Step |
Hyp |
Ref |
Expression |
1 |
|
poimir.0 |
|- ( ph -> N e. NN ) |
2 |
|
poimirlem22.s |
|- S = { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } |
3 |
|
poimirlem22.1 |
|- ( ph -> F : ( 0 ... ( N - 1 ) ) --> ( ( 0 ... K ) ^m ( 1 ... N ) ) ) |
4 |
|
poimirlem12.2 |
|- ( ph -> T e. S ) |
5 |
|
poimirlem11.3 |
|- ( ph -> ( 2nd ` T ) = 0 ) |
6 |
|
poimirlem11.4 |
|- ( ph -> U e. S ) |
7 |
|
poimirlem11.5 |
|- ( ph -> ( 2nd ` U ) = 0 ) |
8 |
|
poimirlem11.6 |
|- ( ph -> M e. ( 1 ... N ) ) |
9 |
|
eldif |
|- ( y e. ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) \ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) <-> ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) |
10 |
|
imassrn |
|- ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) C_ ran ( 2nd ` ( 1st ` T ) ) |
11 |
|
elrabi |
|- ( T e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } -> T e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) |
12 |
11 2
|
eleq2s |
|- ( T e. S -> T e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) |
13 |
4 12
|
syl |
|- ( ph -> T e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) |
14 |
|
xp1st |
|- ( T e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( 1st ` T ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) |
15 |
13 14
|
syl |
|- ( ph -> ( 1st ` T ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) |
16 |
|
xp2nd |
|- ( ( 1st ` T ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 2nd ` ( 1st ` T ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) |
17 |
15 16
|
syl |
|- ( ph -> ( 2nd ` ( 1st ` T ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) |
18 |
|
fvex |
|- ( 2nd ` ( 1st ` T ) ) e. _V |
19 |
|
f1oeq1 |
|- ( f = ( 2nd ` ( 1st ` T ) ) -> ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) ) |
20 |
18 19
|
elab |
|- ( ( 2nd ` ( 1st ` T ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } <-> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
21 |
17 20
|
sylib |
|- ( ph -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
22 |
|
f1of |
|- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 1 ... N ) ) |
23 |
21 22
|
syl |
|- ( ph -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 1 ... N ) ) |
24 |
23
|
frnd |
|- ( ph -> ran ( 2nd ` ( 1st ` T ) ) C_ ( 1 ... N ) ) |
25 |
10 24
|
sstrid |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) C_ ( 1 ... N ) ) |
26 |
|
elrabi |
|- ( U e. { t e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) | F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) } -> U e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) |
27 |
26 2
|
eleq2s |
|- ( U e. S -> U e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) |
28 |
6 27
|
syl |
|- ( ph -> U e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) |
29 |
|
xp1st |
|- ( U e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( 1st ` U ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) |
30 |
28 29
|
syl |
|- ( ph -> ( 1st ` U ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) |
31 |
|
xp2nd |
|- ( ( 1st ` U ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 2nd ` ( 1st ` U ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) |
32 |
30 31
|
syl |
|- ( ph -> ( 2nd ` ( 1st ` U ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) |
33 |
|
fvex |
|- ( 2nd ` ( 1st ` U ) ) e. _V |
34 |
|
f1oeq1 |
|- ( f = ( 2nd ` ( 1st ` U ) ) -> ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) ) |
35 |
33 34
|
elab |
|- ( ( 2nd ` ( 1st ` U ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } <-> ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
36 |
32 35
|
sylib |
|- ( ph -> ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
37 |
|
f1ofo |
|- ( ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) ) |
38 |
36 37
|
syl |
|- ( ph -> ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) ) |
39 |
|
foima |
|- ( ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) -> ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... N ) ) = ( 1 ... N ) ) |
40 |
38 39
|
syl |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... N ) ) = ( 1 ... N ) ) |
41 |
25 40
|
sseqtrrd |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) C_ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... N ) ) ) |
42 |
41
|
ssdifd |
|- ( ph -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) \ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) C_ ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... N ) ) \ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) |
43 |
|
dff1o3 |
|- ( ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) /\ Fun `' ( 2nd ` ( 1st ` U ) ) ) ) |
44 |
43
|
simprbi |
|- ( ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> Fun `' ( 2nd ` ( 1st ` U ) ) ) |
45 |
36 44
|
syl |
|- ( ph -> Fun `' ( 2nd ` ( 1st ` U ) ) ) |
46 |
|
imadif |
|- ( Fun `' ( 2nd ` ( 1st ` U ) ) -> ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... N ) \ ( 1 ... M ) ) ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... N ) ) \ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) |
47 |
45 46
|
syl |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... N ) \ ( 1 ... M ) ) ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... N ) ) \ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) |
48 |
|
difun2 |
|- ( ( ( ( M + 1 ) ... N ) u. ( 1 ... M ) ) \ ( 1 ... M ) ) = ( ( ( M + 1 ) ... N ) \ ( 1 ... M ) ) |
49 |
|
fzsplit |
|- ( M e. ( 1 ... N ) -> ( 1 ... N ) = ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) ) |
50 |
8 49
|
syl |
|- ( ph -> ( 1 ... N ) = ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) ) |
51 |
|
uncom |
|- ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) = ( ( ( M + 1 ) ... N ) u. ( 1 ... M ) ) |
52 |
50 51
|
eqtrdi |
|- ( ph -> ( 1 ... N ) = ( ( ( M + 1 ) ... N ) u. ( 1 ... M ) ) ) |
53 |
52
|
difeq1d |
|- ( ph -> ( ( 1 ... N ) \ ( 1 ... M ) ) = ( ( ( ( M + 1 ) ... N ) u. ( 1 ... M ) ) \ ( 1 ... M ) ) ) |
54 |
|
incom |
|- ( ( ( M + 1 ) ... N ) i^i ( 1 ... M ) ) = ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) |
55 |
|
elfznn |
|- ( M e. ( 1 ... N ) -> M e. NN ) |
56 |
8 55
|
syl |
|- ( ph -> M e. NN ) |
57 |
56
|
nnred |
|- ( ph -> M e. RR ) |
58 |
57
|
ltp1d |
|- ( ph -> M < ( M + 1 ) ) |
59 |
|
fzdisj |
|- ( M < ( M + 1 ) -> ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) = (/) ) |
60 |
58 59
|
syl |
|- ( ph -> ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) = (/) ) |
61 |
54 60
|
syl5eq |
|- ( ph -> ( ( ( M + 1 ) ... N ) i^i ( 1 ... M ) ) = (/) ) |
62 |
|
disj3 |
|- ( ( ( ( M + 1 ) ... N ) i^i ( 1 ... M ) ) = (/) <-> ( ( M + 1 ) ... N ) = ( ( ( M + 1 ) ... N ) \ ( 1 ... M ) ) ) |
63 |
61 62
|
sylib |
|- ( ph -> ( ( M + 1 ) ... N ) = ( ( ( M + 1 ) ... N ) \ ( 1 ... M ) ) ) |
64 |
48 53 63
|
3eqtr4a |
|- ( ph -> ( ( 1 ... N ) \ ( 1 ... M ) ) = ( ( M + 1 ) ... N ) ) |
65 |
64
|
imaeq2d |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... N ) \ ( 1 ... M ) ) ) = ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
66 |
47 65
|
eqtr3d |
|- ( ph -> ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... N ) ) \ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) = ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
67 |
42 66
|
sseqtrd |
|- ( ph -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) \ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) C_ ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
68 |
67
|
sselda |
|- ( ( ph /\ y e. ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) \ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) -> y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
69 |
9 68
|
sylan2br |
|- ( ( ph /\ ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) -> y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
70 |
|
fveq2 |
|- ( t = U -> ( 2nd ` t ) = ( 2nd ` U ) ) |
71 |
70
|
breq2d |
|- ( t = U -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` U ) ) ) |
72 |
71
|
ifbid |
|- ( t = U -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) ) |
73 |
72
|
csbeq1d |
|- ( t = U -> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = [_ if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
74 |
|
2fveq3 |
|- ( t = U -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` U ) ) ) |
75 |
|
2fveq3 |
|- ( t = U -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` U ) ) ) |
76 |
75
|
imaeq1d |
|- ( t = U -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) ) |
77 |
76
|
xpeq1d |
|- ( t = U -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) ) |
78 |
75
|
imaeq1d |
|- ( t = U -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) ) |
79 |
78
|
xpeq1d |
|- ( t = U -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) |
80 |
77 79
|
uneq12d |
|- ( t = U -> ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) = ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) |
81 |
74 80
|
oveq12d |
|- ( t = U -> ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
82 |
81
|
csbeq2dv |
|- ( t = U -> [_ if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = [_ if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
83 |
73 82
|
eqtrd |
|- ( t = U -> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = [_ if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
84 |
83
|
mpteq2dv |
|- ( t = U -> ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) |
85 |
84
|
eqeq2d |
|- ( t = U -> ( F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) <-> F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) ) |
86 |
85 2
|
elrab2 |
|- ( U e. S <-> ( U e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) ) |
87 |
86
|
simprbi |
|- ( U e. S -> F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) |
88 |
6 87
|
syl |
|- ( ph -> F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) |
89 |
|
breq12 |
|- ( ( y = ( M - 1 ) /\ ( 2nd ` U ) = 0 ) -> ( y < ( 2nd ` U ) <-> ( M - 1 ) < 0 ) ) |
90 |
7 89
|
sylan2 |
|- ( ( y = ( M - 1 ) /\ ph ) -> ( y < ( 2nd ` U ) <-> ( M - 1 ) < 0 ) ) |
91 |
90
|
ancoms |
|- ( ( ph /\ y = ( M - 1 ) ) -> ( y < ( 2nd ` U ) <-> ( M - 1 ) < 0 ) ) |
92 |
|
oveq1 |
|- ( y = ( M - 1 ) -> ( y + 1 ) = ( ( M - 1 ) + 1 ) ) |
93 |
56
|
nncnd |
|- ( ph -> M e. CC ) |
94 |
|
npcan1 |
|- ( M e. CC -> ( ( M - 1 ) + 1 ) = M ) |
95 |
93 94
|
syl |
|- ( ph -> ( ( M - 1 ) + 1 ) = M ) |
96 |
92 95
|
sylan9eqr |
|- ( ( ph /\ y = ( M - 1 ) ) -> ( y + 1 ) = M ) |
97 |
91 96
|
ifbieq2d |
|- ( ( ph /\ y = ( M - 1 ) ) -> if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) = if ( ( M - 1 ) < 0 , y , M ) ) |
98 |
56
|
nnzd |
|- ( ph -> M e. ZZ ) |
99 |
1
|
nnzd |
|- ( ph -> N e. ZZ ) |
100 |
|
elfzm1b |
|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M e. ( 1 ... N ) <-> ( M - 1 ) e. ( 0 ... ( N - 1 ) ) ) ) |
101 |
98 99 100
|
syl2anc |
|- ( ph -> ( M e. ( 1 ... N ) <-> ( M - 1 ) e. ( 0 ... ( N - 1 ) ) ) ) |
102 |
8 101
|
mpbid |
|- ( ph -> ( M - 1 ) e. ( 0 ... ( N - 1 ) ) ) |
103 |
|
elfzle1 |
|- ( ( M - 1 ) e. ( 0 ... ( N - 1 ) ) -> 0 <_ ( M - 1 ) ) |
104 |
102 103
|
syl |
|- ( ph -> 0 <_ ( M - 1 ) ) |
105 |
|
0red |
|- ( ph -> 0 e. RR ) |
106 |
|
nnm1nn0 |
|- ( M e. NN -> ( M - 1 ) e. NN0 ) |
107 |
56 106
|
syl |
|- ( ph -> ( M - 1 ) e. NN0 ) |
108 |
107
|
nn0red |
|- ( ph -> ( M - 1 ) e. RR ) |
109 |
105 108
|
lenltd |
|- ( ph -> ( 0 <_ ( M - 1 ) <-> -. ( M - 1 ) < 0 ) ) |
110 |
104 109
|
mpbid |
|- ( ph -> -. ( M - 1 ) < 0 ) |
111 |
110
|
iffalsed |
|- ( ph -> if ( ( M - 1 ) < 0 , y , M ) = M ) |
112 |
111
|
adantr |
|- ( ( ph /\ y = ( M - 1 ) ) -> if ( ( M - 1 ) < 0 , y , M ) = M ) |
113 |
97 112
|
eqtrd |
|- ( ( ph /\ y = ( M - 1 ) ) -> if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) = M ) |
114 |
113
|
csbeq1d |
|- ( ( ph /\ y = ( M - 1 ) ) -> [_ if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = [_ M / j ]_ ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
115 |
|
oveq2 |
|- ( j = M -> ( 1 ... j ) = ( 1 ... M ) ) |
116 |
115
|
imaeq2d |
|- ( j = M -> ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) |
117 |
116
|
xpeq1d |
|- ( j = M -> ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) ) |
118 |
|
oveq1 |
|- ( j = M -> ( j + 1 ) = ( M + 1 ) ) |
119 |
118
|
oveq1d |
|- ( j = M -> ( ( j + 1 ) ... N ) = ( ( M + 1 ) ... N ) ) |
120 |
119
|
imaeq2d |
|- ( j = M -> ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
121 |
120
|
xpeq1d |
|- ( j = M -> ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) |
122 |
117 121
|
uneq12d |
|- ( j = M -> ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) = ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) |
123 |
122
|
oveq2d |
|- ( j = M -> ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
124 |
123
|
adantl |
|- ( ( ph /\ j = M ) -> ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
125 |
8 124
|
csbied |
|- ( ph -> [_ M / j ]_ ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
126 |
125
|
adantr |
|- ( ( ph /\ y = ( M - 1 ) ) -> [_ M / j ]_ ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
127 |
114 126
|
eqtrd |
|- ( ( ph /\ y = ( M - 1 ) ) -> [_ if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
128 |
|
ovexd |
|- ( ph -> ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) e. _V ) |
129 |
88 127 102 128
|
fvmptd |
|- ( ph -> ( F ` ( M - 1 ) ) = ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
130 |
129
|
fveq1d |
|- ( ph -> ( ( F ` ( M - 1 ) ) ` y ) = ( ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ` y ) ) |
131 |
130
|
adantr |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( F ` ( M - 1 ) ) ` y ) = ( ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ` y ) ) |
132 |
|
imassrn |
|- ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) C_ ran ( 2nd ` ( 1st ` U ) ) |
133 |
|
f1of |
|- ( ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) --> ( 1 ... N ) ) |
134 |
36 133
|
syl |
|- ( ph -> ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) --> ( 1 ... N ) ) |
135 |
134
|
frnd |
|- ( ph -> ran ( 2nd ` ( 1st ` U ) ) C_ ( 1 ... N ) ) |
136 |
132 135
|
sstrid |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) C_ ( 1 ... N ) ) |
137 |
136
|
sselda |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> y e. ( 1 ... N ) ) |
138 |
|
xp1st |
|- ( ( 1st ` U ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 1st ` ( 1st ` U ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) ) |
139 |
30 138
|
syl |
|- ( ph -> ( 1st ` ( 1st ` U ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) ) |
140 |
|
elmapfn |
|- ( ( 1st ` ( 1st ` U ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` U ) ) Fn ( 1 ... N ) ) |
141 |
139 140
|
syl |
|- ( ph -> ( 1st ` ( 1st ` U ) ) Fn ( 1 ... N ) ) |
142 |
141
|
adantr |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( 1st ` ( 1st ` U ) ) Fn ( 1 ... N ) ) |
143 |
|
1ex |
|- 1 e. _V |
144 |
|
fnconstg |
|- ( 1 e. _V -> ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) |
145 |
143 144
|
ax-mp |
|- ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) |
146 |
|
c0ex |
|- 0 e. _V |
147 |
|
fnconstg |
|- ( 0 e. _V -> ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
148 |
146 147
|
ax-mp |
|- ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) |
149 |
145 148
|
pm3.2i |
|- ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) /\ ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
150 |
|
imain |
|- ( Fun `' ( 2nd ` ( 1st ` U ) ) -> ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) ) |
151 |
45 150
|
syl |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) ) |
152 |
60
|
imaeq2d |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) ) = ( ( 2nd ` ( 1st ` U ) ) " (/) ) ) |
153 |
|
ima0 |
|- ( ( 2nd ` ( 1st ` U ) ) " (/) ) = (/) |
154 |
152 153
|
eqtrdi |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) ) = (/) ) |
155 |
151 154
|
eqtr3d |
|- ( ph -> ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) = (/) ) |
156 |
|
fnun |
|- ( ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) /\ ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) /\ ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) = (/) ) -> ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) ) |
157 |
149 155 156
|
sylancr |
|- ( ph -> ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) ) |
158 |
|
imaundi |
|- ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) |
159 |
50
|
imaeq2d |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... N ) ) = ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) ) ) |
160 |
159 40
|
eqtr3d |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) " ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) ) = ( 1 ... N ) ) |
161 |
158 160
|
eqtr3id |
|- ( ph -> ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) = ( 1 ... N ) ) |
162 |
161
|
fneq2d |
|- ( ph -> ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) <-> ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( 1 ... N ) ) ) |
163 |
157 162
|
mpbid |
|- ( ph -> ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( 1 ... N ) ) |
164 |
163
|
adantr |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( 1 ... N ) ) |
165 |
|
ovexd |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( 1 ... N ) e. _V ) |
166 |
|
inidm |
|- ( ( 1 ... N ) i^i ( 1 ... N ) ) = ( 1 ... N ) |
167 |
|
eqidd |
|- ( ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` U ) ) ` y ) = ( ( 1st ` ( 1st ` U ) ) ` y ) ) |
168 |
|
fvun2 |
|- ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) /\ ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) /\ ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) = (/) /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) ) -> ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = ( ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ` y ) ) |
169 |
145 148 168
|
mp3an12 |
|- ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) = (/) /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = ( ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ` y ) ) |
170 |
155 169
|
sylan |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = ( ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ` y ) ) |
171 |
146
|
fvconst2 |
|- ( y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) -> ( ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ` y ) = 0 ) |
172 |
171
|
adantl |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ` y ) = 0 ) |
173 |
170 172
|
eqtrd |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = 0 ) |
174 |
173
|
adantr |
|- ( ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) /\ y e. ( 1 ... N ) ) -> ( ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = 0 ) |
175 |
142 164 165 165 166 167 174
|
ofval |
|- ( ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) /\ y e. ( 1 ... N ) ) -> ( ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ` y ) = ( ( ( 1st ` ( 1st ` U ) ) ` y ) + 0 ) ) |
176 |
137 175
|
mpdan |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( ( 1st ` ( 1st ` U ) ) oF + ( ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ` y ) = ( ( ( 1st ` ( 1st ` U ) ) ` y ) + 0 ) ) |
177 |
|
elmapi |
|- ( ( 1st ` ( 1st ` U ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` U ) ) : ( 1 ... N ) --> ( 0 ..^ K ) ) |
178 |
139 177
|
syl |
|- ( ph -> ( 1st ` ( 1st ` U ) ) : ( 1 ... N ) --> ( 0 ..^ K ) ) |
179 |
178
|
ffvelrnda |
|- ( ( ph /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` U ) ) ` y ) e. ( 0 ..^ K ) ) |
180 |
|
elfzonn0 |
|- ( ( ( 1st ` ( 1st ` U ) ) ` y ) e. ( 0 ..^ K ) -> ( ( 1st ` ( 1st ` U ) ) ` y ) e. NN0 ) |
181 |
179 180
|
syl |
|- ( ( ph /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` U ) ) ` y ) e. NN0 ) |
182 |
181
|
nn0cnd |
|- ( ( ph /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` U ) ) ` y ) e. CC ) |
183 |
137 182
|
syldan |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( 1st ` ( 1st ` U ) ) ` y ) e. CC ) |
184 |
183
|
addid1d |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( ( 1st ` ( 1st ` U ) ) ` y ) + 0 ) = ( ( 1st ` ( 1st ` U ) ) ` y ) ) |
185 |
131 176 184
|
3eqtrd |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` U ) ) " ( ( M + 1 ) ... N ) ) ) -> ( ( F ` ( M - 1 ) ) ` y ) = ( ( 1st ` ( 1st ` U ) ) ` y ) ) |
186 |
69 185
|
syldan |
|- ( ( ph /\ ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) -> ( ( F ` ( M - 1 ) ) ` y ) = ( ( 1st ` ( 1st ` U ) ) ` y ) ) |
187 |
|
fveq2 |
|- ( t = T -> ( 2nd ` t ) = ( 2nd ` T ) ) |
188 |
187
|
breq2d |
|- ( t = T -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` T ) ) ) |
189 |
188
|
ifbid |
|- ( t = T -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) ) |
190 |
189
|
csbeq1d |
|- ( t = T -> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = [_ if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
191 |
|
2fveq3 |
|- ( t = T -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` T ) ) ) |
192 |
|
2fveq3 |
|- ( t = T -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` T ) ) ) |
193 |
192
|
imaeq1d |
|- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) ) |
194 |
193
|
xpeq1d |
|- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) ) |
195 |
192
|
imaeq1d |
|- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) ) |
196 |
195
|
xpeq1d |
|- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) |
197 |
194 196
|
uneq12d |
|- ( t = T -> ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) = ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) |
198 |
191 197
|
oveq12d |
|- ( t = T -> ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
199 |
198
|
csbeq2dv |
|- ( t = T -> [_ if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = [_ if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
200 |
190 199
|
eqtrd |
|- ( t = T -> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = [_ if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
201 |
200
|
mpteq2dv |
|- ( t = T -> ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) |
202 |
201
|
eqeq2d |
|- ( t = T -> ( F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) <-> F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) ) |
203 |
202 2
|
elrab2 |
|- ( T e. S <-> ( T e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) /\ F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) ) |
204 |
203
|
simprbi |
|- ( T e. S -> F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) |
205 |
4 204
|
syl |
|- ( ph -> F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) |
206 |
|
breq12 |
|- ( ( y = ( M - 1 ) /\ ( 2nd ` T ) = 0 ) -> ( y < ( 2nd ` T ) <-> ( M - 1 ) < 0 ) ) |
207 |
5 206
|
sylan2 |
|- ( ( y = ( M - 1 ) /\ ph ) -> ( y < ( 2nd ` T ) <-> ( M - 1 ) < 0 ) ) |
208 |
207
|
ancoms |
|- ( ( ph /\ y = ( M - 1 ) ) -> ( y < ( 2nd ` T ) <-> ( M - 1 ) < 0 ) ) |
209 |
208 96
|
ifbieq2d |
|- ( ( ph /\ y = ( M - 1 ) ) -> if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) = if ( ( M - 1 ) < 0 , y , M ) ) |
210 |
209 112
|
eqtrd |
|- ( ( ph /\ y = ( M - 1 ) ) -> if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) = M ) |
211 |
210
|
csbeq1d |
|- ( ( ph /\ y = ( M - 1 ) ) -> [_ if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = [_ M / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
212 |
115
|
imaeq2d |
|- ( j = M -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) |
213 |
212
|
xpeq1d |
|- ( j = M -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) ) |
214 |
119
|
imaeq2d |
|- ( j = M -> ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) |
215 |
214
|
xpeq1d |
|- ( j = M -> ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) |
216 |
213 215
|
uneq12d |
|- ( j = M -> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) = ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) |
217 |
216
|
oveq2d |
|- ( j = M -> ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
218 |
217
|
adantl |
|- ( ( ph /\ j = M ) -> ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
219 |
8 218
|
csbied |
|- ( ph -> [_ M / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
220 |
219
|
adantr |
|- ( ( ph /\ y = ( M - 1 ) ) -> [_ M / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
221 |
211 220
|
eqtrd |
|- ( ( ph /\ y = ( M - 1 ) ) -> [_ if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
222 |
|
ovexd |
|- ( ph -> ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) e. _V ) |
223 |
205 221 102 222
|
fvmptd |
|- ( ph -> ( F ` ( M - 1 ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
224 |
223
|
fveq1d |
|- ( ph -> ( ( F ` ( M - 1 ) ) ` y ) = ( ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ` y ) ) |
225 |
224
|
adantr |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( F ` ( M - 1 ) ) ` y ) = ( ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ` y ) ) |
226 |
25
|
sselda |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> y e. ( 1 ... N ) ) |
227 |
|
xp1st |
|- ( ( 1st ` T ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 1st ` ( 1st ` T ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) ) |
228 |
15 227
|
syl |
|- ( ph -> ( 1st ` ( 1st ` T ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) ) |
229 |
|
elmapfn |
|- ( ( 1st ` ( 1st ` T ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` T ) ) Fn ( 1 ... N ) ) |
230 |
228 229
|
syl |
|- ( ph -> ( 1st ` ( 1st ` T ) ) Fn ( 1 ... N ) ) |
231 |
230
|
adantr |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( 1st ` ( 1st ` T ) ) Fn ( 1 ... N ) ) |
232 |
|
fnconstg |
|- ( 1 e. _V -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) |
233 |
143 232
|
ax-mp |
|- ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) |
234 |
|
fnconstg |
|- ( 0 e. _V -> ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) |
235 |
146 234
|
ax-mp |
|- ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) |
236 |
233 235
|
pm3.2i |
|- ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) |
237 |
|
dff1o3 |
|- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) /\ Fun `' ( 2nd ` ( 1st ` T ) ) ) ) |
238 |
237
|
simprbi |
|- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> Fun `' ( 2nd ` ( 1st ` T ) ) ) |
239 |
21 238
|
syl |
|- ( ph -> Fun `' ( 2nd ` ( 1st ` T ) ) ) |
240 |
|
imain |
|- ( Fun `' ( 2nd ` ( 1st ` T ) ) -> ( ( 2nd ` ( 1st ` T ) ) " ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) ) |
241 |
239 240
|
syl |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) ) |
242 |
60
|
imaeq2d |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) ) = ( ( 2nd ` ( 1st ` T ) ) " (/) ) ) |
243 |
|
ima0 |
|- ( ( 2nd ` ( 1st ` T ) ) " (/) ) = (/) |
244 |
242 243
|
eqtrdi |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( ( 1 ... M ) i^i ( ( M + 1 ) ... N ) ) ) = (/) ) |
245 |
241 244
|
eqtr3d |
|- ( ph -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) = (/) ) |
246 |
|
fnun |
|- ( ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) /\ ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) = (/) ) -> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) ) |
247 |
236 245 246
|
sylancr |
|- ( ph -> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) ) |
248 |
|
imaundi |
|- ( ( 2nd ` ( 1st ` T ) ) " ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) |
249 |
50
|
imaeq2d |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) ) ) |
250 |
|
f1ofo |
|- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) ) |
251 |
21 250
|
syl |
|- ( ph -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) ) |
252 |
|
foima |
|- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) = ( 1 ... N ) ) |
253 |
251 252
|
syl |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) = ( 1 ... N ) ) |
254 |
249 253
|
eqtr3d |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( ( 1 ... M ) u. ( ( M + 1 ) ... N ) ) ) = ( 1 ... N ) ) |
255 |
248 254
|
eqtr3id |
|- ( ph -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) = ( 1 ... N ) ) |
256 |
255
|
fneq2d |
|- ( ph -> ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) u. ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) <-> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( 1 ... N ) ) ) |
257 |
247 256
|
mpbid |
|- ( ph -> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( 1 ... N ) ) |
258 |
257
|
adantr |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) Fn ( 1 ... N ) ) |
259 |
|
ovexd |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( 1 ... N ) e. _V ) |
260 |
|
eqidd |
|- ( ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` y ) = ( ( 1st ` ( 1st ` T ) ) ` y ) ) |
261 |
|
fvun1 |
|- ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) Fn ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) /\ ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) = (/) /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) ) -> ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) ` y ) ) |
262 |
233 235 261
|
mp3an12 |
|- ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) i^i ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) ) = (/) /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) ` y ) ) |
263 |
245 262
|
sylan |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) ` y ) ) |
264 |
143
|
fvconst2 |
|- ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) -> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) ` y ) = 1 ) |
265 |
264
|
adantl |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) ` y ) = 1 ) |
266 |
263 265
|
eqtrd |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = 1 ) |
267 |
266
|
adantr |
|- ( ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) /\ y e. ( 1 ... N ) ) -> ( ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ` y ) = 1 ) |
268 |
231 258 259 259 166 260 267
|
ofval |
|- ( ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) /\ y e. ( 1 ... N ) ) -> ( ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ` y ) = ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) ) |
269 |
226 268
|
mpdan |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( M + 1 ) ... N ) ) X. { 0 } ) ) ) ` y ) = ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) ) |
270 |
225 269
|
eqtrd |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( F ` ( M - 1 ) ) ` y ) = ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) ) |
271 |
270
|
adantrr |
|- ( ( ph /\ ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) -> ( ( F ` ( M - 1 ) ) ` y ) = ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) ) |
272 |
1 2 3 6 7
|
poimirlem10 |
|- ( ph -> ( ( F ` ( N - 1 ) ) oF - ( ( 1 ... N ) X. { 1 } ) ) = ( 1st ` ( 1st ` U ) ) ) |
273 |
1 2 3 4 5
|
poimirlem10 |
|- ( ph -> ( ( F ` ( N - 1 ) ) oF - ( ( 1 ... N ) X. { 1 } ) ) = ( 1st ` ( 1st ` T ) ) ) |
274 |
272 273
|
eqtr3d |
|- ( ph -> ( 1st ` ( 1st ` U ) ) = ( 1st ` ( 1st ` T ) ) ) |
275 |
274
|
fveq1d |
|- ( ph -> ( ( 1st ` ( 1st ` U ) ) ` y ) = ( ( 1st ` ( 1st ` T ) ) ` y ) ) |
276 |
275
|
adantr |
|- ( ( ph /\ ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) -> ( ( 1st ` ( 1st ` U ) ) ` y ) = ( ( 1st ` ( 1st ` T ) ) ` y ) ) |
277 |
186 271 276
|
3eqtr3d |
|- ( ( ph /\ ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) -> ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) = ( ( 1st ` ( 1st ` T ) ) ` y ) ) |
278 |
|
elmapi |
|- ( ( 1st ` ( 1st ` T ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0 ..^ K ) ) |
279 |
228 278
|
syl |
|- ( ph -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0 ..^ K ) ) |
280 |
279
|
ffvelrnda |
|- ( ( ph /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` y ) e. ( 0 ..^ K ) ) |
281 |
|
elfzonn0 |
|- ( ( ( 1st ` ( 1st ` T ) ) ` y ) e. ( 0 ..^ K ) -> ( ( 1st ` ( 1st ` T ) ) ` y ) e. NN0 ) |
282 |
280 281
|
syl |
|- ( ( ph /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` y ) e. NN0 ) |
283 |
282
|
nn0red |
|- ( ( ph /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` y ) e. RR ) |
284 |
283
|
ltp1d |
|- ( ( ph /\ y e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` y ) < ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) ) |
285 |
283 284
|
gtned |
|- ( ( ph /\ y e. ( 1 ... N ) ) -> ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) =/= ( ( 1st ` ( 1st ` T ) ) ` y ) ) |
286 |
226 285
|
syldan |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) =/= ( ( 1st ` ( 1st ` T ) ) ` y ) ) |
287 |
286
|
neneqd |
|- ( ( ph /\ y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) ) -> -. ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) = ( ( 1st ` ( 1st ` T ) ) ` y ) ) |
288 |
287
|
adantrr |
|- ( ( ph /\ ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) -> -. ( ( ( 1st ` ( 1st ` T ) ) ` y ) + 1 ) = ( ( 1st ` ( 1st ` T ) ) ` y ) ) |
289 |
277 288
|
pm2.65da |
|- ( ph -> -. ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) |
290 |
|
iman |
|- ( ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) -> y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) <-> -. ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) /\ -. y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) |
291 |
289 290
|
sylibr |
|- ( ph -> ( y e. ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) -> y e. ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) ) |
292 |
291
|
ssrdv |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... M ) ) C_ ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... M ) ) ) |