Step |
Hyp |
Ref |
Expression |
1 |
|
smat.s |
|- S = ( K ( subMat1 ` A ) L ) |
2 |
|
smat.m |
|- ( ph -> M e. NN ) |
3 |
|
smat.n |
|- ( ph -> N e. NN ) |
4 |
|
smat.k |
|- ( ph -> K e. ( 1 ... M ) ) |
5 |
|
smat.l |
|- ( ph -> L e. ( 1 ... N ) ) |
6 |
|
smat.a |
|- ( ph -> A e. ( B ^m ( ( 1 ... M ) X. ( 1 ... N ) ) ) ) |
7 |
|
elmapi |
|- ( A e. ( B ^m ( ( 1 ... M ) X. ( 1 ... N ) ) ) -> A : ( ( 1 ... M ) X. ( 1 ... N ) ) --> B ) |
8 |
|
ffun |
|- ( A : ( ( 1 ... M ) X. ( 1 ... N ) ) --> B -> Fun A ) |
9 |
6 7 8
|
3syl |
|- ( ph -> Fun A ) |
10 |
|
eqid |
|- ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) = ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) |
11 |
10
|
mpofun |
|- Fun ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) |
12 |
11
|
a1i |
|- ( ph -> Fun ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ) |
13 |
|
funco |
|- ( ( Fun A /\ Fun ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ) -> Fun ( A o. ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ) ) |
14 |
9 12 13
|
syl2anc |
|- ( ph -> Fun ( A o. ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ) ) |
15 |
|
fz1ssnn |
|- ( 1 ... M ) C_ NN |
16 |
15 4
|
sselid |
|- ( ph -> K e. NN ) |
17 |
|
fz1ssnn |
|- ( 1 ... N ) C_ NN |
18 |
17 5
|
sselid |
|- ( ph -> L e. NN ) |
19 |
|
smatfval |
|- ( ( K e. NN /\ L e. NN /\ A e. ( B ^m ( ( 1 ... M ) X. ( 1 ... N ) ) ) ) -> ( K ( subMat1 ` A ) L ) = ( A o. ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ) ) |
20 |
16 18 6 19
|
syl3anc |
|- ( ph -> ( K ( subMat1 ` A ) L ) = ( A o. ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ) ) |
21 |
1 20
|
syl5eq |
|- ( ph -> S = ( A o. ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ) ) |
22 |
21
|
funeqd |
|- ( ph -> ( Fun S <-> Fun ( A o. ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ) ) ) |
23 |
14 22
|
mpbird |
|- ( ph -> Fun S ) |
24 |
|
fdmrn |
|- ( Fun S <-> S : dom S --> ran S ) |
25 |
23 24
|
sylib |
|- ( ph -> S : dom S --> ran S ) |
26 |
21
|
dmeqd |
|- ( ph -> dom S = dom ( A o. ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ) ) |
27 |
|
dmco |
|- dom ( A o. ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ) = ( `' ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) " dom A ) |
28 |
|
fdm |
|- ( A : ( ( 1 ... M ) X. ( 1 ... N ) ) --> B -> dom A = ( ( 1 ... M ) X. ( 1 ... N ) ) ) |
29 |
6 7 28
|
3syl |
|- ( ph -> dom A = ( ( 1 ... M ) X. ( 1 ... N ) ) ) |
30 |
29
|
imaeq2d |
|- ( ph -> ( `' ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) " dom A ) = ( `' ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) " ( ( 1 ... M ) X. ( 1 ... N ) ) ) ) |
31 |
30
|
eleq2d |
|- ( ph -> ( x e. ( `' ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) " dom A ) <-> x e. ( `' ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) " ( ( 1 ... M ) X. ( 1 ... N ) ) ) ) ) |
32 |
|
opex |
|- <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. e. _V |
33 |
10 32
|
fnmpoi |
|- ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) Fn ( NN X. NN ) |
34 |
|
elpreima |
|- ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) Fn ( NN X. NN ) -> ( x e. ( `' ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) " ( ( 1 ... M ) X. ( 1 ... N ) ) ) <-> ( x e. ( NN X. NN ) /\ ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) e. ( ( 1 ... M ) X. ( 1 ... N ) ) ) ) ) |
35 |
33 34
|
ax-mp |
|- ( x e. ( `' ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) " ( ( 1 ... M ) X. ( 1 ... N ) ) ) <-> ( x e. ( NN X. NN ) /\ ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) e. ( ( 1 ... M ) X. ( 1 ... N ) ) ) ) |
36 |
35
|
a1i |
|- ( ph -> ( x e. ( `' ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) " ( ( 1 ... M ) X. ( 1 ... N ) ) ) <-> ( x e. ( NN X. NN ) /\ ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) e. ( ( 1 ... M ) X. ( 1 ... N ) ) ) ) ) |
37 |
|
simplr |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) |
38 |
37
|
fveq2d |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) = ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) ) |
39 |
|
df-ov |
|- ( ( 1st ` x ) ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ( 2nd ` x ) ) = ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` <. ( 1st ` x ) , ( 2nd ` x ) >. ) |
40 |
38 39
|
eqtr4di |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) = ( ( 1st ` x ) ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ( 2nd ` x ) ) ) |
41 |
|
breq1 |
|- ( i = ( 1st ` x ) -> ( i < K <-> ( 1st ` x ) < K ) ) |
42 |
|
id |
|- ( i = ( 1st ` x ) -> i = ( 1st ` x ) ) |
43 |
|
oveq1 |
|- ( i = ( 1st ` x ) -> ( i + 1 ) = ( ( 1st ` x ) + 1 ) ) |
44 |
41 42 43
|
ifbieq12d |
|- ( i = ( 1st ` x ) -> if ( i < K , i , ( i + 1 ) ) = if ( ( 1st ` x ) < K , ( 1st ` x ) , ( ( 1st ` x ) + 1 ) ) ) |
45 |
44
|
opeq1d |
|- ( i = ( 1st ` x ) -> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. = <. if ( ( 1st ` x ) < K , ( 1st ` x ) , ( ( 1st ` x ) + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) |
46 |
|
breq1 |
|- ( j = ( 2nd ` x ) -> ( j < L <-> ( 2nd ` x ) < L ) ) |
47 |
|
id |
|- ( j = ( 2nd ` x ) -> j = ( 2nd ` x ) ) |
48 |
|
oveq1 |
|- ( j = ( 2nd ` x ) -> ( j + 1 ) = ( ( 2nd ` x ) + 1 ) ) |
49 |
46 47 48
|
ifbieq12d |
|- ( j = ( 2nd ` x ) -> if ( j < L , j , ( j + 1 ) ) = if ( ( 2nd ` x ) < L , ( 2nd ` x ) , ( ( 2nd ` x ) + 1 ) ) ) |
50 |
49
|
opeq2d |
|- ( j = ( 2nd ` x ) -> <. if ( ( 1st ` x ) < K , ( 1st ` x ) , ( ( 1st ` x ) + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. = <. if ( ( 1st ` x ) < K , ( 1st ` x ) , ( ( 1st ` x ) + 1 ) ) , if ( ( 2nd ` x ) < L , ( 2nd ` x ) , ( ( 2nd ` x ) + 1 ) ) >. ) |
51 |
|
opex |
|- <. if ( ( 1st ` x ) < K , ( 1st ` x ) , ( ( 1st ` x ) + 1 ) ) , if ( ( 2nd ` x ) < L , ( 2nd ` x ) , ( ( 2nd ` x ) + 1 ) ) >. e. _V |
52 |
45 50 10 51
|
ovmpo |
|- ( ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) -> ( ( 1st ` x ) ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ( 2nd ` x ) ) = <. if ( ( 1st ` x ) < K , ( 1st ` x ) , ( ( 1st ` x ) + 1 ) ) , if ( ( 2nd ` x ) < L , ( 2nd ` x ) , ( ( 2nd ` x ) + 1 ) ) >. ) |
53 |
52
|
adantl |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( 1st ` x ) ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ( 2nd ` x ) ) = <. if ( ( 1st ` x ) < K , ( 1st ` x ) , ( ( 1st ` x ) + 1 ) ) , if ( ( 2nd ` x ) < L , ( 2nd ` x ) , ( ( 2nd ` x ) + 1 ) ) >. ) |
54 |
40 53
|
eqtrd |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) = <. if ( ( 1st ` x ) < K , ( 1st ` x ) , ( ( 1st ` x ) + 1 ) ) , if ( ( 2nd ` x ) < L , ( 2nd ` x ) , ( ( 2nd ` x ) + 1 ) ) >. ) |
55 |
54
|
eleq1d |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) e. ( ( 1 ... M ) X. ( 1 ... N ) ) <-> <. if ( ( 1st ` x ) < K , ( 1st ` x ) , ( ( 1st ` x ) + 1 ) ) , if ( ( 2nd ` x ) < L , ( 2nd ` x ) , ( ( 2nd ` x ) + 1 ) ) >. e. ( ( 1 ... M ) X. ( 1 ... N ) ) ) ) |
56 |
|
opelxp |
|- ( <. if ( ( 1st ` x ) < K , ( 1st ` x ) , ( ( 1st ` x ) + 1 ) ) , if ( ( 2nd ` x ) < L , ( 2nd ` x ) , ( ( 2nd ` x ) + 1 ) ) >. e. ( ( 1 ... M ) X. ( 1 ... N ) ) <-> ( if ( ( 1st ` x ) < K , ( 1st ` x ) , ( ( 1st ` x ) + 1 ) ) e. ( 1 ... M ) /\ if ( ( 2nd ` x ) < L , ( 2nd ` x ) , ( ( 2nd ` x ) + 1 ) ) e. ( 1 ... N ) ) ) |
57 |
55 56
|
bitrdi |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) e. ( ( 1 ... M ) X. ( 1 ... N ) ) <-> ( if ( ( 1st ` x ) < K , ( 1st ` x ) , ( ( 1st ` x ) + 1 ) ) e. ( 1 ... M ) /\ if ( ( 2nd ` x ) < L , ( 2nd ` x ) , ( ( 2nd ` x ) + 1 ) ) e. ( 1 ... N ) ) ) ) |
58 |
|
ifel |
|- ( if ( ( 1st ` x ) < K , ( 1st ` x ) , ( ( 1st ` x ) + 1 ) ) e. ( 1 ... M ) <-> ( ( ( 1st ` x ) < K /\ ( 1st ` x ) e. ( 1 ... M ) ) \/ ( -. ( 1st ` x ) < K /\ ( ( 1st ` x ) + 1 ) e. ( 1 ... M ) ) ) ) |
59 |
|
simplrl |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 1st ` x ) < K ) -> ( 1st ` x ) e. NN ) |
60 |
59
|
nnred |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 1st ` x ) < K ) -> ( 1st ` x ) e. RR ) |
61 |
16
|
nnred |
|- ( ph -> K e. RR ) |
62 |
61
|
ad3antrrr |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 1st ` x ) < K ) -> K e. RR ) |
63 |
2
|
nnred |
|- ( ph -> M e. RR ) |
64 |
63
|
ad3antrrr |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 1st ` x ) < K ) -> M e. RR ) |
65 |
|
simpr |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 1st ` x ) < K ) -> ( 1st ` x ) < K ) |
66 |
60 62 65
|
ltled |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 1st ` x ) < K ) -> ( 1st ` x ) <_ K ) |
67 |
|
elfzle2 |
|- ( K e. ( 1 ... M ) -> K <_ M ) |
68 |
4 67
|
syl |
|- ( ph -> K <_ M ) |
69 |
68
|
ad3antrrr |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 1st ` x ) < K ) -> K <_ M ) |
70 |
60 62 64 66 69
|
letrd |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 1st ` x ) < K ) -> ( 1st ` x ) <_ M ) |
71 |
59 70
|
jca |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 1st ` x ) < K ) -> ( ( 1st ` x ) e. NN /\ ( 1st ` x ) <_ M ) ) |
72 |
2
|
nnzd |
|- ( ph -> M e. ZZ ) |
73 |
|
fznn |
|- ( M e. ZZ -> ( ( 1st ` x ) e. ( 1 ... M ) <-> ( ( 1st ` x ) e. NN /\ ( 1st ` x ) <_ M ) ) ) |
74 |
72 73
|
syl |
|- ( ph -> ( ( 1st ` x ) e. ( 1 ... M ) <-> ( ( 1st ` x ) e. NN /\ ( 1st ` x ) <_ M ) ) ) |
75 |
74
|
ad3antrrr |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 1st ` x ) < K ) -> ( ( 1st ` x ) e. ( 1 ... M ) <-> ( ( 1st ` x ) e. NN /\ ( 1st ` x ) <_ M ) ) ) |
76 |
71 75
|
mpbird |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 1st ` x ) < K ) -> ( 1st ` x ) e. ( 1 ... M ) ) |
77 |
60 62 64 65 69
|
ltletrd |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 1st ` x ) < K ) -> ( 1st ` x ) < M ) |
78 |
2
|
ad3antrrr |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 1st ` x ) < K ) -> M e. NN ) |
79 |
|
nnltlem1 |
|- ( ( ( 1st ` x ) e. NN /\ M e. NN ) -> ( ( 1st ` x ) < M <-> ( 1st ` x ) <_ ( M - 1 ) ) ) |
80 |
59 78 79
|
syl2anc |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 1st ` x ) < K ) -> ( ( 1st ` x ) < M <-> ( 1st ` x ) <_ ( M - 1 ) ) ) |
81 |
77 80
|
mpbid |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 1st ` x ) < K ) -> ( 1st ` x ) <_ ( M - 1 ) ) |
82 |
76 81
|
2thd |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 1st ` x ) < K ) -> ( ( 1st ` x ) e. ( 1 ... M ) <-> ( 1st ` x ) <_ ( M - 1 ) ) ) |
83 |
82
|
pm5.32da |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( ( 1st ` x ) < K /\ ( 1st ` x ) e. ( 1 ... M ) ) <-> ( ( 1st ` x ) < K /\ ( 1st ` x ) <_ ( M - 1 ) ) ) ) |
84 |
|
fznn |
|- ( M e. ZZ -> ( ( ( 1st ` x ) + 1 ) e. ( 1 ... M ) <-> ( ( ( 1st ` x ) + 1 ) e. NN /\ ( ( 1st ` x ) + 1 ) <_ M ) ) ) |
85 |
72 84
|
syl |
|- ( ph -> ( ( ( 1st ` x ) + 1 ) e. ( 1 ... M ) <-> ( ( ( 1st ` x ) + 1 ) e. NN /\ ( ( 1st ` x ) + 1 ) <_ M ) ) ) |
86 |
85
|
ad2antrr |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( ( 1st ` x ) + 1 ) e. ( 1 ... M ) <-> ( ( ( 1st ` x ) + 1 ) e. NN /\ ( ( 1st ` x ) + 1 ) <_ M ) ) ) |
87 |
|
simprl |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( 1st ` x ) e. NN ) |
88 |
87
|
peano2nnd |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( 1st ` x ) + 1 ) e. NN ) |
89 |
88
|
biantrurd |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( ( 1st ` x ) + 1 ) <_ M <-> ( ( ( 1st ` x ) + 1 ) e. NN /\ ( ( 1st ` x ) + 1 ) <_ M ) ) ) |
90 |
87
|
nnzd |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( 1st ` x ) e. ZZ ) |
91 |
72
|
ad2antrr |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> M e. ZZ ) |
92 |
|
zltp1le |
|- ( ( ( 1st ` x ) e. ZZ /\ M e. ZZ ) -> ( ( 1st ` x ) < M <-> ( ( 1st ` x ) + 1 ) <_ M ) ) |
93 |
|
zltlem1 |
|- ( ( ( 1st ` x ) e. ZZ /\ M e. ZZ ) -> ( ( 1st ` x ) < M <-> ( 1st ` x ) <_ ( M - 1 ) ) ) |
94 |
92 93
|
bitr3d |
|- ( ( ( 1st ` x ) e. ZZ /\ M e. ZZ ) -> ( ( ( 1st ` x ) + 1 ) <_ M <-> ( 1st ` x ) <_ ( M - 1 ) ) ) |
95 |
90 91 94
|
syl2anc |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( ( 1st ` x ) + 1 ) <_ M <-> ( 1st ` x ) <_ ( M - 1 ) ) ) |
96 |
86 89 95
|
3bitr2d |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( ( 1st ` x ) + 1 ) e. ( 1 ... M ) <-> ( 1st ` x ) <_ ( M - 1 ) ) ) |
97 |
96
|
anbi2d |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( -. ( 1st ` x ) < K /\ ( ( 1st ` x ) + 1 ) e. ( 1 ... M ) ) <-> ( -. ( 1st ` x ) < K /\ ( 1st ` x ) <_ ( M - 1 ) ) ) ) |
98 |
83 97
|
orbi12d |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( ( ( 1st ` x ) < K /\ ( 1st ` x ) e. ( 1 ... M ) ) \/ ( -. ( 1st ` x ) < K /\ ( ( 1st ` x ) + 1 ) e. ( 1 ... M ) ) ) <-> ( ( ( 1st ` x ) < K /\ ( 1st ` x ) <_ ( M - 1 ) ) \/ ( -. ( 1st ` x ) < K /\ ( 1st ` x ) <_ ( M - 1 ) ) ) ) ) |
99 |
|
pm4.42 |
|- ( ( 1st ` x ) <_ ( M - 1 ) <-> ( ( ( 1st ` x ) <_ ( M - 1 ) /\ ( 1st ` x ) < K ) \/ ( ( 1st ` x ) <_ ( M - 1 ) /\ -. ( 1st ` x ) < K ) ) ) |
100 |
|
ancom |
|- ( ( ( 1st ` x ) <_ ( M - 1 ) /\ ( 1st ` x ) < K ) <-> ( ( 1st ` x ) < K /\ ( 1st ` x ) <_ ( M - 1 ) ) ) |
101 |
|
ancom |
|- ( ( ( 1st ` x ) <_ ( M - 1 ) /\ -. ( 1st ` x ) < K ) <-> ( -. ( 1st ` x ) < K /\ ( 1st ` x ) <_ ( M - 1 ) ) ) |
102 |
100 101
|
orbi12i |
|- ( ( ( ( 1st ` x ) <_ ( M - 1 ) /\ ( 1st ` x ) < K ) \/ ( ( 1st ` x ) <_ ( M - 1 ) /\ -. ( 1st ` x ) < K ) ) <-> ( ( ( 1st ` x ) < K /\ ( 1st ` x ) <_ ( M - 1 ) ) \/ ( -. ( 1st ` x ) < K /\ ( 1st ` x ) <_ ( M - 1 ) ) ) ) |
103 |
99 102
|
bitri |
|- ( ( 1st ` x ) <_ ( M - 1 ) <-> ( ( ( 1st ` x ) < K /\ ( 1st ` x ) <_ ( M - 1 ) ) \/ ( -. ( 1st ` x ) < K /\ ( 1st ` x ) <_ ( M - 1 ) ) ) ) |
104 |
98 103
|
bitr4di |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( ( ( 1st ` x ) < K /\ ( 1st ` x ) e. ( 1 ... M ) ) \/ ( -. ( 1st ` x ) < K /\ ( ( 1st ` x ) + 1 ) e. ( 1 ... M ) ) ) <-> ( 1st ` x ) <_ ( M - 1 ) ) ) |
105 |
58 104
|
syl5bb |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( if ( ( 1st ` x ) < K , ( 1st ` x ) , ( ( 1st ` x ) + 1 ) ) e. ( 1 ... M ) <-> ( 1st ` x ) <_ ( M - 1 ) ) ) |
106 |
|
ifel |
|- ( if ( ( 2nd ` x ) < L , ( 2nd ` x ) , ( ( 2nd ` x ) + 1 ) ) e. ( 1 ... N ) <-> ( ( ( 2nd ` x ) < L /\ ( 2nd ` x ) e. ( 1 ... N ) ) \/ ( -. ( 2nd ` x ) < L /\ ( ( 2nd ` x ) + 1 ) e. ( 1 ... N ) ) ) ) |
107 |
|
simplrr |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 2nd ` x ) < L ) -> ( 2nd ` x ) e. NN ) |
108 |
107
|
nnred |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 2nd ` x ) < L ) -> ( 2nd ` x ) e. RR ) |
109 |
18
|
nnred |
|- ( ph -> L e. RR ) |
110 |
109
|
ad3antrrr |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 2nd ` x ) < L ) -> L e. RR ) |
111 |
3
|
nnred |
|- ( ph -> N e. RR ) |
112 |
111
|
ad3antrrr |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 2nd ` x ) < L ) -> N e. RR ) |
113 |
|
simpr |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 2nd ` x ) < L ) -> ( 2nd ` x ) < L ) |
114 |
108 110 113
|
ltled |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 2nd ` x ) < L ) -> ( 2nd ` x ) <_ L ) |
115 |
|
elfzle2 |
|- ( L e. ( 1 ... N ) -> L <_ N ) |
116 |
5 115
|
syl |
|- ( ph -> L <_ N ) |
117 |
116
|
ad3antrrr |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 2nd ` x ) < L ) -> L <_ N ) |
118 |
108 110 112 114 117
|
letrd |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 2nd ` x ) < L ) -> ( 2nd ` x ) <_ N ) |
119 |
107 118
|
jca |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 2nd ` x ) < L ) -> ( ( 2nd ` x ) e. NN /\ ( 2nd ` x ) <_ N ) ) |
120 |
3
|
nnzd |
|- ( ph -> N e. ZZ ) |
121 |
|
fznn |
|- ( N e. ZZ -> ( ( 2nd ` x ) e. ( 1 ... N ) <-> ( ( 2nd ` x ) e. NN /\ ( 2nd ` x ) <_ N ) ) ) |
122 |
120 121
|
syl |
|- ( ph -> ( ( 2nd ` x ) e. ( 1 ... N ) <-> ( ( 2nd ` x ) e. NN /\ ( 2nd ` x ) <_ N ) ) ) |
123 |
122
|
ad3antrrr |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 2nd ` x ) < L ) -> ( ( 2nd ` x ) e. ( 1 ... N ) <-> ( ( 2nd ` x ) e. NN /\ ( 2nd ` x ) <_ N ) ) ) |
124 |
119 123
|
mpbird |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 2nd ` x ) < L ) -> ( 2nd ` x ) e. ( 1 ... N ) ) |
125 |
108 110 112 113 117
|
ltletrd |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 2nd ` x ) < L ) -> ( 2nd ` x ) < N ) |
126 |
3
|
ad3antrrr |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 2nd ` x ) < L ) -> N e. NN ) |
127 |
|
nnltlem1 |
|- ( ( ( 2nd ` x ) e. NN /\ N e. NN ) -> ( ( 2nd ` x ) < N <-> ( 2nd ` x ) <_ ( N - 1 ) ) ) |
128 |
107 126 127
|
syl2anc |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 2nd ` x ) < L ) -> ( ( 2nd ` x ) < N <-> ( 2nd ` x ) <_ ( N - 1 ) ) ) |
129 |
125 128
|
mpbid |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 2nd ` x ) < L ) -> ( 2nd ` x ) <_ ( N - 1 ) ) |
130 |
124 129
|
2thd |
|- ( ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( 2nd ` x ) < L ) -> ( ( 2nd ` x ) e. ( 1 ... N ) <-> ( 2nd ` x ) <_ ( N - 1 ) ) ) |
131 |
130
|
pm5.32da |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( ( 2nd ` x ) < L /\ ( 2nd ` x ) e. ( 1 ... N ) ) <-> ( ( 2nd ` x ) < L /\ ( 2nd ` x ) <_ ( N - 1 ) ) ) ) |
132 |
|
fznn |
|- ( N e. ZZ -> ( ( ( 2nd ` x ) + 1 ) e. ( 1 ... N ) <-> ( ( ( 2nd ` x ) + 1 ) e. NN /\ ( ( 2nd ` x ) + 1 ) <_ N ) ) ) |
133 |
120 132
|
syl |
|- ( ph -> ( ( ( 2nd ` x ) + 1 ) e. ( 1 ... N ) <-> ( ( ( 2nd ` x ) + 1 ) e. NN /\ ( ( 2nd ` x ) + 1 ) <_ N ) ) ) |
134 |
133
|
ad2antrr |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( ( 2nd ` x ) + 1 ) e. ( 1 ... N ) <-> ( ( ( 2nd ` x ) + 1 ) e. NN /\ ( ( 2nd ` x ) + 1 ) <_ N ) ) ) |
135 |
|
simprr |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( 2nd ` x ) e. NN ) |
136 |
135
|
peano2nnd |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( 2nd ` x ) + 1 ) e. NN ) |
137 |
136
|
biantrurd |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( ( 2nd ` x ) + 1 ) <_ N <-> ( ( ( 2nd ` x ) + 1 ) e. NN /\ ( ( 2nd ` x ) + 1 ) <_ N ) ) ) |
138 |
135
|
nnzd |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( 2nd ` x ) e. ZZ ) |
139 |
120
|
ad2antrr |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> N e. ZZ ) |
140 |
|
zltp1le |
|- ( ( ( 2nd ` x ) e. ZZ /\ N e. ZZ ) -> ( ( 2nd ` x ) < N <-> ( ( 2nd ` x ) + 1 ) <_ N ) ) |
141 |
|
zltlem1 |
|- ( ( ( 2nd ` x ) e. ZZ /\ N e. ZZ ) -> ( ( 2nd ` x ) < N <-> ( 2nd ` x ) <_ ( N - 1 ) ) ) |
142 |
140 141
|
bitr3d |
|- ( ( ( 2nd ` x ) e. ZZ /\ N e. ZZ ) -> ( ( ( 2nd ` x ) + 1 ) <_ N <-> ( 2nd ` x ) <_ ( N - 1 ) ) ) |
143 |
138 139 142
|
syl2anc |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( ( 2nd ` x ) + 1 ) <_ N <-> ( 2nd ` x ) <_ ( N - 1 ) ) ) |
144 |
134 137 143
|
3bitr2d |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( ( 2nd ` x ) + 1 ) e. ( 1 ... N ) <-> ( 2nd ` x ) <_ ( N - 1 ) ) ) |
145 |
144
|
anbi2d |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( -. ( 2nd ` x ) < L /\ ( ( 2nd ` x ) + 1 ) e. ( 1 ... N ) ) <-> ( -. ( 2nd ` x ) < L /\ ( 2nd ` x ) <_ ( N - 1 ) ) ) ) |
146 |
131 145
|
orbi12d |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( ( ( 2nd ` x ) < L /\ ( 2nd ` x ) e. ( 1 ... N ) ) \/ ( -. ( 2nd ` x ) < L /\ ( ( 2nd ` x ) + 1 ) e. ( 1 ... N ) ) ) <-> ( ( ( 2nd ` x ) < L /\ ( 2nd ` x ) <_ ( N - 1 ) ) \/ ( -. ( 2nd ` x ) < L /\ ( 2nd ` x ) <_ ( N - 1 ) ) ) ) ) |
147 |
|
pm4.42 |
|- ( ( 2nd ` x ) <_ ( N - 1 ) <-> ( ( ( 2nd ` x ) <_ ( N - 1 ) /\ ( 2nd ` x ) < L ) \/ ( ( 2nd ` x ) <_ ( N - 1 ) /\ -. ( 2nd ` x ) < L ) ) ) |
148 |
|
ancom |
|- ( ( ( 2nd ` x ) <_ ( N - 1 ) /\ ( 2nd ` x ) < L ) <-> ( ( 2nd ` x ) < L /\ ( 2nd ` x ) <_ ( N - 1 ) ) ) |
149 |
|
ancom |
|- ( ( ( 2nd ` x ) <_ ( N - 1 ) /\ -. ( 2nd ` x ) < L ) <-> ( -. ( 2nd ` x ) < L /\ ( 2nd ` x ) <_ ( N - 1 ) ) ) |
150 |
148 149
|
orbi12i |
|- ( ( ( ( 2nd ` x ) <_ ( N - 1 ) /\ ( 2nd ` x ) < L ) \/ ( ( 2nd ` x ) <_ ( N - 1 ) /\ -. ( 2nd ` x ) < L ) ) <-> ( ( ( 2nd ` x ) < L /\ ( 2nd ` x ) <_ ( N - 1 ) ) \/ ( -. ( 2nd ` x ) < L /\ ( 2nd ` x ) <_ ( N - 1 ) ) ) ) |
151 |
147 150
|
bitri |
|- ( ( 2nd ` x ) <_ ( N - 1 ) <-> ( ( ( 2nd ` x ) < L /\ ( 2nd ` x ) <_ ( N - 1 ) ) \/ ( -. ( 2nd ` x ) < L /\ ( 2nd ` x ) <_ ( N - 1 ) ) ) ) |
152 |
146 151
|
bitr4di |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( ( ( 2nd ` x ) < L /\ ( 2nd ` x ) e. ( 1 ... N ) ) \/ ( -. ( 2nd ` x ) < L /\ ( ( 2nd ` x ) + 1 ) e. ( 1 ... N ) ) ) <-> ( 2nd ` x ) <_ ( N - 1 ) ) ) |
153 |
106 152
|
syl5bb |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( if ( ( 2nd ` x ) < L , ( 2nd ` x ) , ( ( 2nd ` x ) + 1 ) ) e. ( 1 ... N ) <-> ( 2nd ` x ) <_ ( N - 1 ) ) ) |
154 |
105 153
|
anbi12d |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( if ( ( 1st ` x ) < K , ( 1st ` x ) , ( ( 1st ` x ) + 1 ) ) e. ( 1 ... M ) /\ if ( ( 2nd ` x ) < L , ( 2nd ` x ) , ( ( 2nd ` x ) + 1 ) ) e. ( 1 ... N ) ) <-> ( ( 1st ` x ) <_ ( M - 1 ) /\ ( 2nd ` x ) <_ ( N - 1 ) ) ) ) |
155 |
57 154
|
bitrd |
|- ( ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) -> ( ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) e. ( ( 1 ... M ) X. ( 1 ... N ) ) <-> ( ( 1st ` x ) <_ ( M - 1 ) /\ ( 2nd ` x ) <_ ( N - 1 ) ) ) ) |
156 |
155
|
pm5.32da |
|- ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) -> ( ( ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) /\ ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) e. ( ( 1 ... M ) X. ( 1 ... N ) ) ) <-> ( ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) /\ ( ( 1st ` x ) <_ ( M - 1 ) /\ ( 2nd ` x ) <_ ( N - 1 ) ) ) ) ) |
157 |
|
1zzd |
|- ( ph -> 1 e. ZZ ) |
158 |
72 157
|
zsubcld |
|- ( ph -> ( M - 1 ) e. ZZ ) |
159 |
|
fznn |
|- ( ( M - 1 ) e. ZZ -> ( ( 1st ` x ) e. ( 1 ... ( M - 1 ) ) <-> ( ( 1st ` x ) e. NN /\ ( 1st ` x ) <_ ( M - 1 ) ) ) ) |
160 |
158 159
|
syl |
|- ( ph -> ( ( 1st ` x ) e. ( 1 ... ( M - 1 ) ) <-> ( ( 1st ` x ) e. NN /\ ( 1st ` x ) <_ ( M - 1 ) ) ) ) |
161 |
120 157
|
zsubcld |
|- ( ph -> ( N - 1 ) e. ZZ ) |
162 |
|
fznn |
|- ( ( N - 1 ) e. ZZ -> ( ( 2nd ` x ) e. ( 1 ... ( N - 1 ) ) <-> ( ( 2nd ` x ) e. NN /\ ( 2nd ` x ) <_ ( N - 1 ) ) ) ) |
163 |
161 162
|
syl |
|- ( ph -> ( ( 2nd ` x ) e. ( 1 ... ( N - 1 ) ) <-> ( ( 2nd ` x ) e. NN /\ ( 2nd ` x ) <_ ( N - 1 ) ) ) ) |
164 |
160 163
|
anbi12d |
|- ( ph -> ( ( ( 1st ` x ) e. ( 1 ... ( M - 1 ) ) /\ ( 2nd ` x ) e. ( 1 ... ( N - 1 ) ) ) <-> ( ( ( 1st ` x ) e. NN /\ ( 1st ` x ) <_ ( M - 1 ) ) /\ ( ( 2nd ` x ) e. NN /\ ( 2nd ` x ) <_ ( N - 1 ) ) ) ) ) |
165 |
|
an4 |
|- ( ( ( ( 1st ` x ) e. NN /\ ( 1st ` x ) <_ ( M - 1 ) ) /\ ( ( 2nd ` x ) e. NN /\ ( 2nd ` x ) <_ ( N - 1 ) ) ) <-> ( ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) /\ ( ( 1st ` x ) <_ ( M - 1 ) /\ ( 2nd ` x ) <_ ( N - 1 ) ) ) ) |
166 |
164 165
|
bitrdi |
|- ( ph -> ( ( ( 1st ` x ) e. ( 1 ... ( M - 1 ) ) /\ ( 2nd ` x ) e. ( 1 ... ( N - 1 ) ) ) <-> ( ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) /\ ( ( 1st ` x ) <_ ( M - 1 ) /\ ( 2nd ` x ) <_ ( N - 1 ) ) ) ) ) |
167 |
166
|
adantr |
|- ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) -> ( ( ( 1st ` x ) e. ( 1 ... ( M - 1 ) ) /\ ( 2nd ` x ) e. ( 1 ... ( N - 1 ) ) ) <-> ( ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) /\ ( ( 1st ` x ) <_ ( M - 1 ) /\ ( 2nd ` x ) <_ ( N - 1 ) ) ) ) ) |
168 |
156 167
|
bitr4d |
|- ( ( ph /\ x = <. ( 1st ` x ) , ( 2nd ` x ) >. ) -> ( ( ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) /\ ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) e. ( ( 1 ... M ) X. ( 1 ... N ) ) ) <-> ( ( 1st ` x ) e. ( 1 ... ( M - 1 ) ) /\ ( 2nd ` x ) e. ( 1 ... ( N - 1 ) ) ) ) ) |
169 |
168
|
pm5.32da |
|- ( ph -> ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) /\ ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) e. ( ( 1 ... M ) X. ( 1 ... N ) ) ) ) <-> ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ( 1 ... ( M - 1 ) ) /\ ( 2nd ` x ) e. ( 1 ... ( N - 1 ) ) ) ) ) ) |
170 |
|
elxp6 |
|- ( x e. ( NN X. NN ) <-> ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) ) |
171 |
170
|
anbi1i |
|- ( ( x e. ( NN X. NN ) /\ ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) e. ( ( 1 ... M ) X. ( 1 ... N ) ) ) <-> ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) e. ( ( 1 ... M ) X. ( 1 ... N ) ) ) ) |
172 |
|
anass |
|- ( ( ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) ) /\ ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) e. ( ( 1 ... M ) X. ( 1 ... N ) ) ) <-> ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) /\ ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) e. ( ( 1 ... M ) X. ( 1 ... N ) ) ) ) ) |
173 |
171 172
|
bitri |
|- ( ( x e. ( NN X. NN ) /\ ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) e. ( ( 1 ... M ) X. ( 1 ... N ) ) ) <-> ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( ( 1st ` x ) e. NN /\ ( 2nd ` x ) e. NN ) /\ ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) e. ( ( 1 ... M ) X. ( 1 ... N ) ) ) ) ) |
174 |
|
elxp6 |
|- ( x e. ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) <-> ( x = <. ( 1st ` x ) , ( 2nd ` x ) >. /\ ( ( 1st ` x ) e. ( 1 ... ( M - 1 ) ) /\ ( 2nd ` x ) e. ( 1 ... ( N - 1 ) ) ) ) ) |
175 |
169 173 174
|
3bitr4g |
|- ( ph -> ( ( x e. ( NN X. NN ) /\ ( ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ` x ) e. ( ( 1 ... M ) X. ( 1 ... N ) ) ) <-> x e. ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) ) |
176 |
31 36 175
|
3bitrd |
|- ( ph -> ( x e. ( `' ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) " dom A ) <-> x e. ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) ) |
177 |
176
|
eqrdv |
|- ( ph -> ( `' ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) " dom A ) = ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) |
178 |
27 177
|
syl5eq |
|- ( ph -> dom ( A o. ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ) = ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) |
179 |
26 178
|
eqtrd |
|- ( ph -> dom S = ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) |
180 |
179
|
feq2d |
|- ( ph -> ( S : dom S --> ran S <-> S : ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) --> ran S ) ) |
181 |
25 180
|
mpbid |
|- ( ph -> S : ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) --> ran S ) |
182 |
21
|
rneqd |
|- ( ph -> ran S = ran ( A o. ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ) ) |
183 |
|
rncoss |
|- ran ( A o. ( i e. NN , j e. NN |-> <. if ( i < K , i , ( i + 1 ) ) , if ( j < L , j , ( j + 1 ) ) >. ) ) C_ ran A |
184 |
182 183
|
eqsstrdi |
|- ( ph -> ran S C_ ran A ) |
185 |
|
frn |
|- ( A : ( ( 1 ... M ) X. ( 1 ... N ) ) --> B -> ran A C_ B ) |
186 |
6 7 185
|
3syl |
|- ( ph -> ran A C_ B ) |
187 |
184 186
|
sstrd |
|- ( ph -> ran S C_ B ) |
188 |
|
fss |
|- ( ( S : ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) --> ran S /\ ran S C_ B ) -> S : ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) --> B ) |
189 |
181 187 188
|
syl2anc |
|- ( ph -> S : ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) --> B ) |
190 |
|
reldmmap |
|- Rel dom ^m |
191 |
190
|
ovrcl |
|- ( A e. ( B ^m ( ( 1 ... M ) X. ( 1 ... N ) ) ) -> ( B e. _V /\ ( ( 1 ... M ) X. ( 1 ... N ) ) e. _V ) ) |
192 |
6 191
|
syl |
|- ( ph -> ( B e. _V /\ ( ( 1 ... M ) X. ( 1 ... N ) ) e. _V ) ) |
193 |
192
|
simpld |
|- ( ph -> B e. _V ) |
194 |
|
ovex |
|- ( 1 ... ( M - 1 ) ) e. _V |
195 |
|
ovex |
|- ( 1 ... ( N - 1 ) ) e. _V |
196 |
194 195
|
xpex |
|- ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) e. _V |
197 |
|
elmapg |
|- ( ( B e. _V /\ ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) e. _V ) -> ( S e. ( B ^m ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) <-> S : ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) --> B ) ) |
198 |
193 196 197
|
sylancl |
|- ( ph -> ( S e. ( B ^m ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) <-> S : ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) --> B ) ) |
199 |
189 198
|
mpbird |
|- ( ph -> S e. ( B ^m ( ( 1 ... ( M - 1 ) ) X. ( 1 ... ( N - 1 ) ) ) ) ) |