Metamath Proof Explorer

Theorem sticksstones8

Description: Establish mapping between strictly monotone functions and functions that sum to a fixed non-negative integer. (Contributed by metakunt, 1-Oct-2024)

Ref Expression
Hypotheses sticksstones8.1 
|- ( ph -> N e. NN0 )
sticksstones8.2 
|- ( ph -> K e. NN0 )
sticksstones8.3 
|- F = ( a e. A |-> ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) )
sticksstones8.4 
|- A = { g | ( g : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( g ` i ) = N ) }
sticksstones8.5 
|- B = { f | ( f : ( 1 ... K ) --> ( 1 ... ( N + K ) ) /\ A. x e. ( 1 ... K ) A. y e. ( 1 ... K ) ( x < y -> ( f ` x ) < ( f ` y ) ) ) }
Assertion sticksstones8 
|- ( ph -> F : A --> B )

Proof

Step Hyp Ref Expression
1 sticksstones8.1 
 |-  ( ph -> N e. NN0 )
2 sticksstones8.2 
 |-  ( ph -> K e. NN0 )
3 sticksstones8.3 
 |-  F = ( a e. A |-> ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) )
4 sticksstones8.4 
 |-  A = { g | ( g : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( g ` i ) = N ) }
5 sticksstones8.5 
 |-  B = { f | ( f : ( 1 ... K ) --> ( 1 ... ( N + K ) ) /\ A. x e. ( 1 ... K ) A. y e. ( 1 ... K ) ( x < y -> ( f ` x ) < ( f ` y ) ) ) }
6 eqidd 
 |-  ( ( ph /\ a e. A /\ j e. ( 1 ... K ) ) -> ( e e. ( 1 ... K ) |-> ( e + sum_ l e. ( 1 ... e ) ( a ` l ) ) ) = ( e e. ( 1 ... K ) |-> ( e + sum_ l e. ( 1 ... e ) ( a ` l ) ) ) )
7 simpr 
 |-  ( ( ( ph /\ a e. A /\ j e. ( 1 ... K ) ) /\ e = j ) -> e = j )
8 7 oveq2d 
 |-  ( ( ( ph /\ a e. A /\ j e. ( 1 ... K ) ) /\ e = j ) -> ( 1 ... e ) = ( 1 ... j ) )
9 8 sumeq1d 
 |-  ( ( ( ph /\ a e. A /\ j e. ( 1 ... K ) ) /\ e = j ) -> sum_ l e. ( 1 ... e ) ( a ` l ) = sum_ l e. ( 1 ... j ) ( a ` l ) )
10 7 9 oveq12d 
 |-  ( ( ( ph /\ a e. A /\ j e. ( 1 ... K ) ) /\ e = j ) -> ( e + sum_ l e. ( 1 ... e ) ( a ` l ) ) = ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) )
11 simp3 
 |-  ( ( ph /\ a e. A /\ j e. ( 1 ... K ) ) -> j e. ( 1 ... K ) )
12 ovexd 
 |-  ( ( ph /\ a e. A /\ j e. ( 1 ... K ) ) -> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) e. _V )
13 6 10 11 12 fvmptd 
 |-  ( ( ph /\ a e. A /\ j e. ( 1 ... K ) ) -> ( ( e e. ( 1 ... K ) |-> ( e + sum_ l e. ( 1 ... e ) ( a ` l ) ) ) ` j ) = ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) )
14 1 3ad2ant1 
 |-  ( ( ph /\ a e. A /\ j e. ( 1 ... K ) ) -> N e. NN0 )
15 2 3ad2ant1 
 |-  ( ( ph /\ a e. A /\ j e. ( 1 ... K ) ) -> K e. NN0 )
16 simpr 
 |-  ( ( ph /\ a e. A ) -> a e. A )
17 4 a1i 
 |-  ( ( ph /\ a e. A ) -> A = { g | ( g : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( g ` i ) = N ) } )
18 17 eqcomd 
 |-  ( ( ph /\ a e. A ) -> { g | ( g : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( g ` i ) = N ) } = A )
19 16 18 eleqtrrd 
 |-  ( ( ph /\ a e. A ) -> a e. { g | ( g : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( g ` i ) = N ) } )
20 feq1 
 |-  ( g = a -> ( g : ( 1 ... ( K + 1 ) ) --> NN0 <-> a : ( 1 ... ( K + 1 ) ) --> NN0 ) )
21 simpl 
 |-  ( ( g = a /\ i e. ( 1 ... ( K + 1 ) ) ) -> g = a )
22 21 fveq1d 
 |-  ( ( g = a /\ i e. ( 1 ... ( K + 1 ) ) ) -> ( g ` i ) = ( a ` i ) )
23 22 sumeq2dv 
 |-  ( g = a -> sum_ i e. ( 1 ... ( K + 1 ) ) ( g ` i ) = sum_ i e. ( 1 ... ( K + 1 ) ) ( a ` i ) )
24 23 eqeq1d 
 |-  ( g = a -> ( sum_ i e. ( 1 ... ( K + 1 ) ) ( g ` i ) = N <-> sum_ i e. ( 1 ... ( K + 1 ) ) ( a ` i ) = N ) )
25 20 24 anbi12d 
 |-  ( g = a -> ( ( g : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( g ` i ) = N ) <-> ( a : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( a ` i ) = N ) ) )
26 25 elabg 
 |-  ( a e. A -> ( a e. { g | ( g : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( g ` i ) = N ) } <-> ( a : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( a ` i ) = N ) ) )
27 16 26 syl 
 |-  ( ( ph /\ a e. A ) -> ( a e. { g | ( g : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( g ` i ) = N ) } <-> ( a : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( a ` i ) = N ) ) )
28 27 biimpd 
 |-  ( ( ph /\ a e. A ) -> ( a e. { g | ( g : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( g ` i ) = N ) } -> ( a : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( a ` i ) = N ) ) )
29 19 28 mpd 
 |-  ( ( ph /\ a e. A ) -> ( a : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( a ` i ) = N ) )
30 29 simpld 
 |-  ( ( ph /\ a e. A ) -> a : ( 1 ... ( K + 1 ) ) --> NN0 )
31 30 3adant3 
 |-  ( ( ph /\ a e. A /\ j e. ( 1 ... K ) ) -> a : ( 1 ... ( K + 1 ) ) --> NN0 )
32 eqid 
 |-  ( e e. ( 1 ... K ) |-> ( e + sum_ l e. ( 1 ... e ) ( a ` l ) ) ) = ( e e. ( 1 ... K ) |-> ( e + sum_ l e. ( 1 ... e ) ( a ` l ) ) )
33 fveq2 
 |-  ( i = l -> ( a ` i ) = ( a ` l ) )
34 nfcv 
 |-  F/_ l ( a ` i )
35 nfcv 
 |-  F/_ i ( a ` l )
36 33 34 35 cbvsum 
 |-  sum_ i e. ( 1 ... ( K + 1 ) ) ( a ` i ) = sum_ l e. ( 1 ... ( K + 1 ) ) ( a ` l )
37 29 simprd 
 |-  ( ( ph /\ a e. A ) -> sum_ i e. ( 1 ... ( K + 1 ) ) ( a ` i ) = N )
38 36 37 eqtr3id 
 |-  ( ( ph /\ a e. A ) -> sum_ l e. ( 1 ... ( K + 1 ) ) ( a ` l ) = N )
39 38 3adant3 
 |-  ( ( ph /\ a e. A /\ j e. ( 1 ... K ) ) -> sum_ l e. ( 1 ... ( K + 1 ) ) ( a ` l ) = N )
40 14 15 31 11 32 39 sticksstones7 
 |-  ( ( ph /\ a e. A /\ j e. ( 1 ... K ) ) -> ( ( e e. ( 1 ... K ) |-> ( e + sum_ l e. ( 1 ... e ) ( a ` l ) ) ) ` j ) e. ( 1 ... ( N + K ) ) )
41 13 40 eqeltrrd 
 |-  ( ( ph /\ a e. A /\ j e. ( 1 ... K ) ) -> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) e. ( 1 ... ( N + K ) ) )
42 41 3expa 
 |-  ( ( ( ph /\ a e. A ) /\ j e. ( 1 ... K ) ) -> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) e. ( 1 ... ( N + K ) ) )
43 eqid 
 |-  ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) = ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) )
44 42 43 fmptd 
 |-  ( ( ph /\ a e. A ) -> ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) : ( 1 ... K ) --> ( 1 ... ( N + K ) ) )
45 1 ad3antrrr 
 |-  ( ( ( ( ph /\ a e. A ) /\ x e. ( 1 ... K ) ) /\ y e. ( 1 ... K ) ) -> N e. NN0 )
46 45 adantr 
 |-  ( ( ( ( ( ph /\ a e. A ) /\ x e. ( 1 ... K ) ) /\ y e. ( 1 ... K ) ) /\ x < y ) -> N e. NN0 )
47 2 ad3antrrr 
 |-  ( ( ( ( ph /\ a e. A ) /\ x e. ( 1 ... K ) ) /\ y e. ( 1 ... K ) ) -> K e. NN0 )
48 47 adantr 
 |-  ( ( ( ( ( ph /\ a e. A ) /\ x e. ( 1 ... K ) ) /\ y e. ( 1 ... K ) ) /\ x < y ) -> K e. NN0 )
49 26 adantl 
 |-  ( ( ph /\ a e. A ) -> ( a e. { g | ( g : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( g ` i ) = N ) } <-> ( a : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( a ` i ) = N ) ) )
50 49 biimpd 
 |-  ( ( ph /\ a e. A ) -> ( a e. { g | ( g : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( g ` i ) = N ) } -> ( a : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( a ` i ) = N ) ) )
51 19 50 mpd 
 |-  ( ( ph /\ a e. A ) -> ( a : ( 1 ... ( K + 1 ) ) --> NN0 /\ sum_ i e. ( 1 ... ( K + 1 ) ) ( a ` i ) = N ) )
52 51 simpld 
 |-  ( ( ph /\ a e. A ) -> a : ( 1 ... ( K + 1 ) ) --> NN0 )
53 52 adantr 
 |-  ( ( ( ph /\ a e. A ) /\ x e. ( 1 ... K ) ) -> a : ( 1 ... ( K + 1 ) ) --> NN0 )
54 53 adantr 
 |-  ( ( ( ( ph /\ a e. A ) /\ x e. ( 1 ... K ) ) /\ y e. ( 1 ... K ) ) -> a : ( 1 ... ( K + 1 ) ) --> NN0 )
55 54 adantr 
 |-  ( ( ( ( ( ph /\ a e. A ) /\ x e. ( 1 ... K ) ) /\ y e. ( 1 ... K ) ) /\ x < y ) -> a : ( 1 ... ( K + 1 ) ) --> NN0 )
56 simpllr 
 |-  ( ( ( ( ( ph /\ a e. A ) /\ x e. ( 1 ... K ) ) /\ y e. ( 1 ... K ) ) /\ x < y ) -> x e. ( 1 ... K ) )
57 simplr 
 |-  ( ( ( ( ( ph /\ a e. A ) /\ x e. ( 1 ... K ) ) /\ y e. ( 1 ... K ) ) /\ x < y ) -> y e. ( 1 ... K ) )
58 simpr 
 |-  ( ( ( ( ( ph /\ a e. A ) /\ x e. ( 1 ... K ) ) /\ y e. ( 1 ... K ) ) /\ x < y ) -> x < y )
59 46 48 55 56 57 58 43 sticksstones6 
 |-  ( ( ( ( ( ph /\ a e. A ) /\ x e. ( 1 ... K ) ) /\ y e. ( 1 ... K ) ) /\ x < y ) -> ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` x ) < ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` y ) )
60 59 ex 
 |-  ( ( ( ( ph /\ a e. A ) /\ x e. ( 1 ... K ) ) /\ y e. ( 1 ... K ) ) -> ( x < y -> ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` x ) < ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` y ) ) )
61 60 ralrimiva 
 |-  ( ( ( ph /\ a e. A ) /\ x e. ( 1 ... K ) ) -> A. y e. ( 1 ... K ) ( x < y -> ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` x ) < ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` y ) ) )
62 61 ralrimiva 
 |-  ( ( ph /\ a e. A ) -> A. x e. ( 1 ... K ) A. y e. ( 1 ... K ) ( x < y -> ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` x ) < ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` y ) ) )
63 44 62 jca 
 |-  ( ( ph /\ a e. A ) -> ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) : ( 1 ... K ) --> ( 1 ... ( N + K ) ) /\ A. x e. ( 1 ... K ) A. y e. ( 1 ... K ) ( x < y -> ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` x ) < ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` y ) ) ) )
64 fzfid 
 |-  ( ( ph /\ a e. A ) -> ( 1 ... K ) e. Fin )
65 44 64 fexd 
 |-  ( ( ph /\ a e. A ) -> ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) e. _V )
66 feq1 
 |-  ( f = ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) -> ( f : ( 1 ... K ) --> ( 1 ... ( N + K ) ) <-> ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) : ( 1 ... K ) --> ( 1 ... ( N + K ) ) ) )
67 fveq1 
 |-  ( f = ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) -> ( f ` x ) = ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` x ) )
68 fveq1 
 |-  ( f = ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) -> ( f ` y ) = ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` y ) )
69 67 68 breq12d 
 |-  ( f = ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) -> ( ( f ` x ) < ( f ` y ) <-> ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` x ) < ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` y ) ) )
70 69 imbi2d 
 |-  ( f = ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) -> ( ( x < y -> ( f ` x ) < ( f ` y ) ) <-> ( x < y -> ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` x ) < ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` y ) ) ) )
71 70 2ralbidv 
 |-  ( f = ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) -> ( A. x e. ( 1 ... K ) A. y e. ( 1 ... K ) ( x < y -> ( f ` x ) < ( f ` y ) ) <-> A. x e. ( 1 ... K ) A. y e. ( 1 ... K ) ( x < y -> ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` x ) < ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` y ) ) ) )
72 66 71 anbi12d 
 |-  ( f = ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) -> ( ( f : ( 1 ... K ) --> ( 1 ... ( N + K ) ) /\ A. x e. ( 1 ... K ) A. y e. ( 1 ... K ) ( x < y -> ( f ` x ) < ( f ` y ) ) ) <-> ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) : ( 1 ... K ) --> ( 1 ... ( N + K ) ) /\ A. x e. ( 1 ... K ) A. y e. ( 1 ... K ) ( x < y -> ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` x ) < ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` y ) ) ) ) )
73 72 elabg 
 |-  ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) e. _V -> ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) e. { f | ( f : ( 1 ... K ) --> ( 1 ... ( N + K ) ) /\ A. x e. ( 1 ... K ) A. y e. ( 1 ... K ) ( x < y -> ( f ` x ) < ( f ` y ) ) ) } <-> ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) : ( 1 ... K ) --> ( 1 ... ( N + K ) ) /\ A. x e. ( 1 ... K ) A. y e. ( 1 ... K ) ( x < y -> ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` x ) < ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` y ) ) ) ) )
74 65 73 syl 
 |-  ( ( ph /\ a e. A ) -> ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) e. { f | ( f : ( 1 ... K ) --> ( 1 ... ( N + K ) ) /\ A. x e. ( 1 ... K ) A. y e. ( 1 ... K ) ( x < y -> ( f ` x ) < ( f ` y ) ) ) } <-> ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) : ( 1 ... K ) --> ( 1 ... ( N + K ) ) /\ A. x e. ( 1 ... K ) A. y e. ( 1 ... K ) ( x < y -> ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` x ) < ( ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) ` y ) ) ) ) )
75 63 74 mpbird 
 |-  ( ( ph /\ a e. A ) -> ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) e. { f | ( f : ( 1 ... K ) --> ( 1 ... ( N + K ) ) /\ A. x e. ( 1 ... K ) A. y e. ( 1 ... K ) ( x < y -> ( f ` x ) < ( f ` y ) ) ) } )
76 5 a1i 
 |-  ( ( ph /\ a e. A ) -> B = { f | ( f : ( 1 ... K ) --> ( 1 ... ( N + K ) ) /\ A. x e. ( 1 ... K ) A. y e. ( 1 ... K ) ( x < y -> ( f ` x ) < ( f ` y ) ) ) } )
77 75 76 eleqtrrd 
 |-  ( ( ph /\ a e. A ) -> ( j e. ( 1 ... K ) |-> ( j + sum_ l e. ( 1 ... j ) ( a ` l ) ) ) e. B )
78 77 3 fmptd 
 |-  ( ph -> F : A --> B )