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 |
|
poimirlem9.1 |
|- ( ph -> T e. S ) |
4 |
|
poimirlem9.2 |
|- ( ph -> ( 2nd ` T ) e. ( 1 ... ( N - 1 ) ) ) |
5 |
|
poimirlem9.3 |
|- ( ph -> U e. S ) |
6 |
|
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 ) ) ) |
7 |
6 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 ) ) ) |
8 |
5 7
|
syl |
|- ( ph -> U e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) |
9 |
|
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 ) } ) ) |
10 |
8 9
|
syl |
|- ( ph -> ( 1st ` U ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) |
11 |
|
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 ) } ) |
12 |
10 11
|
syl |
|- ( ph -> ( 2nd ` ( 1st ` U ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) |
13 |
|
fvex |
|- ( 2nd ` ( 1st ` U ) ) e. _V |
14 |
|
f1oeq1 |
|- ( f = ( 2nd ` ( 1st ` U ) ) -> ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) ) |
15 |
13 14
|
elab |
|- ( ( 2nd ` ( 1st ` U ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } <-> ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
16 |
12 15
|
sylib |
|- ( ph -> ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
17 |
|
f1ofn |
|- ( ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> ( 2nd ` ( 1st ` U ) ) Fn ( 1 ... N ) ) |
18 |
16 17
|
syl |
|- ( ph -> ( 2nd ` ( 1st ` U ) ) Fn ( 1 ... N ) ) |
19 |
|
difss |
|- ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) C_ ( 1 ... N ) |
20 |
|
fnssres |
|- ( ( ( 2nd ` ( 1st ` U ) ) Fn ( 1 ... N ) /\ ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) C_ ( 1 ... N ) ) -> ( ( 2nd ` ( 1st ` U ) ) |` ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) Fn ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) |
21 |
18 19 20
|
sylancl |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) |` ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) Fn ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) |
22 |
|
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 ) ) ) |
23 |
22 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 ) ) ) |
24 |
3 23
|
syl |
|- ( ph -> T e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) |
25 |
|
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 ) } ) ) |
26 |
24 25
|
syl |
|- ( ph -> ( 1st ` T ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) |
27 |
|
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 ) } ) |
28 |
26 27
|
syl |
|- ( ph -> ( 2nd ` ( 1st ` T ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) |
29 |
|
fvex |
|- ( 2nd ` ( 1st ` T ) ) e. _V |
30 |
|
f1oeq1 |
|- ( f = ( 2nd ` ( 1st ` T ) ) -> ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) ) |
31 |
29 30
|
elab |
|- ( ( 2nd ` ( 1st ` T ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } <-> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
32 |
28 31
|
sylib |
|- ( ph -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
33 |
|
f1ofn |
|- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> ( 2nd ` ( 1st ` T ) ) Fn ( 1 ... N ) ) |
34 |
32 33
|
syl |
|- ( ph -> ( 2nd ` ( 1st ` T ) ) Fn ( 1 ... N ) ) |
35 |
|
fnssres |
|- ( ( ( 2nd ` ( 1st ` T ) ) Fn ( 1 ... N ) /\ ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) C_ ( 1 ... N ) ) -> ( ( 2nd ` ( 1st ` T ) ) |` ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) Fn ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) |
36 |
34 19 35
|
sylancl |
|- ( ph -> ( ( 2nd ` ( 1st ` T ) ) |` ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) Fn ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) |
37 |
|
fzp1elp1 |
|- ( ( 2nd ` T ) e. ( 1 ... ( N - 1 ) ) -> ( ( 2nd ` T ) + 1 ) e. ( 1 ... ( ( N - 1 ) + 1 ) ) ) |
38 |
4 37
|
syl |
|- ( ph -> ( ( 2nd ` T ) + 1 ) e. ( 1 ... ( ( N - 1 ) + 1 ) ) ) |
39 |
1
|
nncnd |
|- ( ph -> N e. CC ) |
40 |
|
npcan1 |
|- ( N e. CC -> ( ( N - 1 ) + 1 ) = N ) |
41 |
39 40
|
syl |
|- ( ph -> ( ( N - 1 ) + 1 ) = N ) |
42 |
41
|
oveq2d |
|- ( ph -> ( 1 ... ( ( N - 1 ) + 1 ) ) = ( 1 ... N ) ) |
43 |
38 42
|
eleqtrd |
|- ( ph -> ( ( 2nd ` T ) + 1 ) e. ( 1 ... N ) ) |
44 |
|
fzsplit |
|- ( ( ( 2nd ` T ) + 1 ) e. ( 1 ... N ) -> ( 1 ... N ) = ( ( 1 ... ( ( 2nd ` T ) + 1 ) ) u. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) ) |
45 |
43 44
|
syl |
|- ( ph -> ( 1 ... N ) = ( ( 1 ... ( ( 2nd ` T ) + 1 ) ) u. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) ) |
46 |
45
|
difeq1d |
|- ( ph -> ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) = ( ( ( 1 ... ( ( 2nd ` T ) + 1 ) ) u. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) |
47 |
|
difundir |
|- ( ( ( 1 ... ( ( 2nd ` T ) + 1 ) ) u. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) = ( ( ( 1 ... ( ( 2nd ` T ) + 1 ) ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) u. ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) |
48 |
|
elfznn |
|- ( ( 2nd ` T ) e. ( 1 ... ( N - 1 ) ) -> ( 2nd ` T ) e. NN ) |
49 |
4 48
|
syl |
|- ( ph -> ( 2nd ` T ) e. NN ) |
50 |
49
|
nncnd |
|- ( ph -> ( 2nd ` T ) e. CC ) |
51 |
|
npcan1 |
|- ( ( 2nd ` T ) e. CC -> ( ( ( 2nd ` T ) - 1 ) + 1 ) = ( 2nd ` T ) ) |
52 |
50 51
|
syl |
|- ( ph -> ( ( ( 2nd ` T ) - 1 ) + 1 ) = ( 2nd ` T ) ) |
53 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
54 |
49 53
|
eleqtrdi |
|- ( ph -> ( 2nd ` T ) e. ( ZZ>= ` 1 ) ) |
55 |
52 54
|
eqeltrd |
|- ( ph -> ( ( ( 2nd ` T ) - 1 ) + 1 ) e. ( ZZ>= ` 1 ) ) |
56 |
49
|
nnzd |
|- ( ph -> ( 2nd ` T ) e. ZZ ) |
57 |
|
peano2zm |
|- ( ( 2nd ` T ) e. ZZ -> ( ( 2nd ` T ) - 1 ) e. ZZ ) |
58 |
56 57
|
syl |
|- ( ph -> ( ( 2nd ` T ) - 1 ) e. ZZ ) |
59 |
|
uzid |
|- ( ( ( 2nd ` T ) - 1 ) e. ZZ -> ( ( 2nd ` T ) - 1 ) e. ( ZZ>= ` ( ( 2nd ` T ) - 1 ) ) ) |
60 |
|
peano2uz |
|- ( ( ( 2nd ` T ) - 1 ) e. ( ZZ>= ` ( ( 2nd ` T ) - 1 ) ) -> ( ( ( 2nd ` T ) - 1 ) + 1 ) e. ( ZZ>= ` ( ( 2nd ` T ) - 1 ) ) ) |
61 |
58 59 60
|
3syl |
|- ( ph -> ( ( ( 2nd ` T ) - 1 ) + 1 ) e. ( ZZ>= ` ( ( 2nd ` T ) - 1 ) ) ) |
62 |
52 61
|
eqeltrrd |
|- ( ph -> ( 2nd ` T ) e. ( ZZ>= ` ( ( 2nd ` T ) - 1 ) ) ) |
63 |
|
peano2uz |
|- ( ( 2nd ` T ) e. ( ZZ>= ` ( ( 2nd ` T ) - 1 ) ) -> ( ( 2nd ` T ) + 1 ) e. ( ZZ>= ` ( ( 2nd ` T ) - 1 ) ) ) |
64 |
62 63
|
syl |
|- ( ph -> ( ( 2nd ` T ) + 1 ) e. ( ZZ>= ` ( ( 2nd ` T ) - 1 ) ) ) |
65 |
|
fzsplit2 |
|- ( ( ( ( ( 2nd ` T ) - 1 ) + 1 ) e. ( ZZ>= ` 1 ) /\ ( ( 2nd ` T ) + 1 ) e. ( ZZ>= ` ( ( 2nd ` T ) - 1 ) ) ) -> ( 1 ... ( ( 2nd ` T ) + 1 ) ) = ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) u. ( ( ( ( 2nd ` T ) - 1 ) + 1 ) ... ( ( 2nd ` T ) + 1 ) ) ) ) |
66 |
55 64 65
|
syl2anc |
|- ( ph -> ( 1 ... ( ( 2nd ` T ) + 1 ) ) = ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) u. ( ( ( ( 2nd ` T ) - 1 ) + 1 ) ... ( ( 2nd ` T ) + 1 ) ) ) ) |
67 |
52
|
oveq1d |
|- ( ph -> ( ( ( ( 2nd ` T ) - 1 ) + 1 ) ... ( ( 2nd ` T ) + 1 ) ) = ( ( 2nd ` T ) ... ( ( 2nd ` T ) + 1 ) ) ) |
68 |
|
fzpr |
|- ( ( 2nd ` T ) e. ZZ -> ( ( 2nd ` T ) ... ( ( 2nd ` T ) + 1 ) ) = { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) |
69 |
56 68
|
syl |
|- ( ph -> ( ( 2nd ` T ) ... ( ( 2nd ` T ) + 1 ) ) = { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) |
70 |
67 69
|
eqtrd |
|- ( ph -> ( ( ( ( 2nd ` T ) - 1 ) + 1 ) ... ( ( 2nd ` T ) + 1 ) ) = { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) |
71 |
70
|
uneq2d |
|- ( ph -> ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) u. ( ( ( ( 2nd ` T ) - 1 ) + 1 ) ... ( ( 2nd ` T ) + 1 ) ) ) = ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) u. { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) |
72 |
66 71
|
eqtrd |
|- ( ph -> ( 1 ... ( ( 2nd ` T ) + 1 ) ) = ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) u. { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) |
73 |
72
|
difeq1d |
|- ( ph -> ( ( 1 ... ( ( 2nd ` T ) + 1 ) ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) = ( ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) u. { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) |
74 |
49
|
nnred |
|- ( ph -> ( 2nd ` T ) e. RR ) |
75 |
74
|
ltm1d |
|- ( ph -> ( ( 2nd ` T ) - 1 ) < ( 2nd ` T ) ) |
76 |
58
|
zred |
|- ( ph -> ( ( 2nd ` T ) - 1 ) e. RR ) |
77 |
76 74
|
ltnled |
|- ( ph -> ( ( ( 2nd ` T ) - 1 ) < ( 2nd ` T ) <-> -. ( 2nd ` T ) <_ ( ( 2nd ` T ) - 1 ) ) ) |
78 |
75 77
|
mpbid |
|- ( ph -> -. ( 2nd ` T ) <_ ( ( 2nd ` T ) - 1 ) ) |
79 |
|
elfzle2 |
|- ( ( 2nd ` T ) e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) -> ( 2nd ` T ) <_ ( ( 2nd ` T ) - 1 ) ) |
80 |
78 79
|
nsyl |
|- ( ph -> -. ( 2nd ` T ) e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) |
81 |
|
difsn |
|- ( -. ( 2nd ` T ) e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) -> ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) \ { ( 2nd ` T ) } ) = ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) |
82 |
80 81
|
syl |
|- ( ph -> ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) \ { ( 2nd ` T ) } ) = ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) |
83 |
|
peano2re |
|- ( ( 2nd ` T ) e. RR -> ( ( 2nd ` T ) + 1 ) e. RR ) |
84 |
74 83
|
syl |
|- ( ph -> ( ( 2nd ` T ) + 1 ) e. RR ) |
85 |
74
|
ltp1d |
|- ( ph -> ( 2nd ` T ) < ( ( 2nd ` T ) + 1 ) ) |
86 |
76 74 84 75 85
|
lttrd |
|- ( ph -> ( ( 2nd ` T ) - 1 ) < ( ( 2nd ` T ) + 1 ) ) |
87 |
76 84
|
ltnled |
|- ( ph -> ( ( ( 2nd ` T ) - 1 ) < ( ( 2nd ` T ) + 1 ) <-> -. ( ( 2nd ` T ) + 1 ) <_ ( ( 2nd ` T ) - 1 ) ) ) |
88 |
86 87
|
mpbid |
|- ( ph -> -. ( ( 2nd ` T ) + 1 ) <_ ( ( 2nd ` T ) - 1 ) ) |
89 |
|
elfzle2 |
|- ( ( ( 2nd ` T ) + 1 ) e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) -> ( ( 2nd ` T ) + 1 ) <_ ( ( 2nd ` T ) - 1 ) ) |
90 |
88 89
|
nsyl |
|- ( ph -> -. ( ( 2nd ` T ) + 1 ) e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) |
91 |
|
difsn |
|- ( -. ( ( 2nd ` T ) + 1 ) e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) -> ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) \ { ( ( 2nd ` T ) + 1 ) } ) = ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) |
92 |
90 91
|
syl |
|- ( ph -> ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) \ { ( ( 2nd ` T ) + 1 ) } ) = ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) |
93 |
82 92
|
ineq12d |
|- ( ph -> ( ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) \ { ( 2nd ` T ) } ) i^i ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) \ { ( ( 2nd ` T ) + 1 ) } ) ) = ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) i^i ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) ) |
94 |
|
difun2 |
|- ( ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) u. { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) = ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) |
95 |
|
df-pr |
|- { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } = ( { ( 2nd ` T ) } u. { ( ( 2nd ` T ) + 1 ) } ) |
96 |
95
|
difeq2i |
|- ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) = ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) \ ( { ( 2nd ` T ) } u. { ( ( 2nd ` T ) + 1 ) } ) ) |
97 |
|
difundi |
|- ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) \ ( { ( 2nd ` T ) } u. { ( ( 2nd ` T ) + 1 ) } ) ) = ( ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) \ { ( 2nd ` T ) } ) i^i ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) \ { ( ( 2nd ` T ) + 1 ) } ) ) |
98 |
94 96 97
|
3eqtrri |
|- ( ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) \ { ( 2nd ` T ) } ) i^i ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) \ { ( ( 2nd ` T ) + 1 ) } ) ) = ( ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) u. { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) |
99 |
|
inidm |
|- ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) i^i ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) = ( 1 ... ( ( 2nd ` T ) - 1 ) ) |
100 |
93 98 99
|
3eqtr3g |
|- ( ph -> ( ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) u. { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) = ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) |
101 |
73 100
|
eqtrd |
|- ( ph -> ( ( 1 ... ( ( 2nd ` T ) + 1 ) ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) = ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) |
102 |
|
peano2re |
|- ( ( ( 2nd ` T ) + 1 ) e. RR -> ( ( ( 2nd ` T ) + 1 ) + 1 ) e. RR ) |
103 |
84 102
|
syl |
|- ( ph -> ( ( ( 2nd ` T ) + 1 ) + 1 ) e. RR ) |
104 |
84
|
ltp1d |
|- ( ph -> ( ( 2nd ` T ) + 1 ) < ( ( ( 2nd ` T ) + 1 ) + 1 ) ) |
105 |
74 84 103 85 104
|
lttrd |
|- ( ph -> ( 2nd ` T ) < ( ( ( 2nd ` T ) + 1 ) + 1 ) ) |
106 |
74 103
|
ltnled |
|- ( ph -> ( ( 2nd ` T ) < ( ( ( 2nd ` T ) + 1 ) + 1 ) <-> -. ( ( ( 2nd ` T ) + 1 ) + 1 ) <_ ( 2nd ` T ) ) ) |
107 |
105 106
|
mpbid |
|- ( ph -> -. ( ( ( 2nd ` T ) + 1 ) + 1 ) <_ ( 2nd ` T ) ) |
108 |
|
elfzle1 |
|- ( ( 2nd ` T ) e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) -> ( ( ( 2nd ` T ) + 1 ) + 1 ) <_ ( 2nd ` T ) ) |
109 |
107 108
|
nsyl |
|- ( ph -> -. ( 2nd ` T ) e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) |
110 |
|
difsn |
|- ( -. ( 2nd ` T ) e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) -> ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) \ { ( 2nd ` T ) } ) = ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) |
111 |
109 110
|
syl |
|- ( ph -> ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) \ { ( 2nd ` T ) } ) = ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) |
112 |
84 103
|
ltnled |
|- ( ph -> ( ( ( 2nd ` T ) + 1 ) < ( ( ( 2nd ` T ) + 1 ) + 1 ) <-> -. ( ( ( 2nd ` T ) + 1 ) + 1 ) <_ ( ( 2nd ` T ) + 1 ) ) ) |
113 |
104 112
|
mpbid |
|- ( ph -> -. ( ( ( 2nd ` T ) + 1 ) + 1 ) <_ ( ( 2nd ` T ) + 1 ) ) |
114 |
|
elfzle1 |
|- ( ( ( 2nd ` T ) + 1 ) e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) -> ( ( ( 2nd ` T ) + 1 ) + 1 ) <_ ( ( 2nd ` T ) + 1 ) ) |
115 |
113 114
|
nsyl |
|- ( ph -> -. ( ( 2nd ` T ) + 1 ) e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) |
116 |
|
difsn |
|- ( -. ( ( 2nd ` T ) + 1 ) e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) -> ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) \ { ( ( 2nd ` T ) + 1 ) } ) = ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) |
117 |
115 116
|
syl |
|- ( ph -> ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) \ { ( ( 2nd ` T ) + 1 ) } ) = ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) |
118 |
111 117
|
ineq12d |
|- ( ph -> ( ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) \ { ( 2nd ` T ) } ) i^i ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) \ { ( ( 2nd ` T ) + 1 ) } ) ) = ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) i^i ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) ) |
119 |
95
|
difeq2i |
|- ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) = ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) \ ( { ( 2nd ` T ) } u. { ( ( 2nd ` T ) + 1 ) } ) ) |
120 |
|
difundi |
|- ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) \ ( { ( 2nd ` T ) } u. { ( ( 2nd ` T ) + 1 ) } ) ) = ( ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) \ { ( 2nd ` T ) } ) i^i ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) \ { ( ( 2nd ` T ) + 1 ) } ) ) |
121 |
119 120
|
eqtr2i |
|- ( ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) \ { ( 2nd ` T ) } ) i^i ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) \ { ( ( 2nd ` T ) + 1 ) } ) ) = ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) |
122 |
|
inidm |
|- ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) i^i ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) = ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) |
123 |
118 121 122
|
3eqtr3g |
|- ( ph -> ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) = ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) |
124 |
101 123
|
uneq12d |
|- ( ph -> ( ( ( 1 ... ( ( 2nd ` T ) + 1 ) ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) u. ( ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) = ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) u. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) ) |
125 |
47 124
|
syl5eq |
|- ( ph -> ( ( ( 1 ... ( ( 2nd ` T ) + 1 ) ) u. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) = ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) u. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) ) |
126 |
46 125
|
eqtrd |
|- ( ph -> ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) = ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) u. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) ) |
127 |
126
|
eleq2d |
|- ( ph -> ( k e. ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) <-> k e. ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) u. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) ) ) |
128 |
|
elun |
|- ( k e. ( ( 1 ... ( ( 2nd ` T ) - 1 ) ) u. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) <-> ( k e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) \/ k e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) ) |
129 |
127 128
|
bitrdi |
|- ( ph -> ( k e. ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) <-> ( k e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) \/ k e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) ) ) |
130 |
129
|
biimpa |
|- ( ( ph /\ k e. ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) -> ( k e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) \/ k e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) ) |
131 |
|
fveq2 |
|- ( t = T -> ( 2nd ` t ) = ( 2nd ` T ) ) |
132 |
131
|
breq2d |
|- ( t = T -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` T ) ) ) |
133 |
132
|
ifbid |
|- ( t = T -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) ) |
134 |
133
|
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 } ) ) ) ) |
135 |
|
2fveq3 |
|- ( t = T -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` T ) ) ) |
136 |
|
2fveq3 |
|- ( t = T -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` T ) ) ) |
137 |
136
|
imaeq1d |
|- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) ) |
138 |
137
|
xpeq1d |
|- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) ) |
139 |
136
|
imaeq1d |
|- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) ) |
140 |
139
|
xpeq1d |
|- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) |
141 |
138 140
|
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 } ) ) ) |
142 |
135 141
|
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 } ) ) ) ) |
143 |
142
|
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 } ) ) ) ) |
144 |
134 143
|
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 } ) ) ) ) |
145 |
144
|
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 } ) ) ) ) ) |
146 |
145
|
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 } ) ) ) ) ) ) |
147 |
146 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 } ) ) ) ) ) ) |
148 |
147
|
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 } ) ) ) ) ) |
149 |
3 148
|
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 } ) ) ) ) ) |
150 |
|
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 ) ) ) |
151 |
26 150
|
syl |
|- ( ph -> ( 1st ` ( 1st ` T ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) ) |
152 |
|
elmapi |
|- ( ( 1st ` ( 1st ` T ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0 ..^ K ) ) |
153 |
151 152
|
syl |
|- ( ph -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0 ..^ K ) ) |
154 |
|
elfzoelz |
|- ( n e. ( 0 ..^ K ) -> n e. ZZ ) |
155 |
154
|
ssriv |
|- ( 0 ..^ K ) C_ ZZ |
156 |
|
fss |
|- ( ( ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0 ..^ K ) /\ ( 0 ..^ K ) C_ ZZ ) -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ZZ ) |
157 |
153 155 156
|
sylancl |
|- ( ph -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ZZ ) |
158 |
1 149 157 32 4
|
poimirlem1 |
|- ( ph -> -. E* n e. ( 1 ... N ) ( ( F ` ( ( 2nd ` T ) - 1 ) ) ` n ) =/= ( ( F ` ( 2nd ` T ) ) ` n ) ) |
159 |
1
|
adantr |
|- ( ( ph /\ ( 2nd ` U ) =/= ( 2nd ` T ) ) -> N e. NN ) |
160 |
|
fveq2 |
|- ( t = U -> ( 2nd ` t ) = ( 2nd ` U ) ) |
161 |
160
|
breq2d |
|- ( t = U -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` U ) ) ) |
162 |
161
|
ifbid |
|- ( t = U -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` U ) , y , ( y + 1 ) ) ) |
163 |
162
|
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 } ) ) ) ) |
164 |
|
2fveq3 |
|- ( t = U -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` U ) ) ) |
165 |
|
2fveq3 |
|- ( t = U -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` U ) ) ) |
166 |
165
|
imaeq1d |
|- ( t = U -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) ) |
167 |
166
|
xpeq1d |
|- ( t = U -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( 1 ... j ) ) X. { 1 } ) ) |
168 |
165
|
imaeq1d |
|- ( t = U -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) ) |
169 |
168
|
xpeq1d |
|- ( t = U -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` U ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) |
170 |
167 169
|
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 } ) ) ) |
171 |
164 170
|
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 } ) ) ) ) |
172 |
171
|
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 } ) ) ) ) |
173 |
163 172
|
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 } ) ) ) ) |
174 |
173
|
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 } ) ) ) ) ) |
175 |
174
|
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 } ) ) ) ) ) ) |
176 |
175 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 } ) ) ) ) ) ) |
177 |
176
|
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 } ) ) ) ) ) |
178 |
5 177
|
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 } ) ) ) ) ) |
179 |
178
|
adantr |
|- ( ( ph /\ ( 2nd ` U ) =/= ( 2nd ` T ) ) -> 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 } ) ) ) ) ) |
180 |
|
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 ) ) ) |
181 |
10 180
|
syl |
|- ( ph -> ( 1st ` ( 1st ` U ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) ) |
182 |
|
elmapi |
|- ( ( 1st ` ( 1st ` U ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` U ) ) : ( 1 ... N ) --> ( 0 ..^ K ) ) |
183 |
181 182
|
syl |
|- ( ph -> ( 1st ` ( 1st ` U ) ) : ( 1 ... N ) --> ( 0 ..^ K ) ) |
184 |
|
fss |
|- ( ( ( 1st ` ( 1st ` U ) ) : ( 1 ... N ) --> ( 0 ..^ K ) /\ ( 0 ..^ K ) C_ ZZ ) -> ( 1st ` ( 1st ` U ) ) : ( 1 ... N ) --> ZZ ) |
185 |
183 155 184
|
sylancl |
|- ( ph -> ( 1st ` ( 1st ` U ) ) : ( 1 ... N ) --> ZZ ) |
186 |
185
|
adantr |
|- ( ( ph /\ ( 2nd ` U ) =/= ( 2nd ` T ) ) -> ( 1st ` ( 1st ` U ) ) : ( 1 ... N ) --> ZZ ) |
187 |
16
|
adantr |
|- ( ( ph /\ ( 2nd ` U ) =/= ( 2nd ` T ) ) -> ( 2nd ` ( 1st ` U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
188 |
4
|
adantr |
|- ( ( ph /\ ( 2nd ` U ) =/= ( 2nd ` T ) ) -> ( 2nd ` T ) e. ( 1 ... ( N - 1 ) ) ) |
189 |
|
xp2nd |
|- ( U e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( 2nd ` U ) e. ( 0 ... N ) ) |
190 |
8 189
|
syl |
|- ( ph -> ( 2nd ` U ) e. ( 0 ... N ) ) |
191 |
|
eldifsn |
|- ( ( 2nd ` U ) e. ( ( 0 ... N ) \ { ( 2nd ` T ) } ) <-> ( ( 2nd ` U ) e. ( 0 ... N ) /\ ( 2nd ` U ) =/= ( 2nd ` T ) ) ) |
192 |
191
|
biimpri |
|- ( ( ( 2nd ` U ) e. ( 0 ... N ) /\ ( 2nd ` U ) =/= ( 2nd ` T ) ) -> ( 2nd ` U ) e. ( ( 0 ... N ) \ { ( 2nd ` T ) } ) ) |
193 |
190 192
|
sylan |
|- ( ( ph /\ ( 2nd ` U ) =/= ( 2nd ` T ) ) -> ( 2nd ` U ) e. ( ( 0 ... N ) \ { ( 2nd ` T ) } ) ) |
194 |
159 179 186 187 188 193
|
poimirlem2 |
|- ( ( ph /\ ( 2nd ` U ) =/= ( 2nd ` T ) ) -> E* n e. ( 1 ... N ) ( ( F ` ( ( 2nd ` T ) - 1 ) ) ` n ) =/= ( ( F ` ( 2nd ` T ) ) ` n ) ) |
195 |
194
|
ex |
|- ( ph -> ( ( 2nd ` U ) =/= ( 2nd ` T ) -> E* n e. ( 1 ... N ) ( ( F ` ( ( 2nd ` T ) - 1 ) ) ` n ) =/= ( ( F ` ( 2nd ` T ) ) ` n ) ) ) |
196 |
195
|
necon1bd |
|- ( ph -> ( -. E* n e. ( 1 ... N ) ( ( F ` ( ( 2nd ` T ) - 1 ) ) ` n ) =/= ( ( F ` ( 2nd ` T ) ) ` n ) -> ( 2nd ` U ) = ( 2nd ` T ) ) ) |
197 |
158 196
|
mpd |
|- ( ph -> ( 2nd ` U ) = ( 2nd ` T ) ) |
198 |
197
|
oveq1d |
|- ( ph -> ( ( 2nd ` U ) - 1 ) = ( ( 2nd ` T ) - 1 ) ) |
199 |
198
|
oveq2d |
|- ( ph -> ( 1 ... ( ( 2nd ` U ) - 1 ) ) = ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) |
200 |
199
|
eleq2d |
|- ( ph -> ( k e. ( 1 ... ( ( 2nd ` U ) - 1 ) ) <-> k e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) ) |
201 |
200
|
biimpar |
|- ( ( ph /\ k e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) -> k e. ( 1 ... ( ( 2nd ` U ) - 1 ) ) ) |
202 |
1
|
adantr |
|- ( ( ph /\ k e. ( 1 ... ( ( 2nd ` U ) - 1 ) ) ) -> N e. NN ) |
203 |
5
|
adantr |
|- ( ( ph /\ k e. ( 1 ... ( ( 2nd ` U ) - 1 ) ) ) -> U e. S ) |
204 |
197 4
|
eqeltrd |
|- ( ph -> ( 2nd ` U ) e. ( 1 ... ( N - 1 ) ) ) |
205 |
204
|
adantr |
|- ( ( ph /\ k e. ( 1 ... ( ( 2nd ` U ) - 1 ) ) ) -> ( 2nd ` U ) e. ( 1 ... ( N - 1 ) ) ) |
206 |
|
simpr |
|- ( ( ph /\ k e. ( 1 ... ( ( 2nd ` U ) - 1 ) ) ) -> k e. ( 1 ... ( ( 2nd ` U ) - 1 ) ) ) |
207 |
202 2 203 205 206
|
poimirlem6 |
|- ( ( ph /\ k e. ( 1 ... ( ( 2nd ` U ) - 1 ) ) ) -> ( iota_ n e. ( 1 ... N ) ( ( F ` ( k - 1 ) ) ` n ) =/= ( ( F ` k ) ` n ) ) = ( ( 2nd ` ( 1st ` U ) ) ` k ) ) |
208 |
201 207
|
syldan |
|- ( ( ph /\ k e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) -> ( iota_ n e. ( 1 ... N ) ( ( F ` ( k - 1 ) ) ` n ) =/= ( ( F ` k ) ` n ) ) = ( ( 2nd ` ( 1st ` U ) ) ` k ) ) |
209 |
1
|
adantr |
|- ( ( ph /\ k e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) -> N e. NN ) |
210 |
3
|
adantr |
|- ( ( ph /\ k e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) -> T e. S ) |
211 |
4
|
adantr |
|- ( ( ph /\ k e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) -> ( 2nd ` T ) e. ( 1 ... ( N - 1 ) ) ) |
212 |
|
simpr |
|- ( ( ph /\ k e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) -> k e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) |
213 |
209 2 210 211 212
|
poimirlem6 |
|- ( ( ph /\ k e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) -> ( iota_ n e. ( 1 ... N ) ( ( F ` ( k - 1 ) ) ` n ) =/= ( ( F ` k ) ` n ) ) = ( ( 2nd ` ( 1st ` T ) ) ` k ) ) |
214 |
208 213
|
eqtr3d |
|- ( ( ph /\ k e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) ) -> ( ( 2nd ` ( 1st ` U ) ) ` k ) = ( ( 2nd ` ( 1st ` T ) ) ` k ) ) |
215 |
197
|
oveq1d |
|- ( ph -> ( ( 2nd ` U ) + 1 ) = ( ( 2nd ` T ) + 1 ) ) |
216 |
215
|
oveq1d |
|- ( ph -> ( ( ( 2nd ` U ) + 1 ) + 1 ) = ( ( ( 2nd ` T ) + 1 ) + 1 ) ) |
217 |
216
|
oveq1d |
|- ( ph -> ( ( ( ( 2nd ` U ) + 1 ) + 1 ) ... N ) = ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) |
218 |
217
|
eleq2d |
|- ( ph -> ( k e. ( ( ( ( 2nd ` U ) + 1 ) + 1 ) ... N ) <-> k e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) ) |
219 |
218
|
biimpar |
|- ( ( ph /\ k e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) -> k e. ( ( ( ( 2nd ` U ) + 1 ) + 1 ) ... N ) ) |
220 |
1
|
adantr |
|- ( ( ph /\ k e. ( ( ( ( 2nd ` U ) + 1 ) + 1 ) ... N ) ) -> N e. NN ) |
221 |
5
|
adantr |
|- ( ( ph /\ k e. ( ( ( ( 2nd ` U ) + 1 ) + 1 ) ... N ) ) -> U e. S ) |
222 |
204
|
adantr |
|- ( ( ph /\ k e. ( ( ( ( 2nd ` U ) + 1 ) + 1 ) ... N ) ) -> ( 2nd ` U ) e. ( 1 ... ( N - 1 ) ) ) |
223 |
|
simpr |
|- ( ( ph /\ k e. ( ( ( ( 2nd ` U ) + 1 ) + 1 ) ... N ) ) -> k e. ( ( ( ( 2nd ` U ) + 1 ) + 1 ) ... N ) ) |
224 |
220 2 221 222 223
|
poimirlem7 |
|- ( ( ph /\ k e. ( ( ( ( 2nd ` U ) + 1 ) + 1 ) ... N ) ) -> ( iota_ n e. ( 1 ... N ) ( ( F ` ( k - 2 ) ) ` n ) =/= ( ( F ` ( k - 1 ) ) ` n ) ) = ( ( 2nd ` ( 1st ` U ) ) ` k ) ) |
225 |
219 224
|
syldan |
|- ( ( ph /\ k e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) -> ( iota_ n e. ( 1 ... N ) ( ( F ` ( k - 2 ) ) ` n ) =/= ( ( F ` ( k - 1 ) ) ` n ) ) = ( ( 2nd ` ( 1st ` U ) ) ` k ) ) |
226 |
1
|
adantr |
|- ( ( ph /\ k e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) -> N e. NN ) |
227 |
3
|
adantr |
|- ( ( ph /\ k e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) -> T e. S ) |
228 |
4
|
adantr |
|- ( ( ph /\ k e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) -> ( 2nd ` T ) e. ( 1 ... ( N - 1 ) ) ) |
229 |
|
simpr |
|- ( ( ph /\ k e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) -> k e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) |
230 |
226 2 227 228 229
|
poimirlem7 |
|- ( ( ph /\ k e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) -> ( iota_ n e. ( 1 ... N ) ( ( F ` ( k - 2 ) ) ` n ) =/= ( ( F ` ( k - 1 ) ) ` n ) ) = ( ( 2nd ` ( 1st ` T ) ) ` k ) ) |
231 |
225 230
|
eqtr3d |
|- ( ( ph /\ k e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) -> ( ( 2nd ` ( 1st ` U ) ) ` k ) = ( ( 2nd ` ( 1st ` T ) ) ` k ) ) |
232 |
214 231
|
jaodan |
|- ( ( ph /\ ( k e. ( 1 ... ( ( 2nd ` T ) - 1 ) ) \/ k e. ( ( ( ( 2nd ` T ) + 1 ) + 1 ) ... N ) ) ) -> ( ( 2nd ` ( 1st ` U ) ) ` k ) = ( ( 2nd ` ( 1st ` T ) ) ` k ) ) |
233 |
130 232
|
syldan |
|- ( ( ph /\ k e. ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) -> ( ( 2nd ` ( 1st ` U ) ) ` k ) = ( ( 2nd ` ( 1st ` T ) ) ` k ) ) |
234 |
|
fvres |
|- ( k e. ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) -> ( ( ( 2nd ` ( 1st ` U ) ) |` ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) ` k ) = ( ( 2nd ` ( 1st ` U ) ) ` k ) ) |
235 |
234
|
adantl |
|- ( ( ph /\ k e. ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) -> ( ( ( 2nd ` ( 1st ` U ) ) |` ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) ` k ) = ( ( 2nd ` ( 1st ` U ) ) ` k ) ) |
236 |
|
fvres |
|- ( k e. ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) -> ( ( ( 2nd ` ( 1st ` T ) ) |` ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) ` k ) = ( ( 2nd ` ( 1st ` T ) ) ` k ) ) |
237 |
236
|
adantl |
|- ( ( ph /\ k e. ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) -> ( ( ( 2nd ` ( 1st ` T ) ) |` ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) ` k ) = ( ( 2nd ` ( 1st ` T ) ) ` k ) ) |
238 |
233 235 237
|
3eqtr4d |
|- ( ( ph /\ k e. ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) -> ( ( ( 2nd ` ( 1st ` U ) ) |` ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) ` k ) = ( ( ( 2nd ` ( 1st ` T ) ) |` ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) ` k ) ) |
239 |
21 36 238
|
eqfnfvd |
|- ( ph -> ( ( 2nd ` ( 1st ` U ) ) |` ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) = ( ( 2nd ` ( 1st ` T ) ) |` ( ( 1 ... N ) \ { ( 2nd ` T ) , ( ( 2nd ` T ) + 1 ) } ) ) ) |