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 |
|
poimirlem22.2 |
|- ( ph -> T e. S ) |
5 |
|
poimirlem18.3 |
|- ( ( ph /\ n e. ( 1 ... N ) ) -> E. p e. ran F ( p ` n ) =/= K ) |
6 |
|
poimirlem18.4 |
|- ( ph -> ( 2nd ` T ) = 0 ) |
7 |
1 2 3 4 5 6
|
poimirlem17 |
|- ( ph -> E. z e. S z =/= T ) |
8 |
6
|
adantr |
|- ( ( ph /\ z e. S ) -> ( 2nd ` T ) = 0 ) |
9 |
|
0nnn |
|- -. 0 e. NN |
10 |
|
elfznn |
|- ( 0 e. ( 1 ... ( N - 1 ) ) -> 0 e. NN ) |
11 |
9 10
|
mto |
|- -. 0 e. ( 1 ... ( N - 1 ) ) |
12 |
|
eleq1 |
|- ( ( 2nd ` z ) = 0 -> ( ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) <-> 0 e. ( 1 ... ( N - 1 ) ) ) ) |
13 |
11 12
|
mtbiri |
|- ( ( 2nd ` z ) = 0 -> -. ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) |
14 |
13
|
necon2ai |
|- ( ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) -> ( 2nd ` z ) =/= 0 ) |
15 |
1
|
ad2antrr |
|- ( ( ( ph /\ z e. S ) /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) -> N e. NN ) |
16 |
|
fveq2 |
|- ( t = z -> ( 2nd ` t ) = ( 2nd ` z ) ) |
17 |
16
|
breq2d |
|- ( t = z -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` z ) ) ) |
18 |
17
|
ifbid |
|- ( t = z -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` z ) , y , ( y + 1 ) ) ) |
19 |
18
|
csbeq1d |
|- ( t = z -> [_ 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 ` z ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
20 |
|
2fveq3 |
|- ( t = z -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` z ) ) ) |
21 |
|
2fveq3 |
|- ( t = z -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` z ) ) ) |
22 |
21
|
imaeq1d |
|- ( t = z -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` z ) ) " ( 1 ... j ) ) ) |
23 |
22
|
xpeq1d |
|- ( t = z -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` z ) ) " ( 1 ... j ) ) X. { 1 } ) ) |
24 |
21
|
imaeq1d |
|- ( t = z -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` z ) ) " ( ( j + 1 ) ... N ) ) ) |
25 |
24
|
xpeq1d |
|- ( t = z -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` z ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) |
26 |
23 25
|
uneq12d |
|- ( t = z -> ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) = ( ( ( ( 2nd ` ( 1st ` z ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` z ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) |
27 |
20 26
|
oveq12d |
|- ( t = z -> ( ( 1st ` ( 1st ` t ) ) oF + ( ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) = ( ( 1st ` ( 1st ` z ) ) oF + ( ( ( ( 2nd ` ( 1st ` z ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` z ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
28 |
27
|
csbeq2dv |
|- ( t = z -> [_ if ( y < ( 2nd ` z ) , 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 ` z ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` z ) ) oF + ( ( ( ( 2nd ` ( 1st ` z ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` z ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
29 |
19 28
|
eqtrd |
|- ( t = z -> [_ 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 ` z ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` z ) ) oF + ( ( ( ( 2nd ` ( 1st ` z ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` z ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) |
30 |
29
|
mpteq2dv |
|- ( t = z -> ( 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 ` z ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` z ) ) oF + ( ( ( ( 2nd ` ( 1st ` z ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` z ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) |
31 |
30
|
eqeq2d |
|- ( t = z -> ( 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 ` z ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` z ) ) oF + ( ( ( ( 2nd ` ( 1st ` z ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` z ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) ) |
32 |
31 2
|
elrab2 |
|- ( z e. S <-> ( z 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 ` z ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` z ) ) oF + ( ( ( ( 2nd ` ( 1st ` z ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` z ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) ) |
33 |
32
|
simprbi |
|- ( z e. S -> F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` z ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` z ) ) oF + ( ( ( ( 2nd ` ( 1st ` z ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` z ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) |
34 |
33
|
ad2antlr |
|- ( ( ( ph /\ z e. S ) /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) -> F = ( y e. ( 0 ... ( N - 1 ) ) |-> [_ if ( y < ( 2nd ` z ) , y , ( y + 1 ) ) / j ]_ ( ( 1st ` ( 1st ` z ) ) oF + ( ( ( ( 2nd ` ( 1st ` z ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` z ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) ) ) ) |
35 |
|
elrabi |
|- ( z 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 } ) ) ) ) } -> z e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) |
36 |
35 2
|
eleq2s |
|- ( z e. S -> z e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) |
37 |
|
xp1st |
|- ( z e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( 1st ` z ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) |
38 |
36 37
|
syl |
|- ( z e. S -> ( 1st ` z ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) |
39 |
|
xp1st |
|- ( ( 1st ` z ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 1st ` ( 1st ` z ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) ) |
40 |
38 39
|
syl |
|- ( z e. S -> ( 1st ` ( 1st ` z ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) ) |
41 |
|
elmapi |
|- ( ( 1st ` ( 1st ` z ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` z ) ) : ( 1 ... N ) --> ( 0 ..^ K ) ) |
42 |
40 41
|
syl |
|- ( z e. S -> ( 1st ` ( 1st ` z ) ) : ( 1 ... N ) --> ( 0 ..^ K ) ) |
43 |
|
elfzoelz |
|- ( n e. ( 0 ..^ K ) -> n e. ZZ ) |
44 |
43
|
ssriv |
|- ( 0 ..^ K ) C_ ZZ |
45 |
|
fss |
|- ( ( ( 1st ` ( 1st ` z ) ) : ( 1 ... N ) --> ( 0 ..^ K ) /\ ( 0 ..^ K ) C_ ZZ ) -> ( 1st ` ( 1st ` z ) ) : ( 1 ... N ) --> ZZ ) |
46 |
42 44 45
|
sylancl |
|- ( z e. S -> ( 1st ` ( 1st ` z ) ) : ( 1 ... N ) --> ZZ ) |
47 |
46
|
ad2antlr |
|- ( ( ( ph /\ z e. S ) /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) -> ( 1st ` ( 1st ` z ) ) : ( 1 ... N ) --> ZZ ) |
48 |
|
xp2nd |
|- ( ( 1st ` z ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) -> ( 2nd ` ( 1st ` z ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) |
49 |
38 48
|
syl |
|- ( z e. S -> ( 2nd ` ( 1st ` z ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) |
50 |
|
fvex |
|- ( 2nd ` ( 1st ` z ) ) e. _V |
51 |
|
f1oeq1 |
|- ( f = ( 2nd ` ( 1st ` z ) ) -> ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( 2nd ` ( 1st ` z ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) ) |
52 |
50 51
|
elab |
|- ( ( 2nd ` ( 1st ` z ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } <-> ( 2nd ` ( 1st ` z ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
53 |
49 52
|
sylib |
|- ( z e. S -> ( 2nd ` ( 1st ` z ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
54 |
53
|
ad2antlr |
|- ( ( ( ph /\ z e. S ) /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) -> ( 2nd ` ( 1st ` z ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
55 |
|
simpr |
|- ( ( ( ph /\ z e. S ) /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) -> ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) |
56 |
15 34 47 54 55
|
poimirlem1 |
|- ( ( ( ph /\ z e. S ) /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) -> -. E* n e. ( 1 ... N ) ( ( F ` ( ( 2nd ` z ) - 1 ) ) ` n ) =/= ( ( F ` ( 2nd ` z ) ) ` n ) ) |
57 |
1
|
ad2antrr |
|- ( ( ( ph /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) /\ ( 2nd ` T ) =/= ( 2nd ` z ) ) -> N e. NN ) |
58 |
|
fveq2 |
|- ( t = T -> ( 2nd ` t ) = ( 2nd ` T ) ) |
59 |
58
|
breq2d |
|- ( t = T -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` T ) ) ) |
60 |
59
|
ifbid |
|- ( t = T -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) ) |
61 |
60
|
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 } ) ) ) ) |
62 |
|
2fveq3 |
|- ( t = T -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` T ) ) ) |
63 |
|
2fveq3 |
|- ( t = T -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` T ) ) ) |
64 |
63
|
imaeq1d |
|- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) ) |
65 |
64
|
xpeq1d |
|- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) ) |
66 |
63
|
imaeq1d |
|- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) ) |
67 |
66
|
xpeq1d |
|- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) |
68 |
65 67
|
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 } ) ) ) |
69 |
62 68
|
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 } ) ) ) ) |
70 |
69
|
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 } ) ) ) ) |
71 |
61 70
|
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 } ) ) ) ) |
72 |
71
|
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 } ) ) ) ) ) |
73 |
72
|
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 } ) ) ) ) ) ) |
74 |
73 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 } ) ) ) ) ) ) |
75 |
74
|
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 } ) ) ) ) ) |
76 |
4 75
|
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 } ) ) ) ) ) |
77 |
76
|
ad2antrr |
|- ( ( ( ph /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) /\ ( 2nd ` T ) =/= ( 2nd ` z ) ) -> 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 } ) ) ) ) ) |
78 |
|
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 ) ) ) |
79 |
78 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 ) ) ) |
80 |
4 79
|
syl |
|- ( ph -> T e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) ) |
81 |
|
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 ) } ) ) |
82 |
80 81
|
syl |
|- ( ph -> ( 1st ` T ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) ) |
83 |
|
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 ) ) ) |
84 |
82 83
|
syl |
|- ( ph -> ( 1st ` ( 1st ` T ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) ) |
85 |
|
elmapi |
|- ( ( 1st ` ( 1st ` T ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0 ..^ K ) ) |
86 |
84 85
|
syl |
|- ( ph -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0 ..^ K ) ) |
87 |
|
fss |
|- ( ( ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0 ..^ K ) /\ ( 0 ..^ K ) C_ ZZ ) -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ZZ ) |
88 |
86 44 87
|
sylancl |
|- ( ph -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ZZ ) |
89 |
88
|
ad2antrr |
|- ( ( ( ph /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) /\ ( 2nd ` T ) =/= ( 2nd ` z ) ) -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ZZ ) |
90 |
|
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 ) } ) |
91 |
82 90
|
syl |
|- ( ph -> ( 2nd ` ( 1st ` T ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) |
92 |
|
fvex |
|- ( 2nd ` ( 1st ` T ) ) e. _V |
93 |
|
f1oeq1 |
|- ( f = ( 2nd ` ( 1st ` T ) ) -> ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) ) |
94 |
92 93
|
elab |
|- ( ( 2nd ` ( 1st ` T ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } <-> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
95 |
91 94
|
sylib |
|- ( ph -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
96 |
95
|
ad2antrr |
|- ( ( ( ph /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) /\ ( 2nd ` T ) =/= ( 2nd ` z ) ) -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) |
97 |
|
simplr |
|- ( ( ( ph /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) /\ ( 2nd ` T ) =/= ( 2nd ` z ) ) -> ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) |
98 |
|
xp2nd |
|- ( T e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( 2nd ` T ) e. ( 0 ... N ) ) |
99 |
80 98
|
syl |
|- ( ph -> ( 2nd ` T ) e. ( 0 ... N ) ) |
100 |
99
|
adantr |
|- ( ( ph /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) -> ( 2nd ` T ) e. ( 0 ... N ) ) |
101 |
|
eldifsn |
|- ( ( 2nd ` T ) e. ( ( 0 ... N ) \ { ( 2nd ` z ) } ) <-> ( ( 2nd ` T ) e. ( 0 ... N ) /\ ( 2nd ` T ) =/= ( 2nd ` z ) ) ) |
102 |
101
|
biimpri |
|- ( ( ( 2nd ` T ) e. ( 0 ... N ) /\ ( 2nd ` T ) =/= ( 2nd ` z ) ) -> ( 2nd ` T ) e. ( ( 0 ... N ) \ { ( 2nd ` z ) } ) ) |
103 |
100 102
|
sylan |
|- ( ( ( ph /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) /\ ( 2nd ` T ) =/= ( 2nd ` z ) ) -> ( 2nd ` T ) e. ( ( 0 ... N ) \ { ( 2nd ` z ) } ) ) |
104 |
57 77 89 96 97 103
|
poimirlem2 |
|- ( ( ( ph /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) /\ ( 2nd ` T ) =/= ( 2nd ` z ) ) -> E* n e. ( 1 ... N ) ( ( F ` ( ( 2nd ` z ) - 1 ) ) ` n ) =/= ( ( F ` ( 2nd ` z ) ) ` n ) ) |
105 |
104
|
ex |
|- ( ( ph /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) -> ( ( 2nd ` T ) =/= ( 2nd ` z ) -> E* n e. ( 1 ... N ) ( ( F ` ( ( 2nd ` z ) - 1 ) ) ` n ) =/= ( ( F ` ( 2nd ` z ) ) ` n ) ) ) |
106 |
105
|
necon1bd |
|- ( ( ph /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) -> ( -. E* n e. ( 1 ... N ) ( ( F ` ( ( 2nd ` z ) - 1 ) ) ` n ) =/= ( ( F ` ( 2nd ` z ) ) ` n ) -> ( 2nd ` T ) = ( 2nd ` z ) ) ) |
107 |
106
|
adantlr |
|- ( ( ( ph /\ z e. S ) /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) -> ( -. E* n e. ( 1 ... N ) ( ( F ` ( ( 2nd ` z ) - 1 ) ) ` n ) =/= ( ( F ` ( 2nd ` z ) ) ` n ) -> ( 2nd ` T ) = ( 2nd ` z ) ) ) |
108 |
56 107
|
mpd |
|- ( ( ( ph /\ z e. S ) /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) -> ( 2nd ` T ) = ( 2nd ` z ) ) |
109 |
108
|
neeq1d |
|- ( ( ( ph /\ z e. S ) /\ ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) -> ( ( 2nd ` T ) =/= 0 <-> ( 2nd ` z ) =/= 0 ) ) |
110 |
109
|
exbiri |
|- ( ( ph /\ z e. S ) -> ( ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) -> ( ( 2nd ` z ) =/= 0 -> ( 2nd ` T ) =/= 0 ) ) ) |
111 |
14 110
|
mpdi |
|- ( ( ph /\ z e. S ) -> ( ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) -> ( 2nd ` T ) =/= 0 ) ) |
112 |
111
|
necon2bd |
|- ( ( ph /\ z e. S ) -> ( ( 2nd ` T ) = 0 -> -. ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) ) |
113 |
8 112
|
mpd |
|- ( ( ph /\ z e. S ) -> -. ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) ) |
114 |
|
xp2nd |
|- ( z e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) -> ( 2nd ` z ) e. ( 0 ... N ) ) |
115 |
36 114
|
syl |
|- ( z e. S -> ( 2nd ` z ) e. ( 0 ... N ) ) |
116 |
1
|
nncnd |
|- ( ph -> N e. CC ) |
117 |
|
npcan1 |
|- ( N e. CC -> ( ( N - 1 ) + 1 ) = N ) |
118 |
116 117
|
syl |
|- ( ph -> ( ( N - 1 ) + 1 ) = N ) |
119 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
120 |
1 119
|
eleqtrdi |
|- ( ph -> N e. ( ZZ>= ` 1 ) ) |
121 |
118 120
|
eqeltrd |
|- ( ph -> ( ( N - 1 ) + 1 ) e. ( ZZ>= ` 1 ) ) |
122 |
1
|
nnzd |
|- ( ph -> N e. ZZ ) |
123 |
|
peano2zm |
|- ( N e. ZZ -> ( N - 1 ) e. ZZ ) |
124 |
122 123
|
syl |
|- ( ph -> ( N - 1 ) e. ZZ ) |
125 |
|
uzid |
|- ( ( N - 1 ) e. ZZ -> ( N - 1 ) e. ( ZZ>= ` ( N - 1 ) ) ) |
126 |
|
peano2uz |
|- ( ( N - 1 ) e. ( ZZ>= ` ( N - 1 ) ) -> ( ( N - 1 ) + 1 ) e. ( ZZ>= ` ( N - 1 ) ) ) |
127 |
124 125 126
|
3syl |
|- ( ph -> ( ( N - 1 ) + 1 ) e. ( ZZ>= ` ( N - 1 ) ) ) |
128 |
118 127
|
eqeltrrd |
|- ( ph -> N e. ( ZZ>= ` ( N - 1 ) ) ) |
129 |
|
fzsplit2 |
|- ( ( ( ( N - 1 ) + 1 ) e. ( ZZ>= ` 1 ) /\ N e. ( ZZ>= ` ( N - 1 ) ) ) -> ( 1 ... N ) = ( ( 1 ... ( N - 1 ) ) u. ( ( ( N - 1 ) + 1 ) ... N ) ) ) |
130 |
121 128 129
|
syl2anc |
|- ( ph -> ( 1 ... N ) = ( ( 1 ... ( N - 1 ) ) u. ( ( ( N - 1 ) + 1 ) ... N ) ) ) |
131 |
118
|
oveq1d |
|- ( ph -> ( ( ( N - 1 ) + 1 ) ... N ) = ( N ... N ) ) |
132 |
|
fzsn |
|- ( N e. ZZ -> ( N ... N ) = { N } ) |
133 |
122 132
|
syl |
|- ( ph -> ( N ... N ) = { N } ) |
134 |
131 133
|
eqtrd |
|- ( ph -> ( ( ( N - 1 ) + 1 ) ... N ) = { N } ) |
135 |
134
|
uneq2d |
|- ( ph -> ( ( 1 ... ( N - 1 ) ) u. ( ( ( N - 1 ) + 1 ) ... N ) ) = ( ( 1 ... ( N - 1 ) ) u. { N } ) ) |
136 |
130 135
|
eqtrd |
|- ( ph -> ( 1 ... N ) = ( ( 1 ... ( N - 1 ) ) u. { N } ) ) |
137 |
136
|
eleq2d |
|- ( ph -> ( ( 2nd ` z ) e. ( 1 ... N ) <-> ( 2nd ` z ) e. ( ( 1 ... ( N - 1 ) ) u. { N } ) ) ) |
138 |
137
|
notbid |
|- ( ph -> ( -. ( 2nd ` z ) e. ( 1 ... N ) <-> -. ( 2nd ` z ) e. ( ( 1 ... ( N - 1 ) ) u. { N } ) ) ) |
139 |
|
ioran |
|- ( -. ( ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) \/ ( 2nd ` z ) = N ) <-> ( -. ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) /\ -. ( 2nd ` z ) = N ) ) |
140 |
|
elun |
|- ( ( 2nd ` z ) e. ( ( 1 ... ( N - 1 ) ) u. { N } ) <-> ( ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) \/ ( 2nd ` z ) e. { N } ) ) |
141 |
|
fvex |
|- ( 2nd ` z ) e. _V |
142 |
141
|
elsn |
|- ( ( 2nd ` z ) e. { N } <-> ( 2nd ` z ) = N ) |
143 |
142
|
orbi2i |
|- ( ( ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) \/ ( 2nd ` z ) e. { N } ) <-> ( ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) \/ ( 2nd ` z ) = N ) ) |
144 |
140 143
|
bitri |
|- ( ( 2nd ` z ) e. ( ( 1 ... ( N - 1 ) ) u. { N } ) <-> ( ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) \/ ( 2nd ` z ) = N ) ) |
145 |
139 144
|
xchnxbir |
|- ( -. ( 2nd ` z ) e. ( ( 1 ... ( N - 1 ) ) u. { N } ) <-> ( -. ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) /\ -. ( 2nd ` z ) = N ) ) |
146 |
138 145
|
bitrdi |
|- ( ph -> ( -. ( 2nd ` z ) e. ( 1 ... N ) <-> ( -. ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) /\ -. ( 2nd ` z ) = N ) ) ) |
147 |
146
|
anbi2d |
|- ( ph -> ( ( ( 2nd ` z ) e. ( 0 ... N ) /\ -. ( 2nd ` z ) e. ( 1 ... N ) ) <-> ( ( 2nd ` z ) e. ( 0 ... N ) /\ ( -. ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) /\ -. ( 2nd ` z ) = N ) ) ) ) |
148 |
1
|
nnnn0d |
|- ( ph -> N e. NN0 ) |
149 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
150 |
148 149
|
eleqtrdi |
|- ( ph -> N e. ( ZZ>= ` 0 ) ) |
151 |
|
fzpred |
|- ( N e. ( ZZ>= ` 0 ) -> ( 0 ... N ) = ( { 0 } u. ( ( 0 + 1 ) ... N ) ) ) |
152 |
150 151
|
syl |
|- ( ph -> ( 0 ... N ) = ( { 0 } u. ( ( 0 + 1 ) ... N ) ) ) |
153 |
152
|
difeq1d |
|- ( ph -> ( ( 0 ... N ) \ ( 1 ... N ) ) = ( ( { 0 } u. ( ( 0 + 1 ) ... N ) ) \ ( 1 ... N ) ) ) |
154 |
|
difun2 |
|- ( ( { 0 } u. ( 1 ... N ) ) \ ( 1 ... N ) ) = ( { 0 } \ ( 1 ... N ) ) |
155 |
|
0p1e1 |
|- ( 0 + 1 ) = 1 |
156 |
155
|
oveq1i |
|- ( ( 0 + 1 ) ... N ) = ( 1 ... N ) |
157 |
156
|
uneq2i |
|- ( { 0 } u. ( ( 0 + 1 ) ... N ) ) = ( { 0 } u. ( 1 ... N ) ) |
158 |
157
|
difeq1i |
|- ( ( { 0 } u. ( ( 0 + 1 ) ... N ) ) \ ( 1 ... N ) ) = ( ( { 0 } u. ( 1 ... N ) ) \ ( 1 ... N ) ) |
159 |
|
incom |
|- ( { 0 } i^i ( 1 ... N ) ) = ( ( 1 ... N ) i^i { 0 } ) |
160 |
|
elfznn |
|- ( 0 e. ( 1 ... N ) -> 0 e. NN ) |
161 |
9 160
|
mto |
|- -. 0 e. ( 1 ... N ) |
162 |
|
disjsn |
|- ( ( ( 1 ... N ) i^i { 0 } ) = (/) <-> -. 0 e. ( 1 ... N ) ) |
163 |
161 162
|
mpbir |
|- ( ( 1 ... N ) i^i { 0 } ) = (/) |
164 |
159 163
|
eqtri |
|- ( { 0 } i^i ( 1 ... N ) ) = (/) |
165 |
|
disj3 |
|- ( ( { 0 } i^i ( 1 ... N ) ) = (/) <-> { 0 } = ( { 0 } \ ( 1 ... N ) ) ) |
166 |
164 165
|
mpbi |
|- { 0 } = ( { 0 } \ ( 1 ... N ) ) |
167 |
154 158 166
|
3eqtr4i |
|- ( ( { 0 } u. ( ( 0 + 1 ) ... N ) ) \ ( 1 ... N ) ) = { 0 } |
168 |
153 167
|
eqtrdi |
|- ( ph -> ( ( 0 ... N ) \ ( 1 ... N ) ) = { 0 } ) |
169 |
168
|
eleq2d |
|- ( ph -> ( ( 2nd ` z ) e. ( ( 0 ... N ) \ ( 1 ... N ) ) <-> ( 2nd ` z ) e. { 0 } ) ) |
170 |
|
eldif |
|- ( ( 2nd ` z ) e. ( ( 0 ... N ) \ ( 1 ... N ) ) <-> ( ( 2nd ` z ) e. ( 0 ... N ) /\ -. ( 2nd ` z ) e. ( 1 ... N ) ) ) |
171 |
141
|
elsn |
|- ( ( 2nd ` z ) e. { 0 } <-> ( 2nd ` z ) = 0 ) |
172 |
169 170 171
|
3bitr3g |
|- ( ph -> ( ( ( 2nd ` z ) e. ( 0 ... N ) /\ -. ( 2nd ` z ) e. ( 1 ... N ) ) <-> ( 2nd ` z ) = 0 ) ) |
173 |
147 172
|
bitr3d |
|- ( ph -> ( ( ( 2nd ` z ) e. ( 0 ... N ) /\ ( -. ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) /\ -. ( 2nd ` z ) = N ) ) <-> ( 2nd ` z ) = 0 ) ) |
174 |
173
|
biimpd |
|- ( ph -> ( ( ( 2nd ` z ) e. ( 0 ... N ) /\ ( -. ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) /\ -. ( 2nd ` z ) = N ) ) -> ( 2nd ` z ) = 0 ) ) |
175 |
174
|
expdimp |
|- ( ( ph /\ ( 2nd ` z ) e. ( 0 ... N ) ) -> ( ( -. ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) /\ -. ( 2nd ` z ) = N ) -> ( 2nd ` z ) = 0 ) ) |
176 |
115 175
|
sylan2 |
|- ( ( ph /\ z e. S ) -> ( ( -. ( 2nd ` z ) e. ( 1 ... ( N - 1 ) ) /\ -. ( 2nd ` z ) = N ) -> ( 2nd ` z ) = 0 ) ) |
177 |
113 176
|
mpand |
|- ( ( ph /\ z e. S ) -> ( -. ( 2nd ` z ) = N -> ( 2nd ` z ) = 0 ) ) |
178 |
1 2 3
|
poimirlem13 |
|- ( ph -> E* z e. S ( 2nd ` z ) = 0 ) |
179 |
|
fveqeq2 |
|- ( z = s -> ( ( 2nd ` z ) = 0 <-> ( 2nd ` s ) = 0 ) ) |
180 |
179
|
rmo4 |
|- ( E* z e. S ( 2nd ` z ) = 0 <-> A. z e. S A. s e. S ( ( ( 2nd ` z ) = 0 /\ ( 2nd ` s ) = 0 ) -> z = s ) ) |
181 |
178 180
|
sylib |
|- ( ph -> A. z e. S A. s e. S ( ( ( 2nd ` z ) = 0 /\ ( 2nd ` s ) = 0 ) -> z = s ) ) |
182 |
181
|
r19.21bi |
|- ( ( ph /\ z e. S ) -> A. s e. S ( ( ( 2nd ` z ) = 0 /\ ( 2nd ` s ) = 0 ) -> z = s ) ) |
183 |
4
|
adantr |
|- ( ( ph /\ z e. S ) -> T e. S ) |
184 |
|
fveqeq2 |
|- ( s = T -> ( ( 2nd ` s ) = 0 <-> ( 2nd ` T ) = 0 ) ) |
185 |
184
|
anbi2d |
|- ( s = T -> ( ( ( 2nd ` z ) = 0 /\ ( 2nd ` s ) = 0 ) <-> ( ( 2nd ` z ) = 0 /\ ( 2nd ` T ) = 0 ) ) ) |
186 |
|
eqeq2 |
|- ( s = T -> ( z = s <-> z = T ) ) |
187 |
185 186
|
imbi12d |
|- ( s = T -> ( ( ( ( 2nd ` z ) = 0 /\ ( 2nd ` s ) = 0 ) -> z = s ) <-> ( ( ( 2nd ` z ) = 0 /\ ( 2nd ` T ) = 0 ) -> z = T ) ) ) |
188 |
187
|
rspccv |
|- ( A. s e. S ( ( ( 2nd ` z ) = 0 /\ ( 2nd ` s ) = 0 ) -> z = s ) -> ( T e. S -> ( ( ( 2nd ` z ) = 0 /\ ( 2nd ` T ) = 0 ) -> z = T ) ) ) |
189 |
182 183 188
|
sylc |
|- ( ( ph /\ z e. S ) -> ( ( ( 2nd ` z ) = 0 /\ ( 2nd ` T ) = 0 ) -> z = T ) ) |
190 |
8 189
|
mpan2d |
|- ( ( ph /\ z e. S ) -> ( ( 2nd ` z ) = 0 -> z = T ) ) |
191 |
177 190
|
syld |
|- ( ( ph /\ z e. S ) -> ( -. ( 2nd ` z ) = N -> z = T ) ) |
192 |
191
|
necon1ad |
|- ( ( ph /\ z e. S ) -> ( z =/= T -> ( 2nd ` z ) = N ) ) |
193 |
192
|
ralrimiva |
|- ( ph -> A. z e. S ( z =/= T -> ( 2nd ` z ) = N ) ) |
194 |
1 2 3
|
poimirlem14 |
|- ( ph -> E* z e. S ( 2nd ` z ) = N ) |
195 |
|
rmoim |
|- ( A. z e. S ( z =/= T -> ( 2nd ` z ) = N ) -> ( E* z e. S ( 2nd ` z ) = N -> E* z e. S z =/= T ) ) |
196 |
193 194 195
|
sylc |
|- ( ph -> E* z e. S z =/= T ) |
197 |
|
reu5 |
|- ( E! z e. S z =/= T <-> ( E. z e. S z =/= T /\ E* z e. S z =/= T ) ) |
198 |
7 196 197
|
sylanbrc |
|- ( ph -> E! z e. S z =/= T ) |