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 |
|
seqf1olem.5 |
|- ( ph -> F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) ) |
7 |
|
seqf1olem.6 |
|- ( ph -> G : ( M ... ( N + 1 ) ) --> C ) |
8 |
|
seqf1olem.7 |
|- J = ( k e. ( M ... N ) |-> ( F ` if ( k < K , k , ( k + 1 ) ) ) ) |
9 |
|
seqf1olem.8 |
|- K = ( `' F ` ( N + 1 ) ) |
10 |
|
fvexd |
|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` if ( k < K , k , ( k + 1 ) ) ) e. _V ) |
11 |
|
fvex |
|- ( `' F ` x ) e. _V |
12 |
|
ovex |
|- ( ( `' F ` x ) - 1 ) e. _V |
13 |
11 12
|
ifex |
|- if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) e. _V |
14 |
13
|
a1i |
|- ( ( ph /\ x e. ( M ... N ) ) -> if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) e. _V ) |
15 |
|
iftrue |
|- ( k < K -> if ( k < K , k , ( k + 1 ) ) = k ) |
16 |
15
|
fveq2d |
|- ( k < K -> ( F ` if ( k < K , k , ( k + 1 ) ) ) = ( F ` k ) ) |
17 |
16
|
eqeq2d |
|- ( k < K -> ( x = ( F ` if ( k < K , k , ( k + 1 ) ) ) <-> x = ( F ` k ) ) ) |
18 |
17
|
adantl |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ k < K ) -> ( x = ( F ` if ( k < K , k , ( k + 1 ) ) ) <-> x = ( F ` k ) ) ) |
19 |
|
simprr |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> x = ( F ` k ) ) |
20 |
|
elfzelz |
|- ( k e. ( M ... N ) -> k e. ZZ ) |
21 |
20
|
zred |
|- ( k e. ( M ... N ) -> k e. RR ) |
22 |
21
|
ad2antlr |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> k e. RR ) |
23 |
|
simprl |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> k < K ) |
24 |
22 23
|
gtned |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> K =/= k ) |
25 |
|
f1of |
|- ( F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) -> F : ( M ... ( N + 1 ) ) --> ( M ... ( N + 1 ) ) ) |
26 |
6 25
|
syl |
|- ( ph -> F : ( M ... ( N + 1 ) ) --> ( M ... ( N + 1 ) ) ) |
27 |
26
|
ad2antrr |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> F : ( M ... ( N + 1 ) ) --> ( M ... ( N + 1 ) ) ) |
28 |
|
fzssp1 |
|- ( M ... N ) C_ ( M ... ( N + 1 ) ) |
29 |
|
simplr |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> k e. ( M ... N ) ) |
30 |
28 29
|
sselid |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> k e. ( M ... ( N + 1 ) ) ) |
31 |
27 30
|
ffvelrnd |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> ( F ` k ) e. ( M ... ( N + 1 ) ) ) |
32 |
|
elfzp1 |
|- ( N e. ( ZZ>= ` M ) -> ( ( F ` k ) e. ( M ... ( N + 1 ) ) <-> ( ( F ` k ) e. ( M ... N ) \/ ( F ` k ) = ( N + 1 ) ) ) ) |
33 |
4 32
|
syl |
|- ( ph -> ( ( F ` k ) e. ( M ... ( N + 1 ) ) <-> ( ( F ` k ) e. ( M ... N ) \/ ( F ` k ) = ( N + 1 ) ) ) ) |
34 |
33
|
ad2antrr |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> ( ( F ` k ) e. ( M ... ( N + 1 ) ) <-> ( ( F ` k ) e. ( M ... N ) \/ ( F ` k ) = ( N + 1 ) ) ) ) |
35 |
31 34
|
mpbid |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> ( ( F ` k ) e. ( M ... N ) \/ ( F ` k ) = ( N + 1 ) ) ) |
36 |
35
|
ord |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> ( -. ( F ` k ) e. ( M ... N ) -> ( F ` k ) = ( N + 1 ) ) ) |
37 |
6
|
ad2antrr |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) ) |
38 |
|
f1ocnvfv |
|- ( ( F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) /\ k e. ( M ... ( N + 1 ) ) ) -> ( ( F ` k ) = ( N + 1 ) -> ( `' F ` ( N + 1 ) ) = k ) ) |
39 |
37 30 38
|
syl2anc |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> ( ( F ` k ) = ( N + 1 ) -> ( `' F ` ( N + 1 ) ) = k ) ) |
40 |
9
|
eqeq1i |
|- ( K = k <-> ( `' F ` ( N + 1 ) ) = k ) |
41 |
39 40
|
syl6ibr |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> ( ( F ` k ) = ( N + 1 ) -> K = k ) ) |
42 |
36 41
|
syld |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> ( -. ( F ` k ) e. ( M ... N ) -> K = k ) ) |
43 |
42
|
necon1ad |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> ( K =/= k -> ( F ` k ) e. ( M ... N ) ) ) |
44 |
24 43
|
mpd |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> ( F ` k ) e. ( M ... N ) ) |
45 |
19 44
|
eqeltrd |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> x e. ( M ... N ) ) |
46 |
19
|
eqcomd |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> ( F ` k ) = x ) |
47 |
|
f1ocnvfv |
|- ( ( F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) /\ k e. ( M ... ( N + 1 ) ) ) -> ( ( F ` k ) = x -> ( `' F ` x ) = k ) ) |
48 |
37 30 47
|
syl2anc |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> ( ( F ` k ) = x -> ( `' F ` x ) = k ) ) |
49 |
46 48
|
mpd |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> ( `' F ` x ) = k ) |
50 |
49 23
|
eqbrtrd |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> ( `' F ` x ) < K ) |
51 |
|
iftrue |
|- ( ( `' F ` x ) < K -> if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) = ( `' F ` x ) ) |
52 |
50 51
|
syl |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) = ( `' F ` x ) ) |
53 |
52 49
|
eqtr2d |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) ) |
54 |
45 53
|
jca |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( k < K /\ x = ( F ` k ) ) ) -> ( x e. ( M ... N ) /\ k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) ) ) |
55 |
54
|
expr |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ k < K ) -> ( x = ( F ` k ) -> ( x e. ( M ... N ) /\ k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) ) ) ) |
56 |
18 55
|
sylbid |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ k < K ) -> ( x = ( F ` if ( k < K , k , ( k + 1 ) ) ) -> ( x e. ( M ... N ) /\ k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) ) ) ) |
57 |
|
iffalse |
|- ( -. k < K -> if ( k < K , k , ( k + 1 ) ) = ( k + 1 ) ) |
58 |
57
|
fveq2d |
|- ( -. k < K -> ( F ` if ( k < K , k , ( k + 1 ) ) ) = ( F ` ( k + 1 ) ) ) |
59 |
58
|
eqeq2d |
|- ( -. k < K -> ( x = ( F ` if ( k < K , k , ( k + 1 ) ) ) <-> x = ( F ` ( k + 1 ) ) ) ) |
60 |
59
|
adantl |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ -. k < K ) -> ( x = ( F ` if ( k < K , k , ( k + 1 ) ) ) <-> x = ( F ` ( k + 1 ) ) ) ) |
61 |
|
simprr |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> x = ( F ` ( k + 1 ) ) ) |
62 |
|
f1ocnv |
|- ( F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) -> `' F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) ) |
63 |
6 62
|
syl |
|- ( ph -> `' F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) ) |
64 |
|
f1of1 |
|- ( `' F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) -> `' F : ( M ... ( N + 1 ) ) -1-1-> ( M ... ( N + 1 ) ) ) |
65 |
63 64
|
syl |
|- ( ph -> `' F : ( M ... ( N + 1 ) ) -1-1-> ( M ... ( N + 1 ) ) ) |
66 |
|
f1f |
|- ( `' F : ( M ... ( N + 1 ) ) -1-1-> ( M ... ( N + 1 ) ) -> `' F : ( M ... ( N + 1 ) ) --> ( M ... ( N + 1 ) ) ) |
67 |
65 66
|
syl |
|- ( ph -> `' F : ( M ... ( N + 1 ) ) --> ( M ... ( N + 1 ) ) ) |
68 |
|
peano2uz |
|- ( N e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( ZZ>= ` M ) ) |
69 |
4 68
|
syl |
|- ( ph -> ( N + 1 ) e. ( ZZ>= ` M ) ) |
70 |
|
eluzfz2 |
|- ( ( N + 1 ) e. ( ZZ>= ` M ) -> ( N + 1 ) e. ( M ... ( N + 1 ) ) ) |
71 |
69 70
|
syl |
|- ( ph -> ( N + 1 ) e. ( M ... ( N + 1 ) ) ) |
72 |
67 71
|
ffvelrnd |
|- ( ph -> ( `' F ` ( N + 1 ) ) e. ( M ... ( N + 1 ) ) ) |
73 |
9 72
|
eqeltrid |
|- ( ph -> K e. ( M ... ( N + 1 ) ) ) |
74 |
73
|
elfzelzd |
|- ( ph -> K e. ZZ ) |
75 |
74
|
zred |
|- ( ph -> K e. RR ) |
76 |
75
|
ad2antrr |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> K e. RR ) |
77 |
21
|
ad2antlr |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> k e. RR ) |
78 |
|
peano2re |
|- ( k e. RR -> ( k + 1 ) e. RR ) |
79 |
77 78
|
syl |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( k + 1 ) e. RR ) |
80 |
|
simprl |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> -. k < K ) |
81 |
76 77 80
|
nltled |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> K <_ k ) |
82 |
77
|
ltp1d |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> k < ( k + 1 ) ) |
83 |
76 77 79 81 82
|
lelttrd |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> K < ( k + 1 ) ) |
84 |
76 83
|
ltned |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> K =/= ( k + 1 ) ) |
85 |
26
|
ad2antrr |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> F : ( M ... ( N + 1 ) ) --> ( M ... ( N + 1 ) ) ) |
86 |
|
fzp1elp1 |
|- ( k e. ( M ... N ) -> ( k + 1 ) e. ( M ... ( N + 1 ) ) ) |
87 |
86
|
ad2antlr |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( k + 1 ) e. ( M ... ( N + 1 ) ) ) |
88 |
85 87
|
ffvelrnd |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( F ` ( k + 1 ) ) e. ( M ... ( N + 1 ) ) ) |
89 |
|
elfzp1 |
|- ( N e. ( ZZ>= ` M ) -> ( ( F ` ( k + 1 ) ) e. ( M ... ( N + 1 ) ) <-> ( ( F ` ( k + 1 ) ) e. ( M ... N ) \/ ( F ` ( k + 1 ) ) = ( N + 1 ) ) ) ) |
90 |
4 89
|
syl |
|- ( ph -> ( ( F ` ( k + 1 ) ) e. ( M ... ( N + 1 ) ) <-> ( ( F ` ( k + 1 ) ) e. ( M ... N ) \/ ( F ` ( k + 1 ) ) = ( N + 1 ) ) ) ) |
91 |
90
|
ad2antrr |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( ( F ` ( k + 1 ) ) e. ( M ... ( N + 1 ) ) <-> ( ( F ` ( k + 1 ) ) e. ( M ... N ) \/ ( F ` ( k + 1 ) ) = ( N + 1 ) ) ) ) |
92 |
88 91
|
mpbid |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( ( F ` ( k + 1 ) ) e. ( M ... N ) \/ ( F ` ( k + 1 ) ) = ( N + 1 ) ) ) |
93 |
92
|
ord |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( -. ( F ` ( k + 1 ) ) e. ( M ... N ) -> ( F ` ( k + 1 ) ) = ( N + 1 ) ) ) |
94 |
6
|
ad2antrr |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) ) |
95 |
|
f1ocnvfv |
|- ( ( F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) /\ ( k + 1 ) e. ( M ... ( N + 1 ) ) ) -> ( ( F ` ( k + 1 ) ) = ( N + 1 ) -> ( `' F ` ( N + 1 ) ) = ( k + 1 ) ) ) |
96 |
94 87 95
|
syl2anc |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( ( F ` ( k + 1 ) ) = ( N + 1 ) -> ( `' F ` ( N + 1 ) ) = ( k + 1 ) ) ) |
97 |
9
|
eqeq1i |
|- ( K = ( k + 1 ) <-> ( `' F ` ( N + 1 ) ) = ( k + 1 ) ) |
98 |
96 97
|
syl6ibr |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( ( F ` ( k + 1 ) ) = ( N + 1 ) -> K = ( k + 1 ) ) ) |
99 |
93 98
|
syld |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( -. ( F ` ( k + 1 ) ) e. ( M ... N ) -> K = ( k + 1 ) ) ) |
100 |
99
|
necon1ad |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( K =/= ( k + 1 ) -> ( F ` ( k + 1 ) ) e. ( M ... N ) ) ) |
101 |
84 100
|
mpd |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( F ` ( k + 1 ) ) e. ( M ... N ) ) |
102 |
61 101
|
eqeltrd |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> x e. ( M ... N ) ) |
103 |
61
|
eqcomd |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( F ` ( k + 1 ) ) = x ) |
104 |
|
f1ocnvfv |
|- ( ( F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) /\ ( k + 1 ) e. ( M ... ( N + 1 ) ) ) -> ( ( F ` ( k + 1 ) ) = x -> ( `' F ` x ) = ( k + 1 ) ) ) |
105 |
94 87 104
|
syl2anc |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( ( F ` ( k + 1 ) ) = x -> ( `' F ` x ) = ( k + 1 ) ) ) |
106 |
103 105
|
mpd |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( `' F ` x ) = ( k + 1 ) ) |
107 |
106
|
breq1d |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( ( `' F ` x ) < K <-> ( k + 1 ) < K ) ) |
108 |
|
lttr |
|- ( ( k e. RR /\ ( k + 1 ) e. RR /\ K e. RR ) -> ( ( k < ( k + 1 ) /\ ( k + 1 ) < K ) -> k < K ) ) |
109 |
77 79 76 108
|
syl3anc |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( ( k < ( k + 1 ) /\ ( k + 1 ) < K ) -> k < K ) ) |
110 |
82 109
|
mpand |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( ( k + 1 ) < K -> k < K ) ) |
111 |
107 110
|
sylbid |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( ( `' F ` x ) < K -> k < K ) ) |
112 |
80 111
|
mtod |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> -. ( `' F ` x ) < K ) |
113 |
|
iffalse |
|- ( -. ( `' F ` x ) < K -> if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) = ( ( `' F ` x ) - 1 ) ) |
114 |
112 113
|
syl |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) = ( ( `' F ` x ) - 1 ) ) |
115 |
106
|
oveq1d |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( ( `' F ` x ) - 1 ) = ( ( k + 1 ) - 1 ) ) |
116 |
77
|
recnd |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> k e. CC ) |
117 |
|
ax-1cn |
|- 1 e. CC |
118 |
|
pncan |
|- ( ( k e. CC /\ 1 e. CC ) -> ( ( k + 1 ) - 1 ) = k ) |
119 |
116 117 118
|
sylancl |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( ( k + 1 ) - 1 ) = k ) |
120 |
114 115 119
|
3eqtrrd |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) ) |
121 |
102 120
|
jca |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ ( -. k < K /\ x = ( F ` ( k + 1 ) ) ) ) -> ( x e. ( M ... N ) /\ k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) ) ) |
122 |
121
|
expr |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ -. k < K ) -> ( x = ( F ` ( k + 1 ) ) -> ( x e. ( M ... N ) /\ k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) ) ) ) |
123 |
60 122
|
sylbid |
|- ( ( ( ph /\ k e. ( M ... N ) ) /\ -. k < K ) -> ( x = ( F ` if ( k < K , k , ( k + 1 ) ) ) -> ( x e. ( M ... N ) /\ k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) ) ) ) |
124 |
56 123
|
pm2.61dan |
|- ( ( ph /\ k e. ( M ... N ) ) -> ( x = ( F ` if ( k < K , k , ( k + 1 ) ) ) -> ( x e. ( M ... N ) /\ k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) ) ) ) |
125 |
124
|
expimpd |
|- ( ph -> ( ( k e. ( M ... N ) /\ x = ( F ` if ( k < K , k , ( k + 1 ) ) ) ) -> ( x e. ( M ... N ) /\ k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) ) ) ) |
126 |
51
|
eqeq2d |
|- ( ( `' F ` x ) < K -> ( k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) <-> k = ( `' F ` x ) ) ) |
127 |
126
|
adantl |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( `' F ` x ) < K ) -> ( k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) <-> k = ( `' F ` x ) ) ) |
128 |
|
eluzel2 |
|- ( N e. ( ZZ>= ` M ) -> M e. ZZ ) |
129 |
4 128
|
syl |
|- ( ph -> M e. ZZ ) |
130 |
129
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> M e. ZZ ) |
131 |
|
eluzelz |
|- ( N e. ( ZZ>= ` M ) -> N e. ZZ ) |
132 |
4 131
|
syl |
|- ( ph -> N e. ZZ ) |
133 |
132
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> N e. ZZ ) |
134 |
|
simprr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> k = ( `' F ` x ) ) |
135 |
67
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> `' F : ( M ... ( N + 1 ) ) --> ( M ... ( N + 1 ) ) ) |
136 |
|
simplr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> x e. ( M ... N ) ) |
137 |
28 136
|
sselid |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> x e. ( M ... ( N + 1 ) ) ) |
138 |
135 137
|
ffvelrnd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> ( `' F ` x ) e. ( M ... ( N + 1 ) ) ) |
139 |
134 138
|
eqeltrd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> k e. ( M ... ( N + 1 ) ) ) |
140 |
139
|
elfzelzd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> k e. ZZ ) |
141 |
|
elfzle1 |
|- ( k e. ( M ... ( N + 1 ) ) -> M <_ k ) |
142 |
139 141
|
syl |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> M <_ k ) |
143 |
140
|
zred |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> k e. RR ) |
144 |
75
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> K e. RR ) |
145 |
132
|
peano2zd |
|- ( ph -> ( N + 1 ) e. ZZ ) |
146 |
145
|
zred |
|- ( ph -> ( N + 1 ) e. RR ) |
147 |
146
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> ( N + 1 ) e. RR ) |
148 |
|
simprl |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> ( `' F ` x ) < K ) |
149 |
134 148
|
eqbrtrd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> k < K ) |
150 |
|
elfzle2 |
|- ( K e. ( M ... ( N + 1 ) ) -> K <_ ( N + 1 ) ) |
151 |
73 150
|
syl |
|- ( ph -> K <_ ( N + 1 ) ) |
152 |
151
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> K <_ ( N + 1 ) ) |
153 |
143 144 147 149 152
|
ltletrd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> k < ( N + 1 ) ) |
154 |
|
zleltp1 |
|- ( ( k e. ZZ /\ N e. ZZ ) -> ( k <_ N <-> k < ( N + 1 ) ) ) |
155 |
140 133 154
|
syl2anc |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> ( k <_ N <-> k < ( N + 1 ) ) ) |
156 |
153 155
|
mpbird |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> k <_ N ) |
157 |
130 133 140 142 156
|
elfzd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> k e. ( M ... N ) ) |
158 |
149 16
|
syl |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> ( F ` if ( k < K , k , ( k + 1 ) ) ) = ( F ` k ) ) |
159 |
134
|
fveq2d |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> ( F ` k ) = ( F ` ( `' F ` x ) ) ) |
160 |
6
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) ) |
161 |
|
f1ocnvfv2 |
|- ( ( F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) /\ x e. ( M ... ( N + 1 ) ) ) -> ( F ` ( `' F ` x ) ) = x ) |
162 |
160 137 161
|
syl2anc |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> ( F ` ( `' F ` x ) ) = x ) |
163 |
158 159 162
|
3eqtrrd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> x = ( F ` if ( k < K , k , ( k + 1 ) ) ) ) |
164 |
157 163
|
jca |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( ( `' F ` x ) < K /\ k = ( `' F ` x ) ) ) -> ( k e. ( M ... N ) /\ x = ( F ` if ( k < K , k , ( k + 1 ) ) ) ) ) |
165 |
164
|
expr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( `' F ` x ) < K ) -> ( k = ( `' F ` x ) -> ( k e. ( M ... N ) /\ x = ( F ` if ( k < K , k , ( k + 1 ) ) ) ) ) ) |
166 |
127 165
|
sylbid |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( `' F ` x ) < K ) -> ( k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) -> ( k e. ( M ... N ) /\ x = ( F ` if ( k < K , k , ( k + 1 ) ) ) ) ) ) |
167 |
113
|
eqeq2d |
|- ( -. ( `' F ` x ) < K -> ( k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) <-> k = ( ( `' F ` x ) - 1 ) ) ) |
168 |
167
|
adantl |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ -. ( `' F ` x ) < K ) -> ( k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) <-> k = ( ( `' F ` x ) - 1 ) ) ) |
169 |
129
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> M e. ZZ ) |
170 |
132
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> N e. ZZ ) |
171 |
|
simprr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> k = ( ( `' F ` x ) - 1 ) ) |
172 |
67
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> `' F : ( M ... ( N + 1 ) ) --> ( M ... ( N + 1 ) ) ) |
173 |
|
simplr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> x e. ( M ... N ) ) |
174 |
28 173
|
sselid |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> x e. ( M ... ( N + 1 ) ) ) |
175 |
172 174
|
ffvelrnd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( `' F ` x ) e. ( M ... ( N + 1 ) ) ) |
176 |
175
|
elfzelzd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( `' F ` x ) e. ZZ ) |
177 |
|
peano2zm |
|- ( ( `' F ` x ) e. ZZ -> ( ( `' F ` x ) - 1 ) e. ZZ ) |
178 |
176 177
|
syl |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( ( `' F ` x ) - 1 ) e. ZZ ) |
179 |
171 178
|
eqeltrd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> k e. ZZ ) |
180 |
129
|
zred |
|- ( ph -> M e. RR ) |
181 |
180
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> M e. RR ) |
182 |
75
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> K e. RR ) |
183 |
179
|
zred |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> k e. RR ) |
184 |
|
elfzle1 |
|- ( K e. ( M ... ( N + 1 ) ) -> M <_ K ) |
185 |
73 184
|
syl |
|- ( ph -> M <_ K ) |
186 |
185
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> M <_ K ) |
187 |
176
|
zred |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( `' F ` x ) e. RR ) |
188 |
|
simprl |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> -. ( `' F ` x ) < K ) |
189 |
182 187 188
|
nltled |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> K <_ ( `' F ` x ) ) |
190 |
|
elfzelz |
|- ( x e. ( M ... N ) -> x e. ZZ ) |
191 |
190
|
adantl |
|- ( ( ph /\ x e. ( M ... N ) ) -> x e. ZZ ) |
192 |
191
|
zred |
|- ( ( ph /\ x e. ( M ... N ) ) -> x e. RR ) |
193 |
132
|
zred |
|- ( ph -> N e. RR ) |
194 |
193
|
adantr |
|- ( ( ph /\ x e. ( M ... N ) ) -> N e. RR ) |
195 |
146
|
adantr |
|- ( ( ph /\ x e. ( M ... N ) ) -> ( N + 1 ) e. RR ) |
196 |
|
elfzle2 |
|- ( x e. ( M ... N ) -> x <_ N ) |
197 |
196
|
adantl |
|- ( ( ph /\ x e. ( M ... N ) ) -> x <_ N ) |
198 |
194
|
ltp1d |
|- ( ( ph /\ x e. ( M ... N ) ) -> N < ( N + 1 ) ) |
199 |
192 194 195 197 198
|
lelttrd |
|- ( ( ph /\ x e. ( M ... N ) ) -> x < ( N + 1 ) ) |
200 |
192 199
|
gtned |
|- ( ( ph /\ x e. ( M ... N ) ) -> ( N + 1 ) =/= x ) |
201 |
200
|
adantr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( N + 1 ) =/= x ) |
202 |
65
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> `' F : ( M ... ( N + 1 ) ) -1-1-> ( M ... ( N + 1 ) ) ) |
203 |
71
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( N + 1 ) e. ( M ... ( N + 1 ) ) ) |
204 |
|
f1fveq |
|- ( ( `' F : ( M ... ( N + 1 ) ) -1-1-> ( M ... ( N + 1 ) ) /\ ( ( N + 1 ) e. ( M ... ( N + 1 ) ) /\ x e. ( M ... ( N + 1 ) ) ) ) -> ( ( `' F ` ( N + 1 ) ) = ( `' F ` x ) <-> ( N + 1 ) = x ) ) |
205 |
202 203 174 204
|
syl12anc |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( ( `' F ` ( N + 1 ) ) = ( `' F ` x ) <-> ( N + 1 ) = x ) ) |
206 |
205
|
necon3bid |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( ( `' F ` ( N + 1 ) ) =/= ( `' F ` x ) <-> ( N + 1 ) =/= x ) ) |
207 |
201 206
|
mpbird |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( `' F ` ( N + 1 ) ) =/= ( `' F ` x ) ) |
208 |
9
|
neeq1i |
|- ( K =/= ( `' F ` x ) <-> ( `' F ` ( N + 1 ) ) =/= ( `' F ` x ) ) |
209 |
207 208
|
sylibr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> K =/= ( `' F ` x ) ) |
210 |
209
|
necomd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( `' F ` x ) =/= K ) |
211 |
182 187 189 210
|
leneltd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> K < ( `' F ` x ) ) |
212 |
74
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> K e. ZZ ) |
213 |
|
zltlem1 |
|- ( ( K e. ZZ /\ ( `' F ` x ) e. ZZ ) -> ( K < ( `' F ` x ) <-> K <_ ( ( `' F ` x ) - 1 ) ) ) |
214 |
212 176 213
|
syl2anc |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( K < ( `' F ` x ) <-> K <_ ( ( `' F ` x ) - 1 ) ) ) |
215 |
211 214
|
mpbid |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> K <_ ( ( `' F ` x ) - 1 ) ) |
216 |
215 171
|
breqtrrd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> K <_ k ) |
217 |
181 182 183 186 216
|
letrd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> M <_ k ) |
218 |
|
elfzle2 |
|- ( ( `' F ` x ) e. ( M ... ( N + 1 ) ) -> ( `' F ` x ) <_ ( N + 1 ) ) |
219 |
175 218
|
syl |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( `' F ` x ) <_ ( N + 1 ) ) |
220 |
193
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> N e. RR ) |
221 |
|
1re |
|- 1 e. RR |
222 |
|
lesubadd |
|- ( ( ( `' F ` x ) e. RR /\ 1 e. RR /\ N e. RR ) -> ( ( ( `' F ` x ) - 1 ) <_ N <-> ( `' F ` x ) <_ ( N + 1 ) ) ) |
223 |
221 222
|
mp3an2 |
|- ( ( ( `' F ` x ) e. RR /\ N e. RR ) -> ( ( ( `' F ` x ) - 1 ) <_ N <-> ( `' F ` x ) <_ ( N + 1 ) ) ) |
224 |
187 220 223
|
syl2anc |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( ( ( `' F ` x ) - 1 ) <_ N <-> ( `' F ` x ) <_ ( N + 1 ) ) ) |
225 |
219 224
|
mpbird |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( ( `' F ` x ) - 1 ) <_ N ) |
226 |
171 225
|
eqbrtrd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> k <_ N ) |
227 |
169 170 179 217 226
|
elfzd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> k e. ( M ... N ) ) |
228 |
182 183 216
|
lensymd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> -. k < K ) |
229 |
228 58
|
syl |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( F ` if ( k < K , k , ( k + 1 ) ) ) = ( F ` ( k + 1 ) ) ) |
230 |
171
|
oveq1d |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( k + 1 ) = ( ( ( `' F ` x ) - 1 ) + 1 ) ) |
231 |
176
|
zcnd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( `' F ` x ) e. CC ) |
232 |
|
npcan |
|- ( ( ( `' F ` x ) e. CC /\ 1 e. CC ) -> ( ( ( `' F ` x ) - 1 ) + 1 ) = ( `' F ` x ) ) |
233 |
231 117 232
|
sylancl |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( ( ( `' F ` x ) - 1 ) + 1 ) = ( `' F ` x ) ) |
234 |
230 233
|
eqtrd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( k + 1 ) = ( `' F ` x ) ) |
235 |
234
|
fveq2d |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( F ` ( k + 1 ) ) = ( F ` ( `' F ` x ) ) ) |
236 |
6
|
ad2antrr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> F : ( M ... ( N + 1 ) ) -1-1-onto-> ( M ... ( N + 1 ) ) ) |
237 |
236 174 161
|
syl2anc |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( F ` ( `' F ` x ) ) = x ) |
238 |
229 235 237
|
3eqtrrd |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> x = ( F ` if ( k < K , k , ( k + 1 ) ) ) ) |
239 |
227 238
|
jca |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ ( -. ( `' F ` x ) < K /\ k = ( ( `' F ` x ) - 1 ) ) ) -> ( k e. ( M ... N ) /\ x = ( F ` if ( k < K , k , ( k + 1 ) ) ) ) ) |
240 |
239
|
expr |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ -. ( `' F ` x ) < K ) -> ( k = ( ( `' F ` x ) - 1 ) -> ( k e. ( M ... N ) /\ x = ( F ` if ( k < K , k , ( k + 1 ) ) ) ) ) ) |
241 |
168 240
|
sylbid |
|- ( ( ( ph /\ x e. ( M ... N ) ) /\ -. ( `' F ` x ) < K ) -> ( k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) -> ( k e. ( M ... N ) /\ x = ( F ` if ( k < K , k , ( k + 1 ) ) ) ) ) ) |
242 |
166 241
|
pm2.61dan |
|- ( ( ph /\ x e. ( M ... N ) ) -> ( k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) -> ( k e. ( M ... N ) /\ x = ( F ` if ( k < K , k , ( k + 1 ) ) ) ) ) ) |
243 |
242
|
expimpd |
|- ( ph -> ( ( x e. ( M ... N ) /\ k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) ) -> ( k e. ( M ... N ) /\ x = ( F ` if ( k < K , k , ( k + 1 ) ) ) ) ) ) |
244 |
125 243
|
impbid |
|- ( ph -> ( ( k e. ( M ... N ) /\ x = ( F ` if ( k < K , k , ( k + 1 ) ) ) ) <-> ( x e. ( M ... N ) /\ k = if ( ( `' F ` x ) < K , ( `' F ` x ) , ( ( `' F ` x ) - 1 ) ) ) ) ) |
245 |
8 10 14 244
|
f1od |
|- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) ) |