ) -1-1-> W ) |
61 |
60
|
ad2antll |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> f : ( 1 ... ( # ` W ) ) -1-1-> W ) |
62 |
|
suppssdm |
|- ( ( F u. H ) supp .0. ) C_ dom ( F u. H ) |
63 |
62
|
a1i |
|- ( ph -> ( ( F u. H ) supp .0. ) C_ dom ( F u. H ) ) |
64 |
9
|
a1i |
|- ( ph -> W = ( ( F u. H ) supp .0. ) ) |
65 |
|
dmun |
|- dom ( F u. H ) = ( dom F u. dom H ) |
66 |
10
|
fdmd |
|- ( ph -> dom F = A ) |
67 |
11
|
fdmd |
|- ( ph -> dom H = A ) |
68 |
66 67
|
uneq12d |
|- ( ph -> ( dom F u. dom H ) = ( A u. A ) ) |
69 |
|
unidm |
|- ( A u. A ) = A |
70 |
68 69
|
eqtrdi |
|- ( ph -> ( dom F u. dom H ) = A ) |
71 |
65 70
|
eqtr2id |
|- ( ph -> A = dom ( F u. H ) ) |
72 |
63 64 71
|
3sstr4d |
|- ( ph -> W C_ A ) |
73 |
72
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> W C_ A ) |
74 |
|
f1ss |
|- ( ( f : ( 1 ... ( # ` W ) ) -1-1-> W /\ W C_ A ) -> f : ( 1 ... ( # ` W ) ) -1-1-> A ) |
75 |
61 73 74
|
syl2anc |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> f : ( 1 ... ( # ` W ) ) -1-1-> A ) |
76 |
|
f1f |
|- ( f : ( 1 ... ( # ` W ) ) -1-1-> A -> f : ( 1 ... ( # ` W ) ) --> A ) |
77 |
75 76
|
syl |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> f : ( 1 ... ( # ` W ) ) --> A ) |
78 |
|
fco |
|- ( ( F : A --> B /\ f : ( 1 ... ( # ` W ) ) --> A ) -> ( F o. f ) : ( 1 ... ( # ` W ) ) --> B ) |
79 |
59 77 78
|
syl2anc |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( F o. f ) : ( 1 ... ( # ` W ) ) --> B ) |
80 |
79
|
ffvelrnda |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ k e. ( 1 ... ( # ` W ) ) ) -> ( ( F o. f ) ` k ) e. B ) |
81 |
11
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> H : A --> B ) |
82 |
|
fco |
|- ( ( H : A --> B /\ f : ( 1 ... ( # ` W ) ) --> A ) -> ( H o. f ) : ( 1 ... ( # ` W ) ) --> B ) |
83 |
81 77 82
|
syl2anc |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( H o. f ) : ( 1 ... ( # ` W ) ) --> B ) |
84 |
83
|
ffvelrnda |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ k e. ( 1 ... ( # ` W ) ) ) -> ( ( H o. f ) ` k ) e. B ) |
85 |
59
|
ffnd |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> F Fn A ) |
86 |
81
|
ffnd |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> H Fn A ) |
87 |
6
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> A e. V ) |
88 |
|
ovexd |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( 1 ... ( # ` W ) ) e. _V ) |
89 |
|
inidm |
|- ( A i^i A ) = A |
90 |
85 86 77 87 87 88 89
|
ofco |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( ( F oF .+ H ) o. f ) = ( ( F o. f ) oF .+ ( H o. f ) ) ) |
91 |
90
|
fveq1d |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( ( ( F oF .+ H ) o. f ) ` k ) = ( ( ( F o. f ) oF .+ ( H o. f ) ) ` k ) ) |
92 |
91
|
adantr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ k e. ( 1 ... ( # ` W ) ) ) -> ( ( ( F oF .+ H ) o. f ) ` k ) = ( ( ( F o. f ) oF .+ ( H o. f ) ) ` k ) ) |
93 |
|
fnfco |
|- ( ( F Fn A /\ f : ( 1 ... ( # ` W ) ) --> A ) -> ( F o. f ) Fn ( 1 ... ( # ` W ) ) ) |
94 |
85 77 93
|
syl2anc |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( F o. f ) Fn ( 1 ... ( # ` W ) ) ) |
95 |
|
fnfco |
|- ( ( H Fn A /\ f : ( 1 ... ( # ` W ) ) --> A ) -> ( H o. f ) Fn ( 1 ... ( # ` W ) ) ) |
96 |
86 77 95
|
syl2anc |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( H o. f ) Fn ( 1 ... ( # ` W ) ) ) |
97 |
|
inidm |
|- ( ( 1 ... ( # ` W ) ) i^i ( 1 ... ( # ` W ) ) ) = ( 1 ... ( # ` W ) ) |
98 |
|
eqidd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ k e. ( 1 ... ( # ` W ) ) ) -> ( ( F o. f ) ` k ) = ( ( F o. f ) ` k ) ) |
99 |
|
eqidd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ k e. ( 1 ... ( # ` W ) ) ) -> ( ( H o. f ) ` k ) = ( ( H o. f ) ` k ) ) |
100 |
94 96 88 88 97 98 99
|
ofval |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ k e. ( 1 ... ( # ` W ) ) ) -> ( ( ( F o. f ) oF .+ ( H o. f ) ) ` k ) = ( ( ( F o. f ) ` k ) .+ ( ( H o. f ) ` k ) ) ) |
101 |
92 100
|
eqtrd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ k e. ( 1 ... ( # ` W ) ) ) -> ( ( ( F oF .+ H ) o. f ) ` k ) = ( ( ( F o. f ) ` k ) .+ ( ( H o. f ) ` k ) ) ) |
102 |
5
|
ad2antrr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> G e. Mnd ) |
103 |
|
elfzouz |
|- ( n e. ( 1 ..^ ( # ` W ) ) -> n e. ( ZZ>= ` 1 ) ) |
104 |
103
|
adantl |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> n e. ( ZZ>= ` 1 ) ) |
105 |
|
elfzouz2 |
|- ( n e. ( 1 ..^ ( # ` W ) ) -> ( # ` W ) e. ( ZZ>= ` n ) ) |
106 |
105
|
adantl |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( # ` W ) e. ( ZZ>= ` n ) ) |
107 |
|
fzss2 |
|- ( ( # ` W ) e. ( ZZ>= ` n ) -> ( 1 ... n ) C_ ( 1 ... ( # ` W ) ) ) |
108 |
106 107
|
syl |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( 1 ... n ) C_ ( 1 ... ( # ` W ) ) ) |
109 |
108
|
sselda |
|- ( ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) /\ k e. ( 1 ... n ) ) -> k e. ( 1 ... ( # ` W ) ) ) |
110 |
80
|
adantlr |
|- ( ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) /\ k e. ( 1 ... ( # ` W ) ) ) -> ( ( F o. f ) ` k ) e. B ) |
111 |
109 110
|
syldan |
|- ( ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) /\ k e. ( 1 ... n ) ) -> ( ( F o. f ) ` k ) e. B ) |
112 |
1 3
|
mndcl |
|- ( ( G e. Mnd /\ k e. B /\ x e. B ) -> ( k .+ x ) e. B ) |
113 |
112
|
3expb |
|- ( ( G e. Mnd /\ ( k e. B /\ x e. B ) ) -> ( k .+ x ) e. B ) |
114 |
102 113
|
sylan |
|- ( ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) /\ ( k e. B /\ x e. B ) ) -> ( k .+ x ) e. B ) |
115 |
104 111 114
|
seqcl |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( seq 1 ( .+ , ( F o. f ) ) ` n ) e. B ) |
116 |
84
|
adantlr |
|- ( ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) /\ k e. ( 1 ... ( # ` W ) ) ) -> ( ( H o. f ) ` k ) e. B ) |
117 |
109 116
|
syldan |
|- ( ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) /\ k e. ( 1 ... n ) ) -> ( ( H o. f ) ` k ) e. B ) |
118 |
104 117 114
|
seqcl |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( seq 1 ( .+ , ( H o. f ) ) ` n ) e. B ) |
119 |
|
fzofzp1 |
|- ( n e. ( 1 ..^ ( # ` W ) ) -> ( n + 1 ) e. ( 1 ... ( # ` W ) ) ) |
120 |
|
ffvelrn |
|- ( ( ( F o. f ) : ( 1 ... ( # ` W ) ) --> B /\ ( n + 1 ) e. ( 1 ... ( # ` W ) ) ) -> ( ( F o. f ) ` ( n + 1 ) ) e. B ) |
121 |
79 119 120
|
syl2an |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( ( F o. f ) ` ( n + 1 ) ) e. B ) |
122 |
|
ffvelrn |
|- ( ( ( H o. f ) : ( 1 ... ( # ` W ) ) --> B /\ ( n + 1 ) e. ( 1 ... ( # ` W ) ) ) -> ( ( H o. f ) ` ( n + 1 ) ) e. B ) |
123 |
83 119 122
|
syl2an |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( ( H o. f ) ` ( n + 1 ) ) e. B ) |
124 |
|
fvco3 |
|- ( ( f : ( 1 ... ( # ` W ) ) --> A /\ ( n + 1 ) e. ( 1 ... ( # ` W ) ) ) -> ( ( F o. f ) ` ( n + 1 ) ) = ( F ` ( f ` ( n + 1 ) ) ) ) |
125 |
77 119 124
|
syl2an |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( ( F o. f ) ` ( n + 1 ) ) = ( F ` ( f ` ( n + 1 ) ) ) ) |
126 |
|
fveq2 |
|- ( k = ( f ` ( n + 1 ) ) -> ( F ` k ) = ( F ` ( f ` ( n + 1 ) ) ) ) |
127 |
126
|
eleq1d |
|- ( k = ( f ` ( n + 1 ) ) -> ( ( F ` k ) e. ( Z ` { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } ) <-> ( F ` ( f ` ( n + 1 ) ) ) e. ( Z ` { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } ) ) ) |
128 |
15
|
expr |
|- ( ( ph /\ x C_ A ) -> ( k e. ( A \ x ) -> ( F ` k ) e. ( Z ` { ( G gsum ( H |` x ) ) } ) ) ) |
129 |
128
|
ralrimiv |
|- ( ( ph /\ x C_ A ) -> A. k e. ( A \ x ) ( F ` k ) e. ( Z ` { ( G gsum ( H |` x ) ) } ) ) |
130 |
129
|
ex |
|- ( ph -> ( x C_ A -> A. k e. ( A \ x ) ( F ` k ) e. ( Z ` { ( G gsum ( H |` x ) ) } ) ) ) |
131 |
130
|
alrimiv |
|- ( ph -> A. x ( x C_ A -> A. k e. ( A \ x ) ( F ` k ) e. ( Z ` { ( G gsum ( H |` x ) ) } ) ) ) |
132 |
131
|
ad2antrr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> A. x ( x C_ A -> A. k e. ( A \ x ) ( F ` k ) e. ( Z ` { ( G gsum ( H |` x ) ) } ) ) ) |
133 |
|
imassrn |
|- ( f " ( 1 ... n ) ) C_ ran f |
134 |
77
|
adantr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> f : ( 1 ... ( # ` W ) ) --> A ) |
135 |
134
|
frnd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ran f C_ A ) |
136 |
133 135
|
sstrid |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( f " ( 1 ... n ) ) C_ A ) |
137 |
|
vex |
|- f e. _V |
138 |
137
|
imaex |
|- ( f " ( 1 ... n ) ) e. _V |
139 |
|
sseq1 |
|- ( x = ( f " ( 1 ... n ) ) -> ( x C_ A <-> ( f " ( 1 ... n ) ) C_ A ) ) |
140 |
|
difeq2 |
|- ( x = ( f " ( 1 ... n ) ) -> ( A \ x ) = ( A \ ( f " ( 1 ... n ) ) ) ) |
141 |
|
reseq2 |
|- ( x = ( f " ( 1 ... n ) ) -> ( H |` x ) = ( H |` ( f " ( 1 ... n ) ) ) ) |
142 |
141
|
oveq2d |
|- ( x = ( f " ( 1 ... n ) ) -> ( G gsum ( H |` x ) ) = ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) ) |
143 |
142
|
sneqd |
|- ( x = ( f " ( 1 ... n ) ) -> { ( G gsum ( H |` x ) ) } = { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } ) |
144 |
143
|
fveq2d |
|- ( x = ( f " ( 1 ... n ) ) -> ( Z ` { ( G gsum ( H |` x ) ) } ) = ( Z ` { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } ) ) |
145 |
144
|
eleq2d |
|- ( x = ( f " ( 1 ... n ) ) -> ( ( F ` k ) e. ( Z ` { ( G gsum ( H |` x ) ) } ) <-> ( F ` k ) e. ( Z ` { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } ) ) ) |
146 |
140 145
|
raleqbidv |
|- ( x = ( f " ( 1 ... n ) ) -> ( A. k e. ( A \ x ) ( F ` k ) e. ( Z ` { ( G gsum ( H |` x ) ) } ) <-> A. k e. ( A \ ( f " ( 1 ... n ) ) ) ( F ` k ) e. ( Z ` { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } ) ) ) |
147 |
139 146
|
imbi12d |
|- ( x = ( f " ( 1 ... n ) ) -> ( ( x C_ A -> A. k e. ( A \ x ) ( F ` k ) e. ( Z ` { ( G gsum ( H |` x ) ) } ) ) <-> ( ( f " ( 1 ... n ) ) C_ A -> A. k e. ( A \ ( f " ( 1 ... n ) ) ) ( F ` k ) e. ( Z ` { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } ) ) ) ) |
148 |
138 147
|
spcv |
|- ( A. x ( x C_ A -> A. k e. ( A \ x ) ( F ` k ) e. ( Z ` { ( G gsum ( H |` x ) ) } ) ) -> ( ( f " ( 1 ... n ) ) C_ A -> A. k e. ( A \ ( f " ( 1 ... n ) ) ) ( F ` k ) e. ( Z ` { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } ) ) ) |
149 |
132 136 148
|
sylc |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> A. k e. ( A \ ( f " ( 1 ... n ) ) ) ( F ` k ) e. ( Z ` { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } ) ) |
150 |
|
ffvelrn |
|- ( ( f : ( 1 ... ( # ` W ) ) --> A /\ ( n + 1 ) e. ( 1 ... ( # ` W ) ) ) -> ( f ` ( n + 1 ) ) e. A ) |
151 |
77 119 150
|
syl2an |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( f ` ( n + 1 ) ) e. A ) |
152 |
|
fzp1nel |
|- -. ( n + 1 ) e. ( 1 ... n ) |
153 |
75
|
adantr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> f : ( 1 ... ( # ` W ) ) -1-1-> A ) |
154 |
119
|
adantl |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( n + 1 ) e. ( 1 ... ( # ` W ) ) ) |
155 |
|
f1elima |
|- ( ( f : ( 1 ... ( # ` W ) ) -1-1-> A /\ ( n + 1 ) e. ( 1 ... ( # ` W ) ) /\ ( 1 ... n ) C_ ( 1 ... ( # ` W ) ) ) -> ( ( f ` ( n + 1 ) ) e. ( f " ( 1 ... n ) ) <-> ( n + 1 ) e. ( 1 ... n ) ) ) |
156 |
153 154 108 155
|
syl3anc |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( ( f ` ( n + 1 ) ) e. ( f " ( 1 ... n ) ) <-> ( n + 1 ) e. ( 1 ... n ) ) ) |
157 |
152 156
|
mtbiri |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> -. ( f ` ( n + 1 ) ) e. ( f " ( 1 ... n ) ) ) |
158 |
151 157
|
eldifd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( f ` ( n + 1 ) ) e. ( A \ ( f " ( 1 ... n ) ) ) ) |
159 |
127 149 158
|
rspcdva |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( F ` ( f ` ( n + 1 ) ) ) e. ( Z ` { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } ) ) |
160 |
125 159
|
eqeltrd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( ( F o. f ) ` ( n + 1 ) ) e. ( Z ` { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } ) ) |
161 |
138
|
a1i |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( f " ( 1 ... n ) ) e. _V ) |
162 |
11
|
ad2antrr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> H : A --> B ) |
163 |
162 136
|
fssresd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( H |` ( f " ( 1 ... n ) ) ) : ( f " ( 1 ... n ) ) --> B ) |
164 |
13
|
ad2antrr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ran H C_ ( Z ` ran H ) ) |
165 |
|
resss |
|- ( H |` ( f " ( 1 ... n ) ) ) C_ H |
166 |
165
|
rnssi |
|- ran ( H |` ( f " ( 1 ... n ) ) ) C_ ran H |
167 |
4
|
cntzidss |
|- ( ( ran H C_ ( Z ` ran H ) /\ ran ( H |` ( f " ( 1 ... n ) ) ) C_ ran H ) -> ran ( H |` ( f " ( 1 ... n ) ) ) C_ ( Z ` ran ( H |` ( f " ( 1 ... n ) ) ) ) ) |
168 |
164 166 167
|
sylancl |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ran ( H |` ( f " ( 1 ... n ) ) ) C_ ( Z ` ran ( H |` ( f " ( 1 ... n ) ) ) ) ) |
169 |
104 57
|
eleqtrrdi |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> n e. NN ) |
170 |
|
f1ores |
|- ( ( f : ( 1 ... ( # ` W ) ) -1-1-> A /\ ( 1 ... n ) C_ ( 1 ... ( # ` W ) ) ) -> ( f |` ( 1 ... n ) ) : ( 1 ... n ) -1-1-onto-> ( f " ( 1 ... n ) ) ) |
171 |
153 108 170
|
syl2anc |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( f |` ( 1 ... n ) ) : ( 1 ... n ) -1-1-onto-> ( f " ( 1 ... n ) ) ) |
172 |
|
f1of1 |
|- ( ( f |` ( 1 ... n ) ) : ( 1 ... n ) -1-1-onto-> ( f " ( 1 ... n ) ) -> ( f |` ( 1 ... n ) ) : ( 1 ... n ) -1-1-> ( f " ( 1 ... n ) ) ) |
173 |
171 172
|
syl |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( f |` ( 1 ... n ) ) : ( 1 ... n ) -1-1-> ( f " ( 1 ... n ) ) ) |
174 |
|
suppssdm |
|- ( ( H |` ( f " ( 1 ... n ) ) ) supp .0. ) C_ dom ( H |` ( f " ( 1 ... n ) ) ) |
175 |
|
dmres |
|- dom ( H |` ( f " ( 1 ... n ) ) ) = ( ( f " ( 1 ... n ) ) i^i dom H ) |
176 |
175
|
a1i |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> dom ( H |` ( f " ( 1 ... n ) ) ) = ( ( f " ( 1 ... n ) ) i^i dom H ) ) |
177 |
174 176
|
sseqtrid |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( ( H |` ( f " ( 1 ... n ) ) ) supp .0. ) C_ ( ( f " ( 1 ... n ) ) i^i dom H ) ) |
178 |
|
inss1 |
|- ( ( f " ( 1 ... n ) ) i^i dom H ) C_ ( f " ( 1 ... n ) ) |
179 |
|
df-ima |
|- ( f " ( 1 ... n ) ) = ran ( f |` ( 1 ... n ) ) |
180 |
179
|
a1i |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( f " ( 1 ... n ) ) = ran ( f |` ( 1 ... n ) ) ) |
181 |
178 180
|
sseqtrid |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( ( f " ( 1 ... n ) ) i^i dom H ) C_ ran ( f |` ( 1 ... n ) ) ) |
182 |
177 181
|
sstrd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( ( H |` ( f " ( 1 ... n ) ) ) supp .0. ) C_ ran ( f |` ( 1 ... n ) ) ) |
183 |
|
eqid |
|- ( ( ( H |` ( f " ( 1 ... n ) ) ) o. ( f |` ( 1 ... n ) ) ) supp .0. ) = ( ( ( H |` ( f " ( 1 ... n ) ) ) o. ( f |` ( 1 ... n ) ) ) supp .0. ) |
184 |
1 2 3 4 102 161 163 168 169 173 182 183
|
gsumval3 |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) = ( seq 1 ( .+ , ( ( H |` ( f " ( 1 ... n ) ) ) o. ( f |` ( 1 ... n ) ) ) ) ` n ) ) |
185 |
179
|
eqimss2i |
|- ran ( f |` ( 1 ... n ) ) C_ ( f " ( 1 ... n ) ) |
186 |
|
cores |
|- ( ran ( f |` ( 1 ... n ) ) C_ ( f " ( 1 ... n ) ) -> ( ( H |` ( f " ( 1 ... n ) ) ) o. ( f |` ( 1 ... n ) ) ) = ( H o. ( f |` ( 1 ... n ) ) ) ) |
187 |
185 186
|
ax-mp |
|- ( ( H |` ( f " ( 1 ... n ) ) ) o. ( f |` ( 1 ... n ) ) ) = ( H o. ( f |` ( 1 ... n ) ) ) |
188 |
|
resco |
|- ( ( H o. f ) |` ( 1 ... n ) ) = ( H o. ( f |` ( 1 ... n ) ) ) |
189 |
187 188
|
eqtr4i |
|- ( ( H |` ( f " ( 1 ... n ) ) ) o. ( f |` ( 1 ... n ) ) ) = ( ( H o. f ) |` ( 1 ... n ) ) |
190 |
189
|
fveq1i |
|- ( ( ( H |` ( f " ( 1 ... n ) ) ) o. ( f |` ( 1 ... n ) ) ) ` k ) = ( ( ( H o. f ) |` ( 1 ... n ) ) ` k ) |
191 |
|
fvres |
|- ( k e. ( 1 ... n ) -> ( ( ( H o. f ) |` ( 1 ... n ) ) ` k ) = ( ( H o. f ) ` k ) ) |
192 |
190 191
|
eqtrid |
|- ( k e. ( 1 ... n ) -> ( ( ( H |` ( f " ( 1 ... n ) ) ) o. ( f |` ( 1 ... n ) ) ) ` k ) = ( ( H o. f ) ` k ) ) |
193 |
192
|
adantl |
|- ( ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) /\ k e. ( 1 ... n ) ) -> ( ( ( H |` ( f " ( 1 ... n ) ) ) o. ( f |` ( 1 ... n ) ) ) ` k ) = ( ( H o. f ) ` k ) ) |
194 |
104 193
|
seqfveq |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( seq 1 ( .+ , ( ( H |` ( f " ( 1 ... n ) ) ) o. ( f |` ( 1 ... n ) ) ) ) ` n ) = ( seq 1 ( .+ , ( H o. f ) ) ` n ) ) |
195 |
184 194
|
eqtr2d |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( seq 1 ( .+ , ( H o. f ) ) ` n ) = ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) ) |
196 |
|
fvex |
|- ( seq 1 ( .+ , ( H o. f ) ) ` n ) e. _V |
197 |
196
|
elsn |
|- ( ( seq 1 ( .+ , ( H o. f ) ) ` n ) e. { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } <-> ( seq 1 ( .+ , ( H o. f ) ) ` n ) = ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) ) |
198 |
195 197
|
sylibr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( seq 1 ( .+ , ( H o. f ) ) ` n ) e. { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } ) |
199 |
3 4
|
cntzi |
|- ( ( ( ( F o. f ) ` ( n + 1 ) ) e. ( Z ` { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } ) /\ ( seq 1 ( .+ , ( H o. f ) ) ` n ) e. { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } ) -> ( ( ( F o. f ) ` ( n + 1 ) ) .+ ( seq 1 ( .+ , ( H o. f ) ) ` n ) ) = ( ( seq 1 ( .+ , ( H o. f ) ) ` n ) .+ ( ( F o. f ) ` ( n + 1 ) ) ) ) |
200 |
160 198 199
|
syl2anc |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( ( ( F o. f ) ` ( n + 1 ) ) .+ ( seq 1 ( .+ , ( H o. f ) ) ` n ) ) = ( ( seq 1 ( .+ , ( H o. f ) ) ` n ) .+ ( ( F o. f ) ` ( n + 1 ) ) ) ) |
201 |
200
|
eqcomd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( ( seq 1 ( .+ , ( H o. f ) ) ` n ) .+ ( ( F o. f ) ` ( n + 1 ) ) ) = ( ( ( F o. f ) ` ( n + 1 ) ) .+ ( seq 1 ( .+ , ( H o. f ) ) ` n ) ) ) |
202 |
1 3 102 115 118 121 123 201
|
mnd4g |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( ( ( seq 1 ( .+ , ( F o. f ) ) ` n ) .+ ( seq 1 ( .+ , ( H o. f ) ) ` n ) ) .+ ( ( ( F o. f ) ` ( n + 1 ) ) .+ ( ( H o. f ) ` ( n + 1 ) ) ) ) = ( ( ( seq 1 ( .+ , ( F o. f ) ) ` n ) .+ ( ( F o. f ) ` ( n + 1 ) ) ) .+ ( ( seq 1 ( .+ , ( H o. f ) ) ` n ) .+ ( ( H o. f ) ` ( n + 1 ) ) ) ) ) |
203 |
55 55 58 80 84 101 202
|
seqcaopr3 |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( seq 1 ( .+ , ( ( F oF .+ H ) o. f ) ) ` ( # ` W ) ) = ( ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` W ) ) .+ ( seq 1 ( .+ , ( H o. f ) ) ` ( # ` W ) ) ) ) |
204 |
54 59 81 87 87 89
|
off |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( F oF .+ H ) : A --> B ) |
205 |
14
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ran ( F oF .+ H ) C_ ( Z ` ran ( F oF .+ H ) ) ) |
206 |
51 113
|
sylan |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ ( k e. B /\ x e. B ) ) -> ( k .+ x ) e. B ) |
207 |
206 59 81 87 87 89
|
off |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( F oF .+ H ) : A --> B ) |
208 |
|
eldifi |
|- ( x e. ( A \ ran f ) -> x e. A ) |
209 |
|
eqidd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ x e. A ) -> ( F ` x ) = ( F ` x ) ) |
210 |
|
eqidd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ x e. A ) -> ( H ` x ) = ( H ` x ) ) |
211 |
85 86 87 87 89 209 210
|
ofval |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ x e. A ) -> ( ( F oF .+ H ) ` x ) = ( ( F ` x ) .+ ( H ` x ) ) ) |
212 |
208 211
|
sylan2 |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ x e. ( A \ ran f ) ) -> ( ( F oF .+ H ) ` x ) = ( ( F ` x ) .+ ( H ` x ) ) ) |
213 |
24
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( F supp .0. ) C_ ( ( F u. H ) supp .0. ) ) |
214 |
|
f1ofo |
|- ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W -> f : ( 1 ... ( # ` W ) ) -onto-> W ) |
215 |
|
forn |
|- ( f : ( 1 ... ( # ` W ) ) -onto-> W -> ran f = W ) |
216 |
214 215
|
syl |
|- ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W -> ran f = W ) |
217 |
216 9
|
eqtrdi |
|- ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W -> ran f = ( ( F u. H ) supp .0. ) ) |
218 |
217
|
sseq2d |
|- ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W -> ( ( F supp .0. ) C_ ran f <-> ( F supp .0. ) C_ ( ( F u. H ) supp .0. ) ) ) |
219 |
218
|
ad2antll |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( ( F supp .0. ) C_ ran f <-> ( F supp .0. ) C_ ( ( F u. H ) supp .0. ) ) ) |
220 |
213 219
|
mpbird |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( F supp .0. ) C_ ran f ) |
221 |
21
|
a1i |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> .0. e. _V ) |
222 |
59 220 87 221
|
suppssr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ x e. ( A \ ran f ) ) -> ( F ` x ) = .0. ) |
223 |
33
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( H supp .0. ) C_ ( ( H u. F ) supp .0. ) ) |
224 |
223 35
|
sseqtrrdi |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( H supp .0. ) C_ ( ( F u. H ) supp .0. ) ) |
225 |
217
|
sseq2d |
|- ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W -> ( ( H supp .0. ) C_ ran f <-> ( H supp .0. ) C_ ( ( F u. H ) supp .0. ) ) ) |
226 |
225
|
ad2antll |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( ( H supp .0. ) C_ ran f <-> ( H supp .0. ) C_ ( ( F u. H ) supp .0. ) ) ) |
227 |
224 226
|
mpbird |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( H supp .0. ) C_ ran f ) |
228 |
81 227 87 221
|
suppssr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ x e. ( A \ ran f ) ) -> ( H ` x ) = .0. ) |
229 |
222 228
|
oveq12d |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ x e. ( A \ ran f ) ) -> ( ( F ` x ) .+ ( H ` x ) ) = ( .0. .+ .0. ) ) |
230 |
19
|
ad2antrr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ x e. ( A \ ran f ) ) -> ( .0. .+ .0. ) = .0. ) |
231 |
212 229 230
|
3eqtrd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ x e. ( A \ ran f ) ) -> ( ( F oF .+ H ) ` x ) = .0. ) |
232 |
207 231
|
suppss |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( ( F oF .+ H ) supp .0. ) C_ ran f ) |
233 |
|
ovex |
|- ( F oF .+ H ) e. _V |
234 |
233 137
|
coex |
|- ( ( F oF .+ H ) o. f ) e. _V |
235 |
|
suppimacnv |
|- ( ( ( ( F oF .+ H ) o. f ) e. _V /\ .0. e. _V ) -> ( ( ( F oF .+ H ) o. f ) supp .0. ) = ( `' ( ( F oF .+ H ) o. f ) " ( _V \ { .0. } ) ) ) |
236 |
235
|
eqcomd |
|- ( ( ( ( F oF .+ H ) o. f ) e. _V /\ .0. e. _V ) -> ( `' ( ( F oF .+ H ) o. f ) " ( _V \ { .0. } ) ) = ( ( ( F oF .+ H ) o. f ) supp .0. ) ) |
237 |
234 21 236
|
mp2an |
|- ( `' ( ( F oF .+ H ) o. f ) " ( _V \ { .0. } ) ) = ( ( ( F oF .+ H ) o. f ) supp .0. ) |
238 |
1 2 3 4 51 87 204 205 56 75 232 237
|
gsumval3 |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( G gsum ( F oF .+ H ) ) = ( seq 1 ( .+ , ( ( F oF .+ H ) o. f ) ) ` ( # ` W ) ) ) |
239 |
12
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ran F C_ ( Z ` ran F ) ) |
240 |
|
eqid |
|- ( ( F o. f ) supp .0. ) = ( ( F o. f ) supp .0. ) |
241 |
1 2 3 4 51 87 59 239 56 75 220 240
|
gsumval3 |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( G gsum F ) = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` W ) ) ) |
242 |
13
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ran H C_ ( Z ` ran H ) ) |
243 |
|
eqid |
|- ( ( H o. f ) supp .0. ) = ( ( H o. f ) supp .0. ) |
244 |
1 2 3 4 51 87 81 242 56 75 227 243
|
gsumval3 |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( G gsum H ) = ( seq 1 ( .+ , ( H o. f ) ) ` ( # ` W ) ) ) |
245 |
241 244
|
oveq12d |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( ( G gsum F ) .+ ( G gsum H ) ) = ( ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` W ) ) .+ ( seq 1 ( .+ , ( H o. f ) ) ` ( # ` W ) ) ) ) |
246 |
203 238 245
|
3eqtr4d |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( G gsum ( F oF .+ H ) ) = ( ( G gsum F ) .+ ( G gsum H ) ) ) |
247 |
246
|
expr |
|- ( ( ph /\ ( # ` W ) e. NN ) -> ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W -> ( G gsum ( F oF .+ H ) ) = ( ( G gsum F ) .+ ( G gsum H ) ) ) ) |
248 |
247
|
exlimdv |
|- ( ( ph /\ ( # ` W ) e. NN ) -> ( E. f f : ( 1 ... ( # ` W ) ) -1-1-onto-> W -> ( G gsum ( F oF .+ H ) ) = ( ( G gsum F ) .+ ( G gsum H ) ) ) ) |
249 |
248
|
expimpd |
|- ( ph -> ( ( ( # ` W ) e. NN /\ E. f f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) -> ( G gsum ( F oF .+ H ) ) = ( ( G gsum F ) .+ ( G gsum H ) ) ) ) |
250 |
7 8
|
fsuppun |
|- ( ph -> ( ( F u. H ) supp .0. ) e. Fin ) |
251 |
9 250
|
eqeltrid |
|- ( ph -> W e. Fin ) |
252 |
|
fz1f1o |
|- ( W e. Fin -> ( W = (/) \/ ( ( # ` W ) e. NN /\ E. f f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) ) |
253 |
251 252
|
syl |
|- ( ph -> ( W = (/) \/ ( ( # ` W ) e. NN /\ E. f f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) ) |
254 |
50 249 253
|
mpjaod |
|- ( ph -> ( G gsum ( F oF .+ H ) ) = ( ( G gsum F ) .+ ( G gsum H ) ) ) |