Step |
Hyp |
Ref |
Expression |
1 |
|
seqhomo.1 |
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) |
2 |
|
seqhomo.2 |
|- ( ( ph /\ x e. ( M ... N ) ) -> ( F ` x ) e. S ) |
3 |
|
seqz.3 |
|- ( ( ph /\ x e. S ) -> ( Z .+ x ) = Z ) |
4 |
|
seqz.4 |
|- ( ( ph /\ x e. S ) -> ( x .+ Z ) = Z ) |
5 |
|
seqz.5 |
|- ( ph -> K e. ( M ... N ) ) |
6 |
|
seqz.6 |
|- ( ph -> N e. V ) |
7 |
|
seqz.7 |
|- ( ph -> ( F ` K ) = Z ) |
8 |
|
elfzuz |
|- ( K e. ( M ... N ) -> K e. ( ZZ>= ` M ) ) |
9 |
5 8
|
syl |
|- ( ph -> K e. ( ZZ>= ` M ) ) |
10 |
5
|
elfzelzd |
|- ( ph -> K e. ZZ ) |
11 |
|
seq1 |
|- ( K e. ZZ -> ( seq K ( .+ , F ) ` K ) = ( F ` K ) ) |
12 |
10 11
|
syl |
|- ( ph -> ( seq K ( .+ , F ) ` K ) = ( F ` K ) ) |
13 |
12 7
|
eqtrd |
|- ( ph -> ( seq K ( .+ , F ) ` K ) = Z ) |
14 |
|
seqeq1 |
|- ( K = M -> seq K ( .+ , F ) = seq M ( .+ , F ) ) |
15 |
14
|
fveq1d |
|- ( K = M -> ( seq K ( .+ , F ) ` K ) = ( seq M ( .+ , F ) ` K ) ) |
16 |
15
|
eqeq1d |
|- ( K = M -> ( ( seq K ( .+ , F ) ` K ) = Z <-> ( seq M ( .+ , F ) ` K ) = Z ) ) |
17 |
13 16
|
syl5ibcom |
|- ( ph -> ( K = M -> ( seq M ( .+ , F ) ` K ) = Z ) ) |
18 |
|
eluzel2 |
|- ( K e. ( ZZ>= ` M ) -> M e. ZZ ) |
19 |
9 18
|
syl |
|- ( ph -> M e. ZZ ) |
20 |
|
seqm1 |
|- ( ( M e. ZZ /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` K ) = ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ ( F ` K ) ) ) |
21 |
19 20
|
sylan |
|- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` K ) = ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ ( F ` K ) ) ) |
22 |
7
|
adantr |
|- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( F ` K ) = Z ) |
23 |
22
|
oveq2d |
|- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ ( F ` K ) ) = ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ Z ) ) |
24 |
|
oveq1 |
|- ( x = ( seq M ( .+ , F ) ` ( K - 1 ) ) -> ( x .+ Z ) = ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ Z ) ) |
25 |
24
|
eqeq1d |
|- ( x = ( seq M ( .+ , F ) ` ( K - 1 ) ) -> ( ( x .+ Z ) = Z <-> ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ Z ) = Z ) ) |
26 |
4
|
ralrimiva |
|- ( ph -> A. x e. S ( x .+ Z ) = Z ) |
27 |
26
|
adantr |
|- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> A. x e. S ( x .+ Z ) = Z ) |
28 |
|
eluzp1m1 |
|- ( ( M e. ZZ /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( K - 1 ) e. ( ZZ>= ` M ) ) |
29 |
19 28
|
sylan |
|- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( K - 1 ) e. ( ZZ>= ` M ) ) |
30 |
|
fzssp1 |
|- ( M ... ( K - 1 ) ) C_ ( M ... ( ( K - 1 ) + 1 ) ) |
31 |
10
|
zcnd |
|- ( ph -> K e. CC ) |
32 |
|
ax-1cn |
|- 1 e. CC |
33 |
|
npcan |
|- ( ( K e. CC /\ 1 e. CC ) -> ( ( K - 1 ) + 1 ) = K ) |
34 |
31 32 33
|
sylancl |
|- ( ph -> ( ( K - 1 ) + 1 ) = K ) |
35 |
34
|
oveq2d |
|- ( ph -> ( M ... ( ( K - 1 ) + 1 ) ) = ( M ... K ) ) |
36 |
30 35
|
sseqtrid |
|- ( ph -> ( M ... ( K - 1 ) ) C_ ( M ... K ) ) |
37 |
|
elfzuz3 |
|- ( K e. ( M ... N ) -> N e. ( ZZ>= ` K ) ) |
38 |
5 37
|
syl |
|- ( ph -> N e. ( ZZ>= ` K ) ) |
39 |
|
fzss2 |
|- ( N e. ( ZZ>= ` K ) -> ( M ... K ) C_ ( M ... N ) ) |
40 |
38 39
|
syl |
|- ( ph -> ( M ... K ) C_ ( M ... N ) ) |
41 |
36 40
|
sstrd |
|- ( ph -> ( M ... ( K - 1 ) ) C_ ( M ... N ) ) |
42 |
41
|
adantr |
|- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( M ... ( K - 1 ) ) C_ ( M ... N ) ) |
43 |
42
|
sselda |
|- ( ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( M ... ( K - 1 ) ) ) -> x e. ( M ... N ) ) |
44 |
2
|
adantlr |
|- ( ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( M ... N ) ) -> ( F ` x ) e. S ) |
45 |
43 44
|
syldan |
|- ( ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) /\ x e. ( M ... ( K - 1 ) ) ) -> ( F ` x ) e. S ) |
46 |
1
|
adantlr |
|- ( ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) |
47 |
29 45 46
|
seqcl |
|- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` ( K - 1 ) ) e. S ) |
48 |
25 27 47
|
rspcdva |
|- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ Z ) = Z ) |
49 |
23 48
|
eqtrd |
|- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( ( seq M ( .+ , F ) ` ( K - 1 ) ) .+ ( F ` K ) ) = Z ) |
50 |
21 49
|
eqtrd |
|- ( ( ph /\ K e. ( ZZ>= ` ( M + 1 ) ) ) -> ( seq M ( .+ , F ) ` K ) = Z ) |
51 |
50
|
ex |
|- ( ph -> ( K e. ( ZZ>= ` ( M + 1 ) ) -> ( seq M ( .+ , F ) ` K ) = Z ) ) |
52 |
|
uzp1 |
|- ( K e. ( ZZ>= ` M ) -> ( K = M \/ K e. ( ZZ>= ` ( M + 1 ) ) ) ) |
53 |
9 52
|
syl |
|- ( ph -> ( K = M \/ K e. ( ZZ>= ` ( M + 1 ) ) ) ) |
54 |
17 51 53
|
mpjaod |
|- ( ph -> ( seq M ( .+ , F ) ` K ) = Z ) |
55 |
54 7
|
eqtr4d |
|- ( ph -> ( seq M ( .+ , F ) ` K ) = ( F ` K ) ) |
56 |
|
eqidd |
|- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> ( F ` x ) = ( F ` x ) ) |
57 |
9 55 38 56
|
seqfveq2 |
|- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq K ( .+ , F ) ` N ) ) |
58 |
|
fvex |
|- ( F ` K ) e. _V |
59 |
58
|
elsn |
|- ( ( F ` K ) e. { Z } <-> ( F ` K ) = Z ) |
60 |
7 59
|
sylibr |
|- ( ph -> ( F ` K ) e. { Z } ) |
61 |
|
simprl |
|- ( ( ph /\ ( x e. { Z } /\ y e. S ) ) -> x e. { Z } ) |
62 |
|
velsn |
|- ( x e. { Z } <-> x = Z ) |
63 |
61 62
|
sylib |
|- ( ( ph /\ ( x e. { Z } /\ y e. S ) ) -> x = Z ) |
64 |
63
|
oveq1d |
|- ( ( ph /\ ( x e. { Z } /\ y e. S ) ) -> ( x .+ y ) = ( Z .+ y ) ) |
65 |
|
oveq2 |
|- ( x = y -> ( Z .+ x ) = ( Z .+ y ) ) |
66 |
65
|
eqeq1d |
|- ( x = y -> ( ( Z .+ x ) = Z <-> ( Z .+ y ) = Z ) ) |
67 |
3
|
ralrimiva |
|- ( ph -> A. x e. S ( Z .+ x ) = Z ) |
68 |
67
|
adantr |
|- ( ( ph /\ ( x e. { Z } /\ y e. S ) ) -> A. x e. S ( Z .+ x ) = Z ) |
69 |
|
simprr |
|- ( ( ph /\ ( x e. { Z } /\ y e. S ) ) -> y e. S ) |
70 |
66 68 69
|
rspcdva |
|- ( ( ph /\ ( x e. { Z } /\ y e. S ) ) -> ( Z .+ y ) = Z ) |
71 |
64 70
|
eqtrd |
|- ( ( ph /\ ( x e. { Z } /\ y e. S ) ) -> ( x .+ y ) = Z ) |
72 |
|
ovex |
|- ( x .+ y ) e. _V |
73 |
72
|
elsn |
|- ( ( x .+ y ) e. { Z } <-> ( x .+ y ) = Z ) |
74 |
71 73
|
sylibr |
|- ( ( ph /\ ( x e. { Z } /\ y e. S ) ) -> ( x .+ y ) e. { Z } ) |
75 |
|
peano2uz |
|- ( K e. ( ZZ>= ` M ) -> ( K + 1 ) e. ( ZZ>= ` M ) ) |
76 |
9 75
|
syl |
|- ( ph -> ( K + 1 ) e. ( ZZ>= ` M ) ) |
77 |
|
fzss1 |
|- ( ( K + 1 ) e. ( ZZ>= ` M ) -> ( ( K + 1 ) ... N ) C_ ( M ... N ) ) |
78 |
76 77
|
syl |
|- ( ph -> ( ( K + 1 ) ... N ) C_ ( M ... N ) ) |
79 |
78
|
sselda |
|- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> x e. ( M ... N ) ) |
80 |
79 2
|
syldan |
|- ( ( ph /\ x e. ( ( K + 1 ) ... N ) ) -> ( F ` x ) e. S ) |
81 |
60 74 38 80
|
seqcl2 |
|- ( ph -> ( seq K ( .+ , F ) ` N ) e. { Z } ) |
82 |
|
elsni |
|- ( ( seq K ( .+ , F ) ` N ) e. { Z } -> ( seq K ( .+ , F ) ` N ) = Z ) |
83 |
81 82
|
syl |
|- ( ph -> ( seq K ( .+ , F ) ` N ) = Z ) |
84 |
57 83
|
eqtrd |
|- ( ph -> ( seq M ( .+ , F ) ` N ) = Z ) |