Step |
Hyp |
Ref |
Expression |
1 |
|
iccshift.1 |
|- ( ph -> A e. RR ) |
2 |
|
iccshift.2 |
|- ( ph -> B e. RR ) |
3 |
|
iccshift.3 |
|- ( ph -> T e. RR ) |
4 |
|
eqeq1 |
|- ( w = x -> ( w = ( z + T ) <-> x = ( z + T ) ) ) |
5 |
4
|
rexbidv |
|- ( w = x -> ( E. z e. ( A [,] B ) w = ( z + T ) <-> E. z e. ( A [,] B ) x = ( z + T ) ) ) |
6 |
5
|
elrab |
|- ( x e. { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } <-> ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) ) |
7 |
|
simprr |
|- ( ( ph /\ ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) ) -> E. z e. ( A [,] B ) x = ( z + T ) ) |
8 |
|
nfv |
|- F/ z ph |
9 |
|
nfv |
|- F/ z x e. CC |
10 |
|
nfre1 |
|- F/ z E. z e. ( A [,] B ) x = ( z + T ) |
11 |
9 10
|
nfan |
|- F/ z ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) |
12 |
8 11
|
nfan |
|- F/ z ( ph /\ ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) ) |
13 |
|
nfv |
|- F/ z x e. ( ( A + T ) [,] ( B + T ) ) |
14 |
|
simp3 |
|- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> x = ( z + T ) ) |
15 |
1 2
|
iccssred |
|- ( ph -> ( A [,] B ) C_ RR ) |
16 |
15
|
sselda |
|- ( ( ph /\ z e. ( A [,] B ) ) -> z e. RR ) |
17 |
3
|
adantr |
|- ( ( ph /\ z e. ( A [,] B ) ) -> T e. RR ) |
18 |
16 17
|
readdcld |
|- ( ( ph /\ z e. ( A [,] B ) ) -> ( z + T ) e. RR ) |
19 |
1
|
adantr |
|- ( ( ph /\ z e. ( A [,] B ) ) -> A e. RR ) |
20 |
|
simpr |
|- ( ( ph /\ z e. ( A [,] B ) ) -> z e. ( A [,] B ) ) |
21 |
2
|
adantr |
|- ( ( ph /\ z e. ( A [,] B ) ) -> B e. RR ) |
22 |
|
elicc2 |
|- ( ( A e. RR /\ B e. RR ) -> ( z e. ( A [,] B ) <-> ( z e. RR /\ A <_ z /\ z <_ B ) ) ) |
23 |
19 21 22
|
syl2anc |
|- ( ( ph /\ z e. ( A [,] B ) ) -> ( z e. ( A [,] B ) <-> ( z e. RR /\ A <_ z /\ z <_ B ) ) ) |
24 |
20 23
|
mpbid |
|- ( ( ph /\ z e. ( A [,] B ) ) -> ( z e. RR /\ A <_ z /\ z <_ B ) ) |
25 |
24
|
simp2d |
|- ( ( ph /\ z e. ( A [,] B ) ) -> A <_ z ) |
26 |
19 16 17 25
|
leadd1dd |
|- ( ( ph /\ z e. ( A [,] B ) ) -> ( A + T ) <_ ( z + T ) ) |
27 |
24
|
simp3d |
|- ( ( ph /\ z e. ( A [,] B ) ) -> z <_ B ) |
28 |
16 21 17 27
|
leadd1dd |
|- ( ( ph /\ z e. ( A [,] B ) ) -> ( z + T ) <_ ( B + T ) ) |
29 |
18 26 28
|
3jca |
|- ( ( ph /\ z e. ( A [,] B ) ) -> ( ( z + T ) e. RR /\ ( A + T ) <_ ( z + T ) /\ ( z + T ) <_ ( B + T ) ) ) |
30 |
29
|
3adant3 |
|- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> ( ( z + T ) e. RR /\ ( A + T ) <_ ( z + T ) /\ ( z + T ) <_ ( B + T ) ) ) |
31 |
1 3
|
readdcld |
|- ( ph -> ( A + T ) e. RR ) |
32 |
31
|
3ad2ant1 |
|- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> ( A + T ) e. RR ) |
33 |
2 3
|
readdcld |
|- ( ph -> ( B + T ) e. RR ) |
34 |
33
|
3ad2ant1 |
|- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> ( B + T ) e. RR ) |
35 |
|
elicc2 |
|- ( ( ( A + T ) e. RR /\ ( B + T ) e. RR ) -> ( ( z + T ) e. ( ( A + T ) [,] ( B + T ) ) <-> ( ( z + T ) e. RR /\ ( A + T ) <_ ( z + T ) /\ ( z + T ) <_ ( B + T ) ) ) ) |
36 |
32 34 35
|
syl2anc |
|- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> ( ( z + T ) e. ( ( A + T ) [,] ( B + T ) ) <-> ( ( z + T ) e. RR /\ ( A + T ) <_ ( z + T ) /\ ( z + T ) <_ ( B + T ) ) ) ) |
37 |
30 36
|
mpbird |
|- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> ( z + T ) e. ( ( A + T ) [,] ( B + T ) ) ) |
38 |
14 37
|
eqeltrd |
|- ( ( ph /\ z e. ( A [,] B ) /\ x = ( z + T ) ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) |
39 |
38
|
3exp |
|- ( ph -> ( z e. ( A [,] B ) -> ( x = ( z + T ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) ) ) |
40 |
39
|
adantr |
|- ( ( ph /\ ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) ) -> ( z e. ( A [,] B ) -> ( x = ( z + T ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) ) ) |
41 |
12 13 40
|
rexlimd |
|- ( ( ph /\ ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) ) -> ( E. z e. ( A [,] B ) x = ( z + T ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) ) |
42 |
7 41
|
mpd |
|- ( ( ph /\ ( x e. CC /\ E. z e. ( A [,] B ) x = ( z + T ) ) ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) |
43 |
6 42
|
sylan2b |
|- ( ( ph /\ x e. { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) |
44 |
31
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( A + T ) e. RR ) |
45 |
33
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( B + T ) e. RR ) |
46 |
|
simpr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. ( ( A + T ) [,] ( B + T ) ) ) |
47 |
|
eliccre |
|- ( ( ( A + T ) e. RR /\ ( B + T ) e. RR /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. RR ) |
48 |
44 45 46 47
|
syl3anc |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. RR ) |
49 |
48
|
recnd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. CC ) |
50 |
1
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> A e. RR ) |
51 |
2
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> B e. RR ) |
52 |
3
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> T e. RR ) |
53 |
48 52
|
resubcld |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x - T ) e. RR ) |
54 |
1
|
recnd |
|- ( ph -> A e. CC ) |
55 |
3
|
recnd |
|- ( ph -> T e. CC ) |
56 |
54 55
|
pncand |
|- ( ph -> ( ( A + T ) - T ) = A ) |
57 |
56
|
eqcomd |
|- ( ph -> A = ( ( A + T ) - T ) ) |
58 |
57
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> A = ( ( A + T ) - T ) ) |
59 |
|
elicc2 |
|- ( ( ( A + T ) e. RR /\ ( B + T ) e. RR ) -> ( x e. ( ( A + T ) [,] ( B + T ) ) <-> ( x e. RR /\ ( A + T ) <_ x /\ x <_ ( B + T ) ) ) ) |
60 |
44 45 59
|
syl2anc |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x e. ( ( A + T ) [,] ( B + T ) ) <-> ( x e. RR /\ ( A + T ) <_ x /\ x <_ ( B + T ) ) ) ) |
61 |
46 60
|
mpbid |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x e. RR /\ ( A + T ) <_ x /\ x <_ ( B + T ) ) ) |
62 |
61
|
simp2d |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( A + T ) <_ x ) |
63 |
44 48 52 62
|
lesub1dd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( ( A + T ) - T ) <_ ( x - T ) ) |
64 |
58 63
|
eqbrtrd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> A <_ ( x - T ) ) |
65 |
61
|
simp3d |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x <_ ( B + T ) ) |
66 |
48 45 52 65
|
lesub1dd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x - T ) <_ ( ( B + T ) - T ) ) |
67 |
2
|
recnd |
|- ( ph -> B e. CC ) |
68 |
67 55
|
pncand |
|- ( ph -> ( ( B + T ) - T ) = B ) |
69 |
68
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( ( B + T ) - T ) = B ) |
70 |
66 69
|
breqtrd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x - T ) <_ B ) |
71 |
50 51 53 64 70
|
eliccd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( x - T ) e. ( A [,] B ) ) |
72 |
55
|
adantr |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> T e. CC ) |
73 |
49 72
|
npcand |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> ( ( x - T ) + T ) = x ) |
74 |
73
|
eqcomd |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x = ( ( x - T ) + T ) ) |
75 |
|
oveq1 |
|- ( z = ( x - T ) -> ( z + T ) = ( ( x - T ) + T ) ) |
76 |
75
|
rspceeqv |
|- ( ( ( x - T ) e. ( A [,] B ) /\ x = ( ( x - T ) + T ) ) -> E. z e. ( A [,] B ) x = ( z + T ) ) |
77 |
71 74 76
|
syl2anc |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> E. z e. ( A [,] B ) x = ( z + T ) ) |
78 |
49 77 6
|
sylanbrc |
|- ( ( ph /\ x e. ( ( A + T ) [,] ( B + T ) ) ) -> x e. { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } ) |
79 |
43 78
|
impbida |
|- ( ph -> ( x e. { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } <-> x e. ( ( A + T ) [,] ( B + T ) ) ) ) |
80 |
79
|
eqrdv |
|- ( ph -> { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } = ( ( A + T ) [,] ( B + T ) ) ) |
81 |
80
|
eqcomd |
|- ( ph -> ( ( A + T ) [,] ( B + T ) ) = { w e. CC | E. z e. ( A [,] B ) w = ( z + T ) } ) |