Step |
Hyp |
Ref |
Expression |
1 |
|
icoshftf1o.1 |
|- F = ( x e. ( A [,) B ) |-> ( x + C ) ) |
2 |
|
icoshft |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( x e. ( A [,) B ) -> ( x + C ) e. ( ( A + C ) [,) ( B + C ) ) ) ) |
3 |
2
|
ralrimiv |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A. x e. ( A [,) B ) ( x + C ) e. ( ( A + C ) [,) ( B + C ) ) ) |
4 |
|
readdcl |
|- ( ( A e. RR /\ C e. RR ) -> ( A + C ) e. RR ) |
5 |
4
|
3adant2 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + C ) e. RR ) |
6 |
|
readdcl |
|- ( ( B e. RR /\ C e. RR ) -> ( B + C ) e. RR ) |
7 |
6
|
3adant1 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. RR ) |
8 |
|
renegcl |
|- ( C e. RR -> -u C e. RR ) |
9 |
8
|
3ad2ant3 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> -u C e. RR ) |
10 |
|
icoshft |
|- ( ( ( A + C ) e. RR /\ ( B + C ) e. RR /\ -u C e. RR ) -> ( y e. ( ( A + C ) [,) ( B + C ) ) -> ( y + -u C ) e. ( ( ( A + C ) + -u C ) [,) ( ( B + C ) + -u C ) ) ) ) |
11 |
5 7 9 10
|
syl3anc |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( y e. ( ( A + C ) [,) ( B + C ) ) -> ( y + -u C ) e. ( ( ( A + C ) + -u C ) [,) ( ( B + C ) + -u C ) ) ) ) |
12 |
11
|
imp |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( y + -u C ) e. ( ( ( A + C ) + -u C ) [,) ( ( B + C ) + -u C ) ) ) |
13 |
7
|
rexrd |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. RR* ) |
14 |
|
icossre |
|- ( ( ( A + C ) e. RR /\ ( B + C ) e. RR* ) -> ( ( A + C ) [,) ( B + C ) ) C_ RR ) |
15 |
5 13 14
|
syl2anc |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) [,) ( B + C ) ) C_ RR ) |
16 |
15
|
sselda |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> y e. RR ) |
17 |
16
|
recnd |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> y e. CC ) |
18 |
|
simpl3 |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> C e. RR ) |
19 |
18
|
recnd |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> C e. CC ) |
20 |
17 19
|
negsubd |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( y + -u C ) = ( y - C ) ) |
21 |
5
|
recnd |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + C ) e. CC ) |
22 |
|
simp3 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR ) |
23 |
22
|
recnd |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. CC ) |
24 |
21 23
|
negsubd |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) + -u C ) = ( ( A + C ) - C ) ) |
25 |
|
simp1 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR ) |
26 |
25
|
recnd |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. CC ) |
27 |
26 23
|
pncand |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) - C ) = A ) |
28 |
24 27
|
eqtrd |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) + -u C ) = A ) |
29 |
7
|
recnd |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) e. CC ) |
30 |
29 23
|
negsubd |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + C ) + -u C ) = ( ( B + C ) - C ) ) |
31 |
|
simp2 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR ) |
32 |
31
|
recnd |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC ) |
33 |
32 23
|
pncand |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + C ) - C ) = B ) |
34 |
30 33
|
eqtrd |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + C ) + -u C ) = B ) |
35 |
28 34
|
oveq12d |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( A + C ) + -u C ) [,) ( ( B + C ) + -u C ) ) = ( A [,) B ) ) |
36 |
35
|
adantr |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( ( ( A + C ) + -u C ) [,) ( ( B + C ) + -u C ) ) = ( A [,) B ) ) |
37 |
12 20 36
|
3eltr3d |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( y - C ) e. ( A [,) B ) ) |
38 |
|
reueq |
|- ( ( y - C ) e. ( A [,) B ) <-> E! x e. ( A [,) B ) x = ( y - C ) ) |
39 |
37 38
|
sylib |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> E! x e. ( A [,) B ) x = ( y - C ) ) |
40 |
16
|
adantr |
|- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) /\ x e. ( A [,) B ) ) -> y e. RR ) |
41 |
40
|
recnd |
|- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) /\ x e. ( A [,) B ) ) -> y e. CC ) |
42 |
|
simpll3 |
|- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) /\ x e. ( A [,) B ) ) -> C e. RR ) |
43 |
42
|
recnd |
|- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) /\ x e. ( A [,) B ) ) -> C e. CC ) |
44 |
|
simpl1 |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> A e. RR ) |
45 |
|
simpl2 |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> B e. RR ) |
46 |
45
|
rexrd |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> B e. RR* ) |
47 |
|
icossre |
|- ( ( A e. RR /\ B e. RR* ) -> ( A [,) B ) C_ RR ) |
48 |
44 46 47
|
syl2anc |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( A [,) B ) C_ RR ) |
49 |
48
|
sselda |
|- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) /\ x e. ( A [,) B ) ) -> x e. RR ) |
50 |
49
|
recnd |
|- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) /\ x e. ( A [,) B ) ) -> x e. CC ) |
51 |
41 43 50
|
subadd2d |
|- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) /\ x e. ( A [,) B ) ) -> ( ( y - C ) = x <-> ( x + C ) = y ) ) |
52 |
|
eqcom |
|- ( x = ( y - C ) <-> ( y - C ) = x ) |
53 |
|
eqcom |
|- ( y = ( x + C ) <-> ( x + C ) = y ) |
54 |
51 52 53
|
3bitr4g |
|- ( ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) /\ x e. ( A [,) B ) ) -> ( x = ( y - C ) <-> y = ( x + C ) ) ) |
55 |
54
|
reubidva |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> ( E! x e. ( A [,) B ) x = ( y - C ) <-> E! x e. ( A [,) B ) y = ( x + C ) ) ) |
56 |
39 55
|
mpbid |
|- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ y e. ( ( A + C ) [,) ( B + C ) ) ) -> E! x e. ( A [,) B ) y = ( x + C ) ) |
57 |
56
|
ralrimiva |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A. y e. ( ( A + C ) [,) ( B + C ) ) E! x e. ( A [,) B ) y = ( x + C ) ) |
58 |
1
|
f1ompt |
|- ( F : ( A [,) B ) -1-1-onto-> ( ( A + C ) [,) ( B + C ) ) <-> ( A. x e. ( A [,) B ) ( x + C ) e. ( ( A + C ) [,) ( B + C ) ) /\ A. y e. ( ( A + C ) [,) ( B + C ) ) E! x e. ( A [,) B ) y = ( x + C ) ) ) |
59 |
3 57 58
|
sylanbrc |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> F : ( A [,) B ) -1-1-onto-> ( ( A + C ) [,) ( B + C ) ) ) |