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 |
|
fex |
|- ( ( H : A --> B /\ A e. V ) -> H e. _V ) |
24 |
11 6 23
|
syl2anc |
|- ( ph -> H e. _V ) |
25 |
24
|
suppun |
|- ( ph -> ( F supp .0. ) C_ ( ( F u. H ) supp .0. ) ) |
26 |
25 9
|
sseqtrrdi |
|- ( ph -> ( F supp .0. ) C_ W ) |
27 |
10 6 22 26
|
gsumcllem |
|- ( ( ph /\ W = (/) ) -> F = ( x e. A |-> .0. ) ) |
28 |
27
|
oveq2d |
|- ( ( ph /\ W = (/) ) -> ( G gsum F ) = ( G gsum ( x e. A |-> .0. ) ) ) |
29 |
2
|
gsumz |
|- ( ( G e. Mnd /\ A e. V ) -> ( G gsum ( x e. A |-> .0. ) ) = .0. ) |
30 |
5 6 29
|
syl2anc |
|- ( ph -> ( G gsum ( x e. A |-> .0. ) ) = .0. ) |
31 |
30
|
adantr |
|- ( ( ph /\ W = (/) ) -> ( G gsum ( x e. A |-> .0. ) ) = .0. ) |
32 |
28 31
|
eqtrd |
|- ( ( ph /\ W = (/) ) -> ( G gsum F ) = .0. ) |
33 |
|
fex |
|- ( ( F : A --> B /\ A e. V ) -> F e. _V ) |
34 |
10 6 33
|
syl2anc |
|- ( ph -> F e. _V ) |
35 |
34
|
suppun |
|- ( ph -> ( H supp .0. ) C_ ( ( H u. F ) supp .0. ) ) |
36 |
|
uncom |
|- ( F u. H ) = ( H u. F ) |
37 |
36
|
oveq1i |
|- ( ( F u. H ) supp .0. ) = ( ( H u. F ) supp .0. ) |
38 |
35 37
|
sseqtrrdi |
|- ( ph -> ( H supp .0. ) C_ ( ( F u. H ) supp .0. ) ) |
39 |
38 9
|
sseqtrrdi |
|- ( ph -> ( H supp .0. ) C_ W ) |
40 |
11 6 22 39
|
gsumcllem |
|- ( ( ph /\ W = (/) ) -> H = ( x e. A |-> .0. ) ) |
41 |
40
|
oveq2d |
|- ( ( ph /\ W = (/) ) -> ( G gsum H ) = ( G gsum ( x e. A |-> .0. ) ) ) |
42 |
41 31
|
eqtrd |
|- ( ( ph /\ W = (/) ) -> ( G gsum H ) = .0. ) |
43 |
32 42
|
oveq12d |
|- ( ( ph /\ W = (/) ) -> ( ( G gsum F ) .+ ( G gsum H ) ) = ( .0. .+ .0. ) ) |
44 |
6
|
adantr |
|- ( ( ph /\ W = (/) ) -> A e. V ) |
45 |
17
|
ad2antrr |
|- ( ( ( ph /\ W = (/) ) /\ x e. A ) -> .0. e. B ) |
46 |
44 45 45 27 40
|
offval2 |
|- ( ( ph /\ W = (/) ) -> ( F oF .+ H ) = ( x e. A |-> ( .0. .+ .0. ) ) ) |
47 |
20
|
mpteq2dv |
|- ( ( ph /\ W = (/) ) -> ( x e. A |-> ( .0. .+ .0. ) ) = ( x e. A |-> .0. ) ) |
48 |
46 47
|
eqtrd |
|- ( ( ph /\ W = (/) ) -> ( F oF .+ H ) = ( x e. A |-> .0. ) ) |
49 |
48
|
oveq2d |
|- ( ( ph /\ W = (/) ) -> ( G gsum ( F oF .+ H ) ) = ( G gsum ( x e. A |-> .0. ) ) ) |
50 |
49 31
|
eqtrd |
|- ( ( ph /\ W = (/) ) -> ( G gsum ( F oF .+ H ) ) = .0. ) |
51 |
20 43 50
|
3eqtr4rd |
|- ( ( ph /\ W = (/) ) -> ( G gsum ( F oF .+ H ) ) = ( ( G gsum F ) .+ ( G gsum H ) ) ) |
52 |
51
|
ex |
|- ( ph -> ( W = (/) -> ( G gsum ( F oF .+ H ) ) = ( ( G gsum F ) .+ ( G gsum H ) ) ) ) |
53 |
5
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> G e. Mnd ) |
54 |
1 3
|
mndcl |
|- ( ( G e. Mnd /\ z e. B /\ w e. B ) -> ( z .+ w ) e. B ) |
55 |
54
|
3expb |
|- ( ( G e. Mnd /\ ( z e. B /\ w e. B ) ) -> ( z .+ w ) e. B ) |
56 |
53 55
|
sylan |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ ( z e. B /\ w e. B ) ) -> ( z .+ w ) e. B ) |
57 |
56
|
caovclg |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ ( x e. B /\ y e. B ) ) -> ( x .+ y ) e. B ) |
58 |
|
simprl |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( # ` W ) e. NN ) |
59 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
60 |
58 59
|
eleqtrdi |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( # ` W ) e. ( ZZ>= ` 1 ) ) |
61 |
10
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> F : A --> B ) |
62 |
|
f1of1 |
|- ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W -> f : ( 1 ... ( # ` W ) ) -1-1-> W ) |
63 |
62
|
ad2antll |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> f : ( 1 ... ( # ` W ) ) -1-1-> W ) |
64 |
|
suppssdm |
|- ( ( F u. H ) supp .0. ) C_ dom ( F u. H ) |
65 |
64
|
a1i |
|- ( ph -> ( ( F u. H ) supp .0. ) C_ dom ( F u. H ) ) |
66 |
9
|
a1i |
|- ( ph -> W = ( ( F u. H ) supp .0. ) ) |
67 |
|
dmun |
|- dom ( F u. H ) = ( dom F u. dom H ) |
68 |
10
|
fdmd |
|- ( ph -> dom F = A ) |
69 |
11
|
fdmd |
|- ( ph -> dom H = A ) |
70 |
68 69
|
uneq12d |
|- ( ph -> ( dom F u. dom H ) = ( A u. A ) ) |
71 |
|
unidm |
|- ( A u. A ) = A |
72 |
70 71
|
eqtrdi |
|- ( ph -> ( dom F u. dom H ) = A ) |
73 |
67 72
|
syl5req |
|- ( ph -> A = dom ( F u. H ) ) |
74 |
65 66 73
|
3sstr4d |
|- ( ph -> W C_ A ) |
75 |
74
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> W C_ A ) |
76 |
|
f1ss |
|- ( ( f : ( 1 ... ( # ` W ) ) -1-1-> W /\ W C_ A ) -> f : ( 1 ... ( # ` W ) ) -1-1-> A ) |
77 |
63 75 76
|
syl2anc |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> f : ( 1 ... ( # ` W ) ) -1-1-> A ) |
78 |
|
f1f |
|- ( f : ( 1 ... ( # ` W ) ) -1-1-> A -> f : ( 1 ... ( # ` W ) ) --> A ) |
79 |
77 78
|
syl |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> f : ( 1 ... ( # ` W ) ) --> A ) |
80 |
|
fco |
|- ( ( F : A --> B /\ f : ( 1 ... ( # ` W ) ) --> A ) -> ( F o. f ) : ( 1 ... ( # ` W ) ) --> B ) |
81 |
61 79 80
|
syl2anc |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( F o. f ) : ( 1 ... ( # ` W ) ) --> B ) |
82 |
81
|
ffvelrnda |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ k e. ( 1 ... ( # ` W ) ) ) -> ( ( F o. f ) ` k ) e. B ) |
83 |
11
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> H : A --> B ) |
84 |
|
fco |
|- ( ( H : A --> B /\ f : ( 1 ... ( # ` W ) ) --> A ) -> ( H o. f ) : ( 1 ... ( # ` W ) ) --> B ) |
85 |
83 79 84
|
syl2anc |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( H o. f ) : ( 1 ... ( # ` W ) ) --> B ) |
86 |
85
|
ffvelrnda |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ k e. ( 1 ... ( # ` W ) ) ) -> ( ( H o. f ) ` k ) e. B ) |
87 |
61
|
ffnd |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> F Fn A ) |
88 |
83
|
ffnd |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> H Fn A ) |
89 |
6
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> A e. V ) |
90 |
|
ovexd |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( 1 ... ( # ` W ) ) e. _V ) |
91 |
|
inidm |
|- ( A i^i A ) = A |
92 |
87 88 79 89 89 90 91
|
ofco |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( ( F oF .+ H ) o. f ) = ( ( F o. f ) oF .+ ( H o. f ) ) ) |
93 |
92
|
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 ) ) |
94 |
93
|
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 ) ) |
95 |
|
fnfco |
|- ( ( F Fn A /\ f : ( 1 ... ( # ` W ) ) --> A ) -> ( F o. f ) Fn ( 1 ... ( # ` W ) ) ) |
96 |
87 79 95
|
syl2anc |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( F o. f ) Fn ( 1 ... ( # ` W ) ) ) |
97 |
|
fnfco |
|- ( ( H Fn A /\ f : ( 1 ... ( # ` W ) ) --> A ) -> ( H o. f ) Fn ( 1 ... ( # ` W ) ) ) |
98 |
88 79 97
|
syl2anc |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( H o. f ) Fn ( 1 ... ( # ` W ) ) ) |
99 |
|
inidm |
|- ( ( 1 ... ( # ` W ) ) i^i ( 1 ... ( # ` W ) ) ) = ( 1 ... ( # ` W ) ) |
100 |
|
eqidd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ k e. ( 1 ... ( # ` W ) ) ) -> ( ( F o. f ) ` k ) = ( ( F o. f ) ` k ) ) |
101 |
|
eqidd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ k e. ( 1 ... ( # ` W ) ) ) -> ( ( H o. f ) ` k ) = ( ( H o. f ) ` k ) ) |
102 |
96 98 90 90 99 100 101
|
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 ) ) ) |
103 |
94 102
|
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 ) ) ) |
104 |
5
|
ad2antrr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> G e. Mnd ) |
105 |
|
elfzouz |
|- ( n e. ( 1 ..^ ( # ` W ) ) -> n e. ( ZZ>= ` 1 ) ) |
106 |
105
|
adantl |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> n e. ( ZZ>= ` 1 ) ) |
107 |
|
elfzouz2 |
|- ( n e. ( 1 ..^ ( # ` W ) ) -> ( # ` W ) e. ( ZZ>= ` n ) ) |
108 |
107
|
adantl |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( # ` W ) e. ( ZZ>= ` n ) ) |
109 |
|
fzss2 |
|- ( ( # ` W ) e. ( ZZ>= ` n ) -> ( 1 ... n ) C_ ( 1 ... ( # ` W ) ) ) |
110 |
108 109
|
syl |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( 1 ... n ) C_ ( 1 ... ( # ` W ) ) ) |
111 |
110
|
sselda |
|- ( ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) /\ k e. ( 1 ... n ) ) -> k e. ( 1 ... ( # ` W ) ) ) |
112 |
82
|
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 ) |
113 |
111 112
|
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 ) |
114 |
1 3
|
mndcl |
|- ( ( G e. Mnd /\ k e. B /\ x e. B ) -> ( k .+ x ) e. B ) |
115 |
114
|
3expb |
|- ( ( G e. Mnd /\ ( k e. B /\ x e. B ) ) -> ( k .+ x ) e. B ) |
116 |
104 115
|
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 ) |
117 |
106 113 116
|
seqcl |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( seq 1 ( .+ , ( F o. f ) ) ` n ) e. B ) |
118 |
86
|
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 ) |
119 |
111 118
|
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 ) |
120 |
106 119 116
|
seqcl |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( seq 1 ( .+ , ( H o. f ) ) ` n ) e. B ) |
121 |
|
fzofzp1 |
|- ( n e. ( 1 ..^ ( # ` W ) ) -> ( n + 1 ) e. ( 1 ... ( # ` W ) ) ) |
122 |
|
ffvelrn |
|- ( ( ( F o. f ) : ( 1 ... ( # ` W ) ) --> B /\ ( n + 1 ) e. ( 1 ... ( # ` W ) ) ) -> ( ( F o. f ) ` ( n + 1 ) ) e. B ) |
123 |
81 121 122
|
syl2an |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( ( F o. f ) ` ( n + 1 ) ) e. B ) |
124 |
|
ffvelrn |
|- ( ( ( H o. f ) : ( 1 ... ( # ` W ) ) --> B /\ ( n + 1 ) e. ( 1 ... ( # ` W ) ) ) -> ( ( H o. f ) ` ( n + 1 ) ) e. B ) |
125 |
85 121 124
|
syl2an |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( ( H o. f ) ` ( n + 1 ) ) e. B ) |
126 |
|
fvco3 |
|- ( ( f : ( 1 ... ( # ` W ) ) --> A /\ ( n + 1 ) e. ( 1 ... ( # ` W ) ) ) -> ( ( F o. f ) ` ( n + 1 ) ) = ( F ` ( f ` ( n + 1 ) ) ) ) |
127 |
79 121 126
|
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 ) ) ) ) |
128 |
|
fveq2 |
|- ( k = ( f ` ( n + 1 ) ) -> ( F ` k ) = ( F ` ( f ` ( n + 1 ) ) ) ) |
129 |
128
|
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 ) ) ) ) } ) ) ) |
130 |
15
|
expr |
|- ( ( ph /\ x C_ A ) -> ( k e. ( A \ x ) -> ( F ` k ) e. ( Z ` { ( G gsum ( H |` x ) ) } ) ) ) |
131 |
130
|
ralrimiv |
|- ( ( ph /\ x C_ A ) -> A. k e. ( A \ x ) ( F ` k ) e. ( Z ` { ( G gsum ( H |` x ) ) } ) ) |
132 |
131
|
ex |
|- ( ph -> ( x C_ A -> A. k e. ( A \ x ) ( F ` k ) e. ( Z ` { ( G gsum ( H |` x ) ) } ) ) ) |
133 |
132
|
alrimiv |
|- ( ph -> A. x ( x C_ A -> A. k e. ( A \ x ) ( F ` k ) e. ( Z ` { ( G gsum ( H |` x ) ) } ) ) ) |
134 |
133
|
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 ) ) } ) ) ) |
135 |
|
imassrn |
|- ( f " ( 1 ... n ) ) C_ ran f |
136 |
79
|
adantr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> f : ( 1 ... ( # ` W ) ) --> A ) |
137 |
136
|
frnd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ran f C_ A ) |
138 |
135 137
|
sstrid |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( f " ( 1 ... n ) ) C_ A ) |
139 |
|
vex |
|- f e. _V |
140 |
139
|
imaex |
|- ( f " ( 1 ... n ) ) e. _V |
141 |
|
sseq1 |
|- ( x = ( f " ( 1 ... n ) ) -> ( x C_ A <-> ( f " ( 1 ... n ) ) C_ A ) ) |
142 |
|
difeq2 |
|- ( x = ( f " ( 1 ... n ) ) -> ( A \ x ) = ( A \ ( f " ( 1 ... n ) ) ) ) |
143 |
|
reseq2 |
|- ( x = ( f " ( 1 ... n ) ) -> ( H |` x ) = ( H |` ( f " ( 1 ... n ) ) ) ) |
144 |
143
|
oveq2d |
|- ( x = ( f " ( 1 ... n ) ) -> ( G gsum ( H |` x ) ) = ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) ) |
145 |
144
|
sneqd |
|- ( x = ( f " ( 1 ... n ) ) -> { ( G gsum ( H |` x ) ) } = { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } ) |
146 |
145
|
fveq2d |
|- ( x = ( f " ( 1 ... n ) ) -> ( Z ` { ( G gsum ( H |` x ) ) } ) = ( Z ` { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } ) ) |
147 |
146
|
eleq2d |
|- ( x = ( f " ( 1 ... n ) ) -> ( ( F ` k ) e. ( Z ` { ( G gsum ( H |` x ) ) } ) <-> ( F ` k ) e. ( Z ` { ( G gsum ( H |` ( f " ( 1 ... n ) ) ) ) } ) ) ) |
148 |
142 147
|
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 ) ) ) ) } ) ) ) |
149 |
141 148
|
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 ) ) ) ) } ) ) ) ) |
150 |
140 149
|
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 ) ) ) ) } ) ) ) |
151 |
134 138 150
|
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 ) ) ) ) } ) ) |
152 |
|
ffvelrn |
|- ( ( f : ( 1 ... ( # ` W ) ) --> A /\ ( n + 1 ) e. ( 1 ... ( # ` W ) ) ) -> ( f ` ( n + 1 ) ) e. A ) |
153 |
79 121 152
|
syl2an |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( f ` ( n + 1 ) ) e. A ) |
154 |
|
fzp1nel |
|- -. ( n + 1 ) e. ( 1 ... n ) |
155 |
77
|
adantr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> f : ( 1 ... ( # ` W ) ) -1-1-> A ) |
156 |
121
|
adantl |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( n + 1 ) e. ( 1 ... ( # ` W ) ) ) |
157 |
|
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 ) ) ) |
158 |
155 156 110 157
|
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 ) ) ) |
159 |
154 158
|
mtbiri |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> -. ( f ` ( n + 1 ) ) e. ( f " ( 1 ... n ) ) ) |
160 |
153 159
|
eldifd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( f ` ( n + 1 ) ) e. ( A \ ( f " ( 1 ... n ) ) ) ) |
161 |
129 151 160
|
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 ) ) ) ) } ) ) |
162 |
127 161
|
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 ) ) ) ) } ) ) |
163 |
140
|
a1i |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( f " ( 1 ... n ) ) e. _V ) |
164 |
11
|
ad2antrr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> H : A --> B ) |
165 |
164 138
|
fssresd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( H |` ( f " ( 1 ... n ) ) ) : ( f " ( 1 ... n ) ) --> B ) |
166 |
13
|
ad2antrr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ran H C_ ( Z ` ran H ) ) |
167 |
|
resss |
|- ( H |` ( f " ( 1 ... n ) ) ) C_ H |
168 |
167
|
rnssi |
|- ran ( H |` ( f " ( 1 ... n ) ) ) C_ ran H |
169 |
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 ) ) ) ) ) |
170 |
166 168 169
|
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 ) ) ) ) ) |
171 |
106 59
|
eleqtrrdi |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> n e. NN ) |
172 |
|
f1ores |
|- ( ( f : ( 1 ... ( # ` W ) ) -1-1-> A /\ ( 1 ... n ) C_ ( 1 ... ( # ` W ) ) ) -> ( f |` ( 1 ... n ) ) : ( 1 ... n ) -1-1-onto-> ( f " ( 1 ... n ) ) ) |
173 |
155 110 172
|
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 ) ) ) |
174 |
|
f1of1 |
|- ( ( f |` ( 1 ... n ) ) : ( 1 ... n ) -1-1-onto-> ( f " ( 1 ... n ) ) -> ( f |` ( 1 ... n ) ) : ( 1 ... n ) -1-1-> ( f " ( 1 ... n ) ) ) |
175 |
173 174
|
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 ) ) ) |
176 |
|
suppssdm |
|- ( ( H |` ( f " ( 1 ... n ) ) ) supp .0. ) C_ dom ( H |` ( f " ( 1 ... n ) ) ) |
177 |
|
dmres |
|- dom ( H |` ( f " ( 1 ... n ) ) ) = ( ( f " ( 1 ... n ) ) i^i dom H ) |
178 |
177
|
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 ) ) |
179 |
176 178
|
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 ) ) |
180 |
|
inss1 |
|- ( ( f " ( 1 ... n ) ) i^i dom H ) C_ ( f " ( 1 ... n ) ) |
181 |
|
df-ima |
|- ( f " ( 1 ... n ) ) = ran ( f |` ( 1 ... n ) ) |
182 |
181
|
a1i |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ n e. ( 1 ..^ ( # ` W ) ) ) -> ( f " ( 1 ... n ) ) = ran ( f |` ( 1 ... n ) ) ) |
183 |
180 182
|
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 ) ) ) |
184 |
179 183
|
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 ) ) ) |
185 |
|
eqid |
|- ( ( ( H |` ( f " ( 1 ... n ) ) ) o. ( f |` ( 1 ... n ) ) ) supp .0. ) = ( ( ( H |` ( f " ( 1 ... n ) ) ) o. ( f |` ( 1 ... n ) ) ) supp .0. ) |
186 |
1 2 3 4 104 163 165 170 171 175 184 185
|
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 ) ) |
187 |
181
|
eqimss2i |
|- ran ( f |` ( 1 ... n ) ) C_ ( f " ( 1 ... n ) ) |
188 |
|
cores |
|- ( ran ( f |` ( 1 ... n ) ) C_ ( f " ( 1 ... n ) ) -> ( ( H |` ( f " ( 1 ... n ) ) ) o. ( f |` ( 1 ... n ) ) ) = ( H o. ( f |` ( 1 ... n ) ) ) ) |
189 |
187 188
|
ax-mp |
|- ( ( H |` ( f " ( 1 ... n ) ) ) o. ( f |` ( 1 ... n ) ) ) = ( H o. ( f |` ( 1 ... n ) ) ) |
190 |
|
resco |
|- ( ( H o. f ) |` ( 1 ... n ) ) = ( H o. ( f |` ( 1 ... n ) ) ) |
191 |
189 190
|
eqtr4i |
|- ( ( H |` ( f " ( 1 ... n ) ) ) o. ( f |` ( 1 ... n ) ) ) = ( ( H o. f ) |` ( 1 ... n ) ) |
192 |
191
|
fveq1i |
|- ( ( ( H |` ( f " ( 1 ... n ) ) ) o. ( f |` ( 1 ... n ) ) ) ` k ) = ( ( ( H o. f ) |` ( 1 ... n ) ) ` k ) |
193 |
|
fvres |
|- ( k e. ( 1 ... n ) -> ( ( ( H o. f ) |` ( 1 ... n ) ) ` k ) = ( ( H o. f ) ` k ) ) |
194 |
192 193
|
syl5eq |
|- ( k e. ( 1 ... n ) -> ( ( ( H |` ( f " ( 1 ... n ) ) ) o. ( f |` ( 1 ... n ) ) ) ` k ) = ( ( H o. f ) ` k ) ) |
195 |
194
|
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 ) ) |
196 |
106 195
|
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 ) ) |
197 |
186 196
|
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 ) ) ) ) ) |
198 |
|
fvex |
|- ( seq 1 ( .+ , ( H o. f ) ) ` n ) e. _V |
199 |
198
|
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 ) ) ) ) ) |
200 |
197 199
|
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 ) ) ) ) } ) |
201 |
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 ) ) ) ) |
202 |
162 200 201
|
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 ) ) ) ) |
203 |
202
|
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 ) ) ) |
204 |
1 3 104 117 120 123 125 203
|
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 ) ) ) ) ) |
205 |
57 57 60 82 86 103 204
|
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 ) ) ) ) |
206 |
56 61 83 89 89 91
|
off |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( F oF .+ H ) : A --> B ) |
207 |
14
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ran ( F oF .+ H ) C_ ( Z ` ran ( F oF .+ H ) ) ) |
208 |
53 115
|
sylan |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ ( k e. B /\ x e. B ) ) -> ( k .+ x ) e. B ) |
209 |
208 61 83 89 89 91
|
off |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( F oF .+ H ) : A --> B ) |
210 |
|
eldifi |
|- ( x e. ( A \ ran f ) -> x e. A ) |
211 |
|
eqidd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ x e. A ) -> ( F ` x ) = ( F ` x ) ) |
212 |
|
eqidd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ x e. A ) -> ( H ` x ) = ( H ` x ) ) |
213 |
87 88 89 89 91 211 212
|
ofval |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ x e. A ) -> ( ( F oF .+ H ) ` x ) = ( ( F ` x ) .+ ( H ` x ) ) ) |
214 |
210 213
|
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 ) ) ) |
215 |
25
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( F supp .0. ) C_ ( ( F u. H ) supp .0. ) ) |
216 |
|
f1ofo |
|- ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W -> f : ( 1 ... ( # ` W ) ) -onto-> W ) |
217 |
|
forn |
|- ( f : ( 1 ... ( # ` W ) ) -onto-> W -> ran f = W ) |
218 |
216 217
|
syl |
|- ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W -> ran f = W ) |
219 |
218 9
|
eqtrdi |
|- ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W -> ran f = ( ( F u. H ) supp .0. ) ) |
220 |
219
|
sseq2d |
|- ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W -> ( ( F supp .0. ) C_ ran f <-> ( F supp .0. ) C_ ( ( F u. H ) supp .0. ) ) ) |
221 |
220
|
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. ) ) ) |
222 |
215 221
|
mpbird |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( F supp .0. ) C_ ran f ) |
223 |
21
|
a1i |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> .0. e. _V ) |
224 |
61 222 89 223
|
suppssr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ x e. ( A \ ran f ) ) -> ( F ` x ) = .0. ) |
225 |
35
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( H supp .0. ) C_ ( ( H u. F ) supp .0. ) ) |
226 |
225 37
|
sseqtrrdi |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( H supp .0. ) C_ ( ( F u. H ) supp .0. ) ) |
227 |
219
|
sseq2d |
|- ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W -> ( ( H supp .0. ) C_ ran f <-> ( H supp .0. ) C_ ( ( F u. H ) supp .0. ) ) ) |
228 |
227
|
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. ) ) ) |
229 |
226 228
|
mpbird |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( H supp .0. ) C_ ran f ) |
230 |
83 229 89 223
|
suppssr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ x e. ( A \ ran f ) ) -> ( H ` x ) = .0. ) |
231 |
224 230
|
oveq12d |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ x e. ( A \ ran f ) ) -> ( ( F ` x ) .+ ( H ` x ) ) = ( .0. .+ .0. ) ) |
232 |
19
|
ad2antrr |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ x e. ( A \ ran f ) ) -> ( .0. .+ .0. ) = .0. ) |
233 |
214 231 232
|
3eqtrd |
|- ( ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) /\ x e. ( A \ ran f ) ) -> ( ( F oF .+ H ) ` x ) = .0. ) |
234 |
209 233
|
suppss |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( ( F oF .+ H ) supp .0. ) C_ ran f ) |
235 |
|
ovex |
|- ( F oF .+ H ) e. _V |
236 |
235 139
|
coex |
|- ( ( F oF .+ H ) o. f ) e. _V |
237 |
|
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. } ) ) ) |
238 |
237
|
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. ) ) |
239 |
236 21 238
|
mp2an |
|- ( `' ( ( F oF .+ H ) o. f ) " ( _V \ { .0. } ) ) = ( ( ( F oF .+ H ) o. f ) supp .0. ) |
240 |
1 2 3 4 53 89 206 207 58 77 234 239
|
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 ) ) ) |
241 |
12
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ran F C_ ( Z ` ran F ) ) |
242 |
|
eqid |
|- ( ( F o. f ) supp .0. ) = ( ( F o. f ) supp .0. ) |
243 |
1 2 3 4 53 89 61 241 58 77 222 242
|
gsumval3 |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( G gsum F ) = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` W ) ) ) |
244 |
13
|
adantr |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ran H C_ ( Z ` ran H ) ) |
245 |
|
eqid |
|- ( ( H o. f ) supp .0. ) = ( ( H o. f ) supp .0. ) |
246 |
1 2 3 4 53 89 83 244 58 77 229 245
|
gsumval3 |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( G gsum H ) = ( seq 1 ( .+ , ( H o. f ) ) ` ( # ` W ) ) ) |
247 |
243 246
|
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 ) ) ) ) |
248 |
205 240 247
|
3eqtr4d |
|- ( ( ph /\ ( ( # ` W ) e. NN /\ f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) -> ( G gsum ( F oF .+ H ) ) = ( ( G gsum F ) .+ ( G gsum H ) ) ) |
249 |
248
|
expr |
|- ( ( ph /\ ( # ` W ) e. NN ) -> ( f : ( 1 ... ( # ` W ) ) -1-1-onto-> W -> ( G gsum ( F oF .+ H ) ) = ( ( G gsum F ) .+ ( G gsum H ) ) ) ) |
250 |
249
|
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 ) ) ) ) |
251 |
250
|
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 ) ) ) ) |
252 |
7 8
|
fsuppun |
|- ( ph -> ( ( F u. H ) supp .0. ) e. Fin ) |
253 |
9 252
|
eqeltrid |
|- ( ph -> W e. Fin ) |
254 |
|
fz1f1o |
|- ( W e. Fin -> ( W = (/) \/ ( ( # ` W ) e. NN /\ E. f f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) ) |
255 |
253 254
|
syl |
|- ( ph -> ( W = (/) \/ ( ( # ` W ) e. NN /\ E. f f : ( 1 ... ( # ` W ) ) -1-1-onto-> W ) ) ) |
256 |
52 251 255
|
mpjaod |
|- ( ph -> ( G gsum ( F oF .+ H ) ) = ( ( G gsum F ) .+ ( G gsum H ) ) ) |