Step |
Hyp |
Ref |
Expression |
1 |
|
flltp1 |
|- ( A e. RR -> A < ( ( |_ ` A ) + 1 ) ) |
2 |
1
|
ad3antrrr |
|- ( ( ( ( A e. RR /\ B e. RR ) /\ ( |_ ` A ) <_ B ) /\ B < A ) -> A < ( ( |_ ` A ) + 1 ) ) |
3 |
|
flval |
|- ( B e. RR -> ( |_ ` B ) = ( iota_ x e. ZZ ( x <_ B /\ B < ( x + 1 ) ) ) ) |
4 |
3
|
ad3antlr |
|- ( ( ( ( A e. RR /\ B e. RR ) /\ ( |_ ` A ) <_ B ) /\ B < A ) -> ( |_ ` B ) = ( iota_ x e. ZZ ( x <_ B /\ B < ( x + 1 ) ) ) ) |
5 |
|
simplr |
|- ( ( ( ( A e. RR /\ B e. RR ) /\ ( |_ ` A ) <_ B ) /\ B < A ) -> ( |_ ` A ) <_ B ) |
6 |
1
|
adantr |
|- ( ( A e. RR /\ B e. RR ) -> A < ( ( |_ ` A ) + 1 ) ) |
7 |
|
reflcl |
|- ( A e. RR -> ( |_ ` A ) e. RR ) |
8 |
|
peano2re |
|- ( ( |_ ` A ) e. RR -> ( ( |_ ` A ) + 1 ) e. RR ) |
9 |
7 8
|
syl |
|- ( A e. RR -> ( ( |_ ` A ) + 1 ) e. RR ) |
10 |
9
|
adantl |
|- ( ( B e. RR /\ A e. RR ) -> ( ( |_ ` A ) + 1 ) e. RR ) |
11 |
|
lttr |
|- ( ( B e. RR /\ A e. RR /\ ( ( |_ ` A ) + 1 ) e. RR ) -> ( ( B < A /\ A < ( ( |_ ` A ) + 1 ) ) -> B < ( ( |_ ` A ) + 1 ) ) ) |
12 |
10 11
|
mpd3an3 |
|- ( ( B e. RR /\ A e. RR ) -> ( ( B < A /\ A < ( ( |_ ` A ) + 1 ) ) -> B < ( ( |_ ` A ) + 1 ) ) ) |
13 |
12
|
ancoms |
|- ( ( A e. RR /\ B e. RR ) -> ( ( B < A /\ A < ( ( |_ ` A ) + 1 ) ) -> B < ( ( |_ ` A ) + 1 ) ) ) |
14 |
6 13
|
mpan2d |
|- ( ( A e. RR /\ B e. RR ) -> ( B < A -> B < ( ( |_ ` A ) + 1 ) ) ) |
15 |
14
|
imp |
|- ( ( ( A e. RR /\ B e. RR ) /\ B < A ) -> B < ( ( |_ ` A ) + 1 ) ) |
16 |
15
|
adantlr |
|- ( ( ( ( A e. RR /\ B e. RR ) /\ ( |_ ` A ) <_ B ) /\ B < A ) -> B < ( ( |_ ` A ) + 1 ) ) |
17 |
|
flcl |
|- ( A e. RR -> ( |_ ` A ) e. ZZ ) |
18 |
|
rebtwnz |
|- ( B e. RR -> E! x e. ZZ ( x <_ B /\ B < ( x + 1 ) ) ) |
19 |
|
breq1 |
|- ( x = ( |_ ` A ) -> ( x <_ B <-> ( |_ ` A ) <_ B ) ) |
20 |
|
oveq1 |
|- ( x = ( |_ ` A ) -> ( x + 1 ) = ( ( |_ ` A ) + 1 ) ) |
21 |
20
|
breq2d |
|- ( x = ( |_ ` A ) -> ( B < ( x + 1 ) <-> B < ( ( |_ ` A ) + 1 ) ) ) |
22 |
19 21
|
anbi12d |
|- ( x = ( |_ ` A ) -> ( ( x <_ B /\ B < ( x + 1 ) ) <-> ( ( |_ ` A ) <_ B /\ B < ( ( |_ ` A ) + 1 ) ) ) ) |
23 |
22
|
riota2 |
|- ( ( ( |_ ` A ) e. ZZ /\ E! x e. ZZ ( x <_ B /\ B < ( x + 1 ) ) ) -> ( ( ( |_ ` A ) <_ B /\ B < ( ( |_ ` A ) + 1 ) ) <-> ( iota_ x e. ZZ ( x <_ B /\ B < ( x + 1 ) ) ) = ( |_ ` A ) ) ) |
24 |
17 18 23
|
syl2an |
|- ( ( A e. RR /\ B e. RR ) -> ( ( ( |_ ` A ) <_ B /\ B < ( ( |_ ` A ) + 1 ) ) <-> ( iota_ x e. ZZ ( x <_ B /\ B < ( x + 1 ) ) ) = ( |_ ` A ) ) ) |
25 |
24
|
ad2antrr |
|- ( ( ( ( A e. RR /\ B e. RR ) /\ ( |_ ` A ) <_ B ) /\ B < A ) -> ( ( ( |_ ` A ) <_ B /\ B < ( ( |_ ` A ) + 1 ) ) <-> ( iota_ x e. ZZ ( x <_ B /\ B < ( x + 1 ) ) ) = ( |_ ` A ) ) ) |
26 |
5 16 25
|
mpbi2and |
|- ( ( ( ( A e. RR /\ B e. RR ) /\ ( |_ ` A ) <_ B ) /\ B < A ) -> ( iota_ x e. ZZ ( x <_ B /\ B < ( x + 1 ) ) ) = ( |_ ` A ) ) |
27 |
4 26
|
eqtrd |
|- ( ( ( ( A e. RR /\ B e. RR ) /\ ( |_ ` A ) <_ B ) /\ B < A ) -> ( |_ ` B ) = ( |_ ` A ) ) |
28 |
27
|
oveq1d |
|- ( ( ( ( A e. RR /\ B e. RR ) /\ ( |_ ` A ) <_ B ) /\ B < A ) -> ( ( |_ ` B ) + 1 ) = ( ( |_ ` A ) + 1 ) ) |
29 |
2 28
|
breqtrrd |
|- ( ( ( ( A e. RR /\ B e. RR ) /\ ( |_ ` A ) <_ B ) /\ B < A ) -> A < ( ( |_ ` B ) + 1 ) ) |
30 |
29
|
ex |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( |_ ` A ) <_ B ) -> ( B < A -> A < ( ( |_ ` B ) + 1 ) ) ) |
31 |
|
lenlt |
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -. B < A ) ) |
32 |
|
flltp1 |
|- ( B e. RR -> B < ( ( |_ ` B ) + 1 ) ) |
33 |
32
|
adantl |
|- ( ( A e. RR /\ B e. RR ) -> B < ( ( |_ ` B ) + 1 ) ) |
34 |
|
reflcl |
|- ( B e. RR -> ( |_ ` B ) e. RR ) |
35 |
|
peano2re |
|- ( ( |_ ` B ) e. RR -> ( ( |_ ` B ) + 1 ) e. RR ) |
36 |
34 35
|
syl |
|- ( B e. RR -> ( ( |_ ` B ) + 1 ) e. RR ) |
37 |
36
|
adantl |
|- ( ( A e. RR /\ B e. RR ) -> ( ( |_ ` B ) + 1 ) e. RR ) |
38 |
|
lelttr |
|- ( ( A e. RR /\ B e. RR /\ ( ( |_ ` B ) + 1 ) e. RR ) -> ( ( A <_ B /\ B < ( ( |_ ` B ) + 1 ) ) -> A < ( ( |_ ` B ) + 1 ) ) ) |
39 |
37 38
|
mpd3an3 |
|- ( ( A e. RR /\ B e. RR ) -> ( ( A <_ B /\ B < ( ( |_ ` B ) + 1 ) ) -> A < ( ( |_ ` B ) + 1 ) ) ) |
40 |
33 39
|
mpan2d |
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B -> A < ( ( |_ ` B ) + 1 ) ) ) |
41 |
31 40
|
sylbird |
|- ( ( A e. RR /\ B e. RR ) -> ( -. B < A -> A < ( ( |_ ` B ) + 1 ) ) ) |
42 |
41
|
adantr |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( |_ ` A ) <_ B ) -> ( -. B < A -> A < ( ( |_ ` B ) + 1 ) ) ) |
43 |
30 42
|
pm2.61d |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( |_ ` A ) <_ B ) -> A < ( ( |_ ` B ) + 1 ) ) |
44 |
|
flval |
|- ( A e. RR -> ( |_ ` A ) = ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) ) |
45 |
44
|
ad3antrrr |
|- ( ( ( ( A e. RR /\ B e. RR ) /\ A < ( ( |_ ` B ) + 1 ) ) /\ B < A ) -> ( |_ ` A ) = ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) ) |
46 |
34
|
ad2antlr |
|- ( ( ( A e. RR /\ B e. RR ) /\ B < A ) -> ( |_ ` B ) e. RR ) |
47 |
|
simpll |
|- ( ( ( A e. RR /\ B e. RR ) /\ B < A ) -> A e. RR ) |
48 |
|
simplr |
|- ( ( ( A e. RR /\ B e. RR ) /\ B < A ) -> B e. RR ) |
49 |
|
flle |
|- ( B e. RR -> ( |_ ` B ) <_ B ) |
50 |
49
|
ad2antlr |
|- ( ( ( A e. RR /\ B e. RR ) /\ B < A ) -> ( |_ ` B ) <_ B ) |
51 |
|
simpr |
|- ( ( ( A e. RR /\ B e. RR ) /\ B < A ) -> B < A ) |
52 |
46 48 47 50 51
|
lelttrd |
|- ( ( ( A e. RR /\ B e. RR ) /\ B < A ) -> ( |_ ` B ) < A ) |
53 |
46 47 52
|
ltled |
|- ( ( ( A e. RR /\ B e. RR ) /\ B < A ) -> ( |_ ` B ) <_ A ) |
54 |
53
|
adantlr |
|- ( ( ( ( A e. RR /\ B e. RR ) /\ A < ( ( |_ ` B ) + 1 ) ) /\ B < A ) -> ( |_ ` B ) <_ A ) |
55 |
|
simplr |
|- ( ( ( ( A e. RR /\ B e. RR ) /\ A < ( ( |_ ` B ) + 1 ) ) /\ B < A ) -> A < ( ( |_ ` B ) + 1 ) ) |
56 |
|
flcl |
|- ( B e. RR -> ( |_ ` B ) e. ZZ ) |
57 |
|
rebtwnz |
|- ( A e. RR -> E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) |
58 |
|
breq1 |
|- ( x = ( |_ ` B ) -> ( x <_ A <-> ( |_ ` B ) <_ A ) ) |
59 |
|
oveq1 |
|- ( x = ( |_ ` B ) -> ( x + 1 ) = ( ( |_ ` B ) + 1 ) ) |
60 |
59
|
breq2d |
|- ( x = ( |_ ` B ) -> ( A < ( x + 1 ) <-> A < ( ( |_ ` B ) + 1 ) ) ) |
61 |
58 60
|
anbi12d |
|- ( x = ( |_ ` B ) -> ( ( x <_ A /\ A < ( x + 1 ) ) <-> ( ( |_ ` B ) <_ A /\ A < ( ( |_ ` B ) + 1 ) ) ) ) |
62 |
61
|
riota2 |
|- ( ( ( |_ ` B ) e. ZZ /\ E! x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) -> ( ( ( |_ ` B ) <_ A /\ A < ( ( |_ ` B ) + 1 ) ) <-> ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) = ( |_ ` B ) ) ) |
63 |
56 57 62
|
syl2anr |
|- ( ( A e. RR /\ B e. RR ) -> ( ( ( |_ ` B ) <_ A /\ A < ( ( |_ ` B ) + 1 ) ) <-> ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) = ( |_ ` B ) ) ) |
64 |
63
|
ad2antrr |
|- ( ( ( ( A e. RR /\ B e. RR ) /\ A < ( ( |_ ` B ) + 1 ) ) /\ B < A ) -> ( ( ( |_ ` B ) <_ A /\ A < ( ( |_ ` B ) + 1 ) ) <-> ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) = ( |_ ` B ) ) ) |
65 |
54 55 64
|
mpbi2and |
|- ( ( ( ( A e. RR /\ B e. RR ) /\ A < ( ( |_ ` B ) + 1 ) ) /\ B < A ) -> ( iota_ x e. ZZ ( x <_ A /\ A < ( x + 1 ) ) ) = ( |_ ` B ) ) |
66 |
45 65
|
eqtrd |
|- ( ( ( ( A e. RR /\ B e. RR ) /\ A < ( ( |_ ` B ) + 1 ) ) /\ B < A ) -> ( |_ ` A ) = ( |_ ` B ) ) |
67 |
49
|
ad3antlr |
|- ( ( ( ( A e. RR /\ B e. RR ) /\ A < ( ( |_ ` B ) + 1 ) ) /\ B < A ) -> ( |_ ` B ) <_ B ) |
68 |
66 67
|
eqbrtrd |
|- ( ( ( ( A e. RR /\ B e. RR ) /\ A < ( ( |_ ` B ) + 1 ) ) /\ B < A ) -> ( |_ ` A ) <_ B ) |
69 |
68
|
ex |
|- ( ( ( A e. RR /\ B e. RR ) /\ A < ( ( |_ ` B ) + 1 ) ) -> ( B < A -> ( |_ ` A ) <_ B ) ) |
70 |
|
flle |
|- ( A e. RR -> ( |_ ` A ) <_ A ) |
71 |
70
|
adantr |
|- ( ( A e. RR /\ B e. RR ) -> ( |_ ` A ) <_ A ) |
72 |
7
|
adantr |
|- ( ( A e. RR /\ B e. RR ) -> ( |_ ` A ) e. RR ) |
73 |
|
letr |
|- ( ( ( |_ ` A ) e. RR /\ A e. RR /\ B e. RR ) -> ( ( ( |_ ` A ) <_ A /\ A <_ B ) -> ( |_ ` A ) <_ B ) ) |
74 |
73
|
3coml |
|- ( ( A e. RR /\ B e. RR /\ ( |_ ` A ) e. RR ) -> ( ( ( |_ ` A ) <_ A /\ A <_ B ) -> ( |_ ` A ) <_ B ) ) |
75 |
72 74
|
mpd3an3 |
|- ( ( A e. RR /\ B e. RR ) -> ( ( ( |_ ` A ) <_ A /\ A <_ B ) -> ( |_ ` A ) <_ B ) ) |
76 |
71 75
|
mpand |
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B -> ( |_ ` A ) <_ B ) ) |
77 |
31 76
|
sylbird |
|- ( ( A e. RR /\ B e. RR ) -> ( -. B < A -> ( |_ ` A ) <_ B ) ) |
78 |
77
|
adantr |
|- ( ( ( A e. RR /\ B e. RR ) /\ A < ( ( |_ ` B ) + 1 ) ) -> ( -. B < A -> ( |_ ` A ) <_ B ) ) |
79 |
69 78
|
pm2.61d |
|- ( ( ( A e. RR /\ B e. RR ) /\ A < ( ( |_ ` B ) + 1 ) ) -> ( |_ ` A ) <_ B ) |
80 |
43 79
|
impbida |
|- ( ( A e. RR /\ B e. RR ) -> ( ( |_ ` A ) <_ B <-> A < ( ( |_ ` B ) + 1 ) ) ) |