Step |
Hyp |
Ref |
Expression |
1 |
|
metakunt7.1 |
|- ( ph -> M e. NN ) |
2 |
|
metakunt7.2 |
|- ( ph -> I e. NN ) |
3 |
|
metakunt7.3 |
|- ( ph -> I <_ M ) |
4 |
|
metakunt7.4 |
|- A = ( x e. ( 1 ... M ) |-> if ( x = I , M , if ( x < I , x , ( x - 1 ) ) ) ) |
5 |
|
metakunt7.5 |
|- C = ( y e. ( 1 ... M ) |-> if ( y = M , I , if ( y < I , y , ( y + 1 ) ) ) ) |
6 |
|
metakunt7.6 |
|- ( ph -> X e. ( 1 ... M ) ) |
7 |
4
|
a1i |
|- ( ( ph /\ I < X ) -> A = ( x e. ( 1 ... M ) |-> if ( x = I , M , if ( x < I , x , ( x - 1 ) ) ) ) ) |
8 |
|
eqeq1 |
|- ( x = X -> ( x = I <-> X = I ) ) |
9 |
|
breq1 |
|- ( x = X -> ( x < I <-> X < I ) ) |
10 |
|
id |
|- ( x = X -> x = X ) |
11 |
|
oveq1 |
|- ( x = X -> ( x - 1 ) = ( X - 1 ) ) |
12 |
9 10 11
|
ifbieq12d |
|- ( x = X -> if ( x < I , x , ( x - 1 ) ) = if ( X < I , X , ( X - 1 ) ) ) |
13 |
8 12
|
ifbieq2d |
|- ( x = X -> if ( x = I , M , if ( x < I , x , ( x - 1 ) ) ) = if ( X = I , M , if ( X < I , X , ( X - 1 ) ) ) ) |
14 |
13
|
adantl |
|- ( ( ( ph /\ I < X ) /\ x = X ) -> if ( x = I , M , if ( x < I , x , ( x - 1 ) ) ) = if ( X = I , M , if ( X < I , X , ( X - 1 ) ) ) ) |
15 |
2
|
nnred |
|- ( ph -> I e. RR ) |
16 |
15
|
adantr |
|- ( ( ph /\ I < X ) -> I e. RR ) |
17 |
|
simpr |
|- ( ( ph /\ I < X ) -> I < X ) |
18 |
16 17
|
ltned |
|- ( ( ph /\ I < X ) -> I =/= X ) |
19 |
18
|
necomd |
|- ( ( ph /\ I < X ) -> X =/= I ) |
20 |
|
df-ne |
|- ( X =/= I <-> -. X = I ) |
21 |
19 20
|
sylib |
|- ( ( ph /\ I < X ) -> -. X = I ) |
22 |
|
iffalse |
|- ( -. X = I -> if ( X = I , M , if ( X < I , X , ( X - 1 ) ) ) = if ( X < I , X , ( X - 1 ) ) ) |
23 |
21 22
|
syl |
|- ( ( ph /\ I < X ) -> if ( X = I , M , if ( X < I , X , ( X - 1 ) ) ) = if ( X < I , X , ( X - 1 ) ) ) |
24 |
|
elfznn |
|- ( X e. ( 1 ... M ) -> X e. NN ) |
25 |
6 24
|
syl |
|- ( ph -> X e. NN ) |
26 |
25
|
nnred |
|- ( ph -> X e. RR ) |
27 |
26
|
adantr |
|- ( ( ph /\ I < X ) -> X e. RR ) |
28 |
16 27 17
|
ltled |
|- ( ( ph /\ I < X ) -> I <_ X ) |
29 |
16 27
|
lenltd |
|- ( ( ph /\ I < X ) -> ( I <_ X <-> -. X < I ) ) |
30 |
28 29
|
mpbid |
|- ( ( ph /\ I < X ) -> -. X < I ) |
31 |
|
iffalse |
|- ( -. X < I -> if ( X < I , X , ( X - 1 ) ) = ( X - 1 ) ) |
32 |
30 31
|
syl |
|- ( ( ph /\ I < X ) -> if ( X < I , X , ( X - 1 ) ) = ( X - 1 ) ) |
33 |
23 32
|
eqtrd |
|- ( ( ph /\ I < X ) -> if ( X = I , M , if ( X < I , X , ( X - 1 ) ) ) = ( X - 1 ) ) |
34 |
33
|
adantr |
|- ( ( ( ph /\ I < X ) /\ x = X ) -> if ( X = I , M , if ( X < I , X , ( X - 1 ) ) ) = ( X - 1 ) ) |
35 |
14 34
|
eqtrd |
|- ( ( ( ph /\ I < X ) /\ x = X ) -> if ( x = I , M , if ( x < I , x , ( x - 1 ) ) ) = ( X - 1 ) ) |
36 |
6
|
adantr |
|- ( ( ph /\ I < X ) -> X e. ( 1 ... M ) ) |
37 |
36
|
elfzelzd |
|- ( ( ph /\ I < X ) -> X e. ZZ ) |
38 |
|
1zzd |
|- ( ( ph /\ I < X ) -> 1 e. ZZ ) |
39 |
37 38
|
zsubcld |
|- ( ( ph /\ I < X ) -> ( X - 1 ) e. ZZ ) |
40 |
7 35 36 39
|
fvmptd |
|- ( ( ph /\ I < X ) -> ( A ` X ) = ( X - 1 ) ) |
41 |
|
1red |
|- ( ph -> 1 e. RR ) |
42 |
26 41
|
resubcld |
|- ( ph -> ( X - 1 ) e. RR ) |
43 |
|
elfzle2 |
|- ( X e. ( 1 ... M ) -> X <_ M ) |
44 |
6 43
|
syl |
|- ( ph -> X <_ M ) |
45 |
6
|
elfzelzd |
|- ( ph -> X e. ZZ ) |
46 |
1
|
nnzd |
|- ( ph -> M e. ZZ ) |
47 |
|
zlem1lt |
|- ( ( X e. ZZ /\ M e. ZZ ) -> ( X <_ M <-> ( X - 1 ) < M ) ) |
48 |
45 46 47
|
syl2anc |
|- ( ph -> ( X <_ M <-> ( X - 1 ) < M ) ) |
49 |
44 48
|
mpbid |
|- ( ph -> ( X - 1 ) < M ) |
50 |
42 49
|
ltned |
|- ( ph -> ( X - 1 ) =/= M ) |
51 |
50
|
adantr |
|- ( ( ph /\ I < X ) -> ( X - 1 ) =/= M ) |
52 |
40 51
|
eqnetrd |
|- ( ( ph /\ I < X ) -> ( A ` X ) =/= M ) |
53 |
52
|
neneqd |
|- ( ( ph /\ I < X ) -> -. ( A ` X ) = M ) |
54 |
2
|
nnzd |
|- ( ph -> I e. ZZ ) |
55 |
|
zltlem1 |
|- ( ( I e. ZZ /\ X e. ZZ ) -> ( I < X <-> I <_ ( X - 1 ) ) ) |
56 |
55
|
biimpd |
|- ( ( I e. ZZ /\ X e. ZZ ) -> ( I < X -> I <_ ( X - 1 ) ) ) |
57 |
54 45 56
|
syl2anc |
|- ( ph -> ( I < X -> I <_ ( X - 1 ) ) ) |
58 |
57
|
imp |
|- ( ( ph /\ I < X ) -> I <_ ( X - 1 ) ) |
59 |
58 40
|
breqtrrd |
|- ( ( ph /\ I < X ) -> I <_ ( A ` X ) ) |
60 |
39
|
zred |
|- ( ( ph /\ I < X ) -> ( X - 1 ) e. RR ) |
61 |
40 60
|
eqeltrd |
|- ( ( ph /\ I < X ) -> ( A ` X ) e. RR ) |
62 |
16 61
|
lenltd |
|- ( ( ph /\ I < X ) -> ( I <_ ( A ` X ) <-> -. ( A ` X ) < I ) ) |
63 |
59 62
|
mpbid |
|- ( ( ph /\ I < X ) -> -. ( A ` X ) < I ) |
64 |
40 53 63
|
3jca |
|- ( ( ph /\ I < X ) -> ( ( A ` X ) = ( X - 1 ) /\ -. ( A ` X ) = M /\ -. ( A ` X ) < I ) ) |