Step |
Hyp |
Ref |
Expression |
1 |
|
seqf1o.1 |
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) |
2 |
|
seqf1o.2 |
|- ( ( ph /\ ( x e. C /\ y e. C ) ) -> ( x .+ y ) = ( y .+ x ) ) |
3 |
|
seqf1o.3 |
|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) ) |
4 |
|
seqf1o.4 |
|- ( ph -> N e. ( ZZ>= ` M ) ) |
5 |
|
seqf1o.5 |
|- ( ph -> C C_ S ) |
6 |
|
seqf1olem2a.1 |
|- ( ph -> G : A --> C ) |
7 |
|
seqf1olem2a.3 |
|- ( ph -> K e. A ) |
8 |
|
seqf1olem2a.4 |
|- ( ph -> ( M ... N ) C_ A ) |
9 |
|
eluzfz2 |
|- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) ) |
10 |
4 9
|
syl |
|- ( ph -> N e. ( M ... N ) ) |
11 |
|
fveq2 |
|- ( m = M -> ( seq M ( .+ , G ) ` m ) = ( seq M ( .+ , G ) ` M ) ) |
12 |
11
|
oveq2d |
|- ( m = M -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` m ) ) = ( ( G ` K ) .+ ( seq M ( .+ , G ) ` M ) ) ) |
13 |
11
|
oveq1d |
|- ( m = M -> ( ( seq M ( .+ , G ) ` m ) .+ ( G ` K ) ) = ( ( seq M ( .+ , G ) ` M ) .+ ( G ` K ) ) ) |
14 |
12 13
|
eqeq12d |
|- ( m = M -> ( ( ( G ` K ) .+ ( seq M ( .+ , G ) ` m ) ) = ( ( seq M ( .+ , G ) ` m ) .+ ( G ` K ) ) <-> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` M ) ) = ( ( seq M ( .+ , G ) ` M ) .+ ( G ` K ) ) ) ) |
15 |
14
|
imbi2d |
|- ( m = M -> ( ( ph -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` m ) ) = ( ( seq M ( .+ , G ) ` m ) .+ ( G ` K ) ) ) <-> ( ph -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` M ) ) = ( ( seq M ( .+ , G ) ` M ) .+ ( G ` K ) ) ) ) ) |
16 |
|
fveq2 |
|- ( m = n -> ( seq M ( .+ , G ) ` m ) = ( seq M ( .+ , G ) ` n ) ) |
17 |
16
|
oveq2d |
|- ( m = n -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` m ) ) = ( ( G ` K ) .+ ( seq M ( .+ , G ) ` n ) ) ) |
18 |
16
|
oveq1d |
|- ( m = n -> ( ( seq M ( .+ , G ) ` m ) .+ ( G ` K ) ) = ( ( seq M ( .+ , G ) ` n ) .+ ( G ` K ) ) ) |
19 |
17 18
|
eqeq12d |
|- ( m = n -> ( ( ( G ` K ) .+ ( seq M ( .+ , G ) ` m ) ) = ( ( seq M ( .+ , G ) ` m ) .+ ( G ` K ) ) <-> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` n ) ) = ( ( seq M ( .+ , G ) ` n ) .+ ( G ` K ) ) ) ) |
20 |
19
|
imbi2d |
|- ( m = n -> ( ( ph -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` m ) ) = ( ( seq M ( .+ , G ) ` m ) .+ ( G ` K ) ) ) <-> ( ph -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` n ) ) = ( ( seq M ( .+ , G ) ` n ) .+ ( G ` K ) ) ) ) ) |
21 |
|
fveq2 |
|- ( m = ( n + 1 ) -> ( seq M ( .+ , G ) ` m ) = ( seq M ( .+ , G ) ` ( n + 1 ) ) ) |
22 |
21
|
oveq2d |
|- ( m = ( n + 1 ) -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` m ) ) = ( ( G ` K ) .+ ( seq M ( .+ , G ) ` ( n + 1 ) ) ) ) |
23 |
21
|
oveq1d |
|- ( m = ( n + 1 ) -> ( ( seq M ( .+ , G ) ` m ) .+ ( G ` K ) ) = ( ( seq M ( .+ , G ) ` ( n + 1 ) ) .+ ( G ` K ) ) ) |
24 |
22 23
|
eqeq12d |
|- ( m = ( n + 1 ) -> ( ( ( G ` K ) .+ ( seq M ( .+ , G ) ` m ) ) = ( ( seq M ( .+ , G ) ` m ) .+ ( G ` K ) ) <-> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` ( n + 1 ) ) ) = ( ( seq M ( .+ , G ) ` ( n + 1 ) ) .+ ( G ` K ) ) ) ) |
25 |
24
|
imbi2d |
|- ( m = ( n + 1 ) -> ( ( ph -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` m ) ) = ( ( seq M ( .+ , G ) ` m ) .+ ( G ` K ) ) ) <-> ( ph -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` ( n + 1 ) ) ) = ( ( seq M ( .+ , G ) ` ( n + 1 ) ) .+ ( G ` K ) ) ) ) ) |
26 |
|
fveq2 |
|- ( m = N -> ( seq M ( .+ , G ) ` m ) = ( seq M ( .+ , G ) ` N ) ) |
27 |
26
|
oveq2d |
|- ( m = N -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` m ) ) = ( ( G ` K ) .+ ( seq M ( .+ , G ) ` N ) ) ) |
28 |
26
|
oveq1d |
|- ( m = N -> ( ( seq M ( .+ , G ) ` m ) .+ ( G ` K ) ) = ( ( seq M ( .+ , G ) ` N ) .+ ( G ` K ) ) ) |
29 |
27 28
|
eqeq12d |
|- ( m = N -> ( ( ( G ` K ) .+ ( seq M ( .+ , G ) ` m ) ) = ( ( seq M ( .+ , G ) ` m ) .+ ( G ` K ) ) <-> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` N ) ) = ( ( seq M ( .+ , G ) ` N ) .+ ( G ` K ) ) ) ) |
30 |
29
|
imbi2d |
|- ( m = N -> ( ( ph -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` m ) ) = ( ( seq M ( .+ , G ) ` m ) .+ ( G ` K ) ) ) <-> ( ph -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` N ) ) = ( ( seq M ( .+ , G ) ` N ) .+ ( G ` K ) ) ) ) ) |
31 |
6 7
|
ffvelrnd |
|- ( ph -> ( G ` K ) e. C ) |
32 |
|
eluzel2 |
|- ( N e. ( ZZ>= ` M ) -> M e. ZZ ) |
33 |
|
seq1 |
|- ( M e. ZZ -> ( seq M ( .+ , G ) ` M ) = ( G ` M ) ) |
34 |
4 32 33
|
3syl |
|- ( ph -> ( seq M ( .+ , G ) ` M ) = ( G ` M ) ) |
35 |
|
eluzfz1 |
|- ( N e. ( ZZ>= ` M ) -> M e. ( M ... N ) ) |
36 |
4 35
|
syl |
|- ( ph -> M e. ( M ... N ) ) |
37 |
8 36
|
sseldd |
|- ( ph -> M e. A ) |
38 |
6 37
|
ffvelrnd |
|- ( ph -> ( G ` M ) e. C ) |
39 |
34 38
|
eqeltrd |
|- ( ph -> ( seq M ( .+ , G ) ` M ) e. C ) |
40 |
2 31 39
|
caovcomd |
|- ( ph -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` M ) ) = ( ( seq M ( .+ , G ) ` M ) .+ ( G ` K ) ) ) |
41 |
40
|
a1i |
|- ( N e. ( ZZ>= ` M ) -> ( ph -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` M ) ) = ( ( seq M ( .+ , G ) ` M ) .+ ( G ` K ) ) ) ) |
42 |
|
oveq1 |
|- ( ( ( G ` K ) .+ ( seq M ( .+ , G ) ` n ) ) = ( ( seq M ( .+ , G ) ` n ) .+ ( G ` K ) ) -> ( ( ( G ` K ) .+ ( seq M ( .+ , G ) ` n ) ) .+ ( G ` ( n + 1 ) ) ) = ( ( ( seq M ( .+ , G ) ` n ) .+ ( G ` K ) ) .+ ( G ` ( n + 1 ) ) ) ) |
43 |
|
elfzouz |
|- ( n e. ( M ..^ N ) -> n e. ( ZZ>= ` M ) ) |
44 |
43
|
adantl |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> n e. ( ZZ>= ` M ) ) |
45 |
|
seqp1 |
|- ( n e. ( ZZ>= ` M ) -> ( seq M ( .+ , G ) ` ( n + 1 ) ) = ( ( seq M ( .+ , G ) ` n ) .+ ( G ` ( n + 1 ) ) ) ) |
46 |
44 45
|
syl |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( seq M ( .+ , G ) ` ( n + 1 ) ) = ( ( seq M ( .+ , G ) ` n ) .+ ( G ` ( n + 1 ) ) ) ) |
47 |
46
|
oveq2d |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` ( n + 1 ) ) ) = ( ( G ` K ) .+ ( ( seq M ( .+ , G ) ` n ) .+ ( G ` ( n + 1 ) ) ) ) ) |
48 |
3
|
adantlr |
|- ( ( ( ph /\ n e. ( M ..^ N ) ) /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) ) |
49 |
5 31
|
sseldd |
|- ( ph -> ( G ` K ) e. S ) |
50 |
49
|
adantr |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( G ` K ) e. S ) |
51 |
5
|
adantr |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> C C_ S ) |
52 |
51
|
adantr |
|- ( ( ( ph /\ n e. ( M ..^ N ) ) /\ x e. ( M ... n ) ) -> C C_ S ) |
53 |
6
|
adantr |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> G : A --> C ) |
54 |
53
|
adantr |
|- ( ( ( ph /\ n e. ( M ..^ N ) ) /\ x e. ( M ... n ) ) -> G : A --> C ) |
55 |
|
elfzouz2 |
|- ( n e. ( M ..^ N ) -> N e. ( ZZ>= ` n ) ) |
56 |
55
|
adantl |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> N e. ( ZZ>= ` n ) ) |
57 |
|
fzss2 |
|- ( N e. ( ZZ>= ` n ) -> ( M ... n ) C_ ( M ... N ) ) |
58 |
56 57
|
syl |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( M ... n ) C_ ( M ... N ) ) |
59 |
8
|
adantr |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( M ... N ) C_ A ) |
60 |
58 59
|
sstrd |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( M ... n ) C_ A ) |
61 |
60
|
sselda |
|- ( ( ( ph /\ n e. ( M ..^ N ) ) /\ x e. ( M ... n ) ) -> x e. A ) |
62 |
54 61
|
ffvelrnd |
|- ( ( ( ph /\ n e. ( M ..^ N ) ) /\ x e. ( M ... n ) ) -> ( G ` x ) e. C ) |
63 |
52 62
|
sseldd |
|- ( ( ( ph /\ n e. ( M ..^ N ) ) /\ x e. ( M ... n ) ) -> ( G ` x ) e. S ) |
64 |
1
|
adantlr |
|- ( ( ( ph /\ n e. ( M ..^ N ) ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S ) |
65 |
44 63 64
|
seqcl |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( seq M ( .+ , G ) ` n ) e. S ) |
66 |
|
fzofzp1 |
|- ( n e. ( M ..^ N ) -> ( n + 1 ) e. ( M ... N ) ) |
67 |
66
|
adantl |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( n + 1 ) e. ( M ... N ) ) |
68 |
59 67
|
sseldd |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( n + 1 ) e. A ) |
69 |
53 68
|
ffvelrnd |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( G ` ( n + 1 ) ) e. C ) |
70 |
51 69
|
sseldd |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( G ` ( n + 1 ) ) e. S ) |
71 |
48 50 65 70
|
caovassd |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( ( ( G ` K ) .+ ( seq M ( .+ , G ) ` n ) ) .+ ( G ` ( n + 1 ) ) ) = ( ( G ` K ) .+ ( ( seq M ( .+ , G ) ` n ) .+ ( G ` ( n + 1 ) ) ) ) ) |
72 |
47 71
|
eqtr4d |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` ( n + 1 ) ) ) = ( ( ( G ` K ) .+ ( seq M ( .+ , G ) ` n ) ) .+ ( G ` ( n + 1 ) ) ) ) |
73 |
48 65 70 50
|
caovassd |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( ( ( seq M ( .+ , G ) ` n ) .+ ( G ` ( n + 1 ) ) ) .+ ( G ` K ) ) = ( ( seq M ( .+ , G ) ` n ) .+ ( ( G ` ( n + 1 ) ) .+ ( G ` K ) ) ) ) |
74 |
46
|
oveq1d |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( ( seq M ( .+ , G ) ` ( n + 1 ) ) .+ ( G ` K ) ) = ( ( ( seq M ( .+ , G ) ` n ) .+ ( G ` ( n + 1 ) ) ) .+ ( G ` K ) ) ) |
75 |
48 65 50 70
|
caovassd |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( ( ( seq M ( .+ , G ) ` n ) .+ ( G ` K ) ) .+ ( G ` ( n + 1 ) ) ) = ( ( seq M ( .+ , G ) ` n ) .+ ( ( G ` K ) .+ ( G ` ( n + 1 ) ) ) ) ) |
76 |
2
|
adantlr |
|- ( ( ( ph /\ n e. ( M ..^ N ) ) /\ ( x e. C /\ y e. C ) ) -> ( x .+ y ) = ( y .+ x ) ) |
77 |
31
|
adantr |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( G ` K ) e. C ) |
78 |
76 69 77
|
caovcomd |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( ( G ` ( n + 1 ) ) .+ ( G ` K ) ) = ( ( G ` K ) .+ ( G ` ( n + 1 ) ) ) ) |
79 |
78
|
oveq2d |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( ( seq M ( .+ , G ) ` n ) .+ ( ( G ` ( n + 1 ) ) .+ ( G ` K ) ) ) = ( ( seq M ( .+ , G ) ` n ) .+ ( ( G ` K ) .+ ( G ` ( n + 1 ) ) ) ) ) |
80 |
75 79
|
eqtr4d |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( ( ( seq M ( .+ , G ) ` n ) .+ ( G ` K ) ) .+ ( G ` ( n + 1 ) ) ) = ( ( seq M ( .+ , G ) ` n ) .+ ( ( G ` ( n + 1 ) ) .+ ( G ` K ) ) ) ) |
81 |
73 74 80
|
3eqtr4d |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( ( seq M ( .+ , G ) ` ( n + 1 ) ) .+ ( G ` K ) ) = ( ( ( seq M ( .+ , G ) ` n ) .+ ( G ` K ) ) .+ ( G ` ( n + 1 ) ) ) ) |
82 |
72 81
|
eqeq12d |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( ( ( G ` K ) .+ ( seq M ( .+ , G ) ` ( n + 1 ) ) ) = ( ( seq M ( .+ , G ) ` ( n + 1 ) ) .+ ( G ` K ) ) <-> ( ( ( G ` K ) .+ ( seq M ( .+ , G ) ` n ) ) .+ ( G ` ( n + 1 ) ) ) = ( ( ( seq M ( .+ , G ) ` n ) .+ ( G ` K ) ) .+ ( G ` ( n + 1 ) ) ) ) ) |
83 |
42 82
|
syl5ibr |
|- ( ( ph /\ n e. ( M ..^ N ) ) -> ( ( ( G ` K ) .+ ( seq M ( .+ , G ) ` n ) ) = ( ( seq M ( .+ , G ) ` n ) .+ ( G ` K ) ) -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` ( n + 1 ) ) ) = ( ( seq M ( .+ , G ) ` ( n + 1 ) ) .+ ( G ` K ) ) ) ) |
84 |
83
|
expcom |
|- ( n e. ( M ..^ N ) -> ( ph -> ( ( ( G ` K ) .+ ( seq M ( .+ , G ) ` n ) ) = ( ( seq M ( .+ , G ) ` n ) .+ ( G ` K ) ) -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` ( n + 1 ) ) ) = ( ( seq M ( .+ , G ) ` ( n + 1 ) ) .+ ( G ` K ) ) ) ) ) |
85 |
84
|
a2d |
|- ( n e. ( M ..^ N ) -> ( ( ph -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` n ) ) = ( ( seq M ( .+ , G ) ` n ) .+ ( G ` K ) ) ) -> ( ph -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` ( n + 1 ) ) ) = ( ( seq M ( .+ , G ) ` ( n + 1 ) ) .+ ( G ` K ) ) ) ) ) |
86 |
15 20 25 30 41 85
|
fzind2 |
|- ( N e. ( M ... N ) -> ( ph -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` N ) ) = ( ( seq M ( .+ , G ) ` N ) .+ ( G ` K ) ) ) ) |
87 |
10 86
|
mpcom |
|- ( ph -> ( ( G ` K ) .+ ( seq M ( .+ , G ) ` N ) ) = ( ( seq M ( .+ , G ) ` N ) .+ ( G ` K ) ) ) |