Step |
Hyp |
Ref |
Expression |
1 |
|
vitali.1 |
|- .~ = { <. x , y >. | ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) /\ ( x - y ) e. QQ ) } |
2 |
1
|
relopabiv |
|- Rel .~ |
3 |
|
simplr |
|- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> v e. ( 0 [,] 1 ) ) |
4 |
|
simpll |
|- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> u e. ( 0 [,] 1 ) ) |
5 |
|
unitssre |
|- ( 0 [,] 1 ) C_ RR |
6 |
5
|
sseli |
|- ( u e. ( 0 [,] 1 ) -> u e. RR ) |
7 |
6
|
recnd |
|- ( u e. ( 0 [,] 1 ) -> u e. CC ) |
8 |
7
|
ad2antrr |
|- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> u e. CC ) |
9 |
5
|
sseli |
|- ( v e. ( 0 [,] 1 ) -> v e. RR ) |
10 |
9
|
recnd |
|- ( v e. ( 0 [,] 1 ) -> v e. CC ) |
11 |
10
|
ad2antlr |
|- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> v e. CC ) |
12 |
8 11
|
negsubdi2d |
|- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> -u ( u - v ) = ( v - u ) ) |
13 |
|
qnegcl |
|- ( ( u - v ) e. QQ -> -u ( u - v ) e. QQ ) |
14 |
13
|
adantl |
|- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> -u ( u - v ) e. QQ ) |
15 |
12 14
|
eqeltrrd |
|- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> ( v - u ) e. QQ ) |
16 |
3 4 15
|
jca31 |
|- ( ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) -> ( ( v e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) /\ ( v - u ) e. QQ ) ) |
17 |
|
oveq12 |
|- ( ( x = u /\ y = v ) -> ( x - y ) = ( u - v ) ) |
18 |
17
|
eleq1d |
|- ( ( x = u /\ y = v ) -> ( ( x - y ) e. QQ <-> ( u - v ) e. QQ ) ) |
19 |
18 1
|
brab2a |
|- ( u .~ v <-> ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) ) |
20 |
|
oveq12 |
|- ( ( x = v /\ y = u ) -> ( x - y ) = ( v - u ) ) |
21 |
20
|
eleq1d |
|- ( ( x = v /\ y = u ) -> ( ( x - y ) e. QQ <-> ( v - u ) e. QQ ) ) |
22 |
21 1
|
brab2a |
|- ( v .~ u <-> ( ( v e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) /\ ( v - u ) e. QQ ) ) |
23 |
16 19 22
|
3imtr4i |
|- ( u .~ v -> v .~ u ) |
24 |
|
simpl |
|- ( ( u .~ v /\ v .~ w ) -> u .~ v ) |
25 |
24 19
|
sylib |
|- ( ( u .~ v /\ v .~ w ) -> ( ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) /\ ( u - v ) e. QQ ) ) |
26 |
25
|
simpld |
|- ( ( u .~ v /\ v .~ w ) -> ( u e. ( 0 [,] 1 ) /\ v e. ( 0 [,] 1 ) ) ) |
27 |
26
|
simpld |
|- ( ( u .~ v /\ v .~ w ) -> u e. ( 0 [,] 1 ) ) |
28 |
|
simpr |
|- ( ( u .~ v /\ v .~ w ) -> v .~ w ) |
29 |
|
oveq12 |
|- ( ( x = v /\ y = w ) -> ( x - y ) = ( v - w ) ) |
30 |
29
|
eleq1d |
|- ( ( x = v /\ y = w ) -> ( ( x - y ) e. QQ <-> ( v - w ) e. QQ ) ) |
31 |
30 1
|
brab2a |
|- ( v .~ w <-> ( ( v e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) ) /\ ( v - w ) e. QQ ) ) |
32 |
28 31
|
sylib |
|- ( ( u .~ v /\ v .~ w ) -> ( ( v e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) ) /\ ( v - w ) e. QQ ) ) |
33 |
32
|
simpld |
|- ( ( u .~ v /\ v .~ w ) -> ( v e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) ) ) |
34 |
33
|
simprd |
|- ( ( u .~ v /\ v .~ w ) -> w e. ( 0 [,] 1 ) ) |
35 |
27 7
|
syl |
|- ( ( u .~ v /\ v .~ w ) -> u e. CC ) |
36 |
25 11
|
syl |
|- ( ( u .~ v /\ v .~ w ) -> v e. CC ) |
37 |
5 34
|
sselid |
|- ( ( u .~ v /\ v .~ w ) -> w e. RR ) |
38 |
37
|
recnd |
|- ( ( u .~ v /\ v .~ w ) -> w e. CC ) |
39 |
35 36 38
|
npncand |
|- ( ( u .~ v /\ v .~ w ) -> ( ( u - v ) + ( v - w ) ) = ( u - w ) ) |
40 |
25
|
simprd |
|- ( ( u .~ v /\ v .~ w ) -> ( u - v ) e. QQ ) |
41 |
32
|
simprd |
|- ( ( u .~ v /\ v .~ w ) -> ( v - w ) e. QQ ) |
42 |
|
qaddcl |
|- ( ( ( u - v ) e. QQ /\ ( v - w ) e. QQ ) -> ( ( u - v ) + ( v - w ) ) e. QQ ) |
43 |
40 41 42
|
syl2anc |
|- ( ( u .~ v /\ v .~ w ) -> ( ( u - v ) + ( v - w ) ) e. QQ ) |
44 |
39 43
|
eqeltrrd |
|- ( ( u .~ v /\ v .~ w ) -> ( u - w ) e. QQ ) |
45 |
|
oveq12 |
|- ( ( x = u /\ y = w ) -> ( x - y ) = ( u - w ) ) |
46 |
45
|
eleq1d |
|- ( ( x = u /\ y = w ) -> ( ( x - y ) e. QQ <-> ( u - w ) e. QQ ) ) |
47 |
46 1
|
brab2a |
|- ( u .~ w <-> ( ( u e. ( 0 [,] 1 ) /\ w e. ( 0 [,] 1 ) ) /\ ( u - w ) e. QQ ) ) |
48 |
27 34 44 47
|
syl21anbrc |
|- ( ( u .~ v /\ v .~ w ) -> u .~ w ) |
49 |
7
|
subidd |
|- ( u e. ( 0 [,] 1 ) -> ( u - u ) = 0 ) |
50 |
|
0z |
|- 0 e. ZZ |
51 |
|
zq |
|- ( 0 e. ZZ -> 0 e. QQ ) |
52 |
50 51
|
ax-mp |
|- 0 e. QQ |
53 |
49 52
|
eqeltrdi |
|- ( u e. ( 0 [,] 1 ) -> ( u - u ) e. QQ ) |
54 |
53
|
adantr |
|- ( ( u e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) -> ( u - u ) e. QQ ) |
55 |
54
|
pm4.71i |
|- ( ( u e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) <-> ( ( u e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) /\ ( u - u ) e. QQ ) ) |
56 |
|
pm4.24 |
|- ( u e. ( 0 [,] 1 ) <-> ( u e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) ) |
57 |
|
oveq12 |
|- ( ( x = u /\ y = u ) -> ( x - y ) = ( u - u ) ) |
58 |
57
|
eleq1d |
|- ( ( x = u /\ y = u ) -> ( ( x - y ) e. QQ <-> ( u - u ) e. QQ ) ) |
59 |
58 1
|
brab2a |
|- ( u .~ u <-> ( ( u e. ( 0 [,] 1 ) /\ u e. ( 0 [,] 1 ) ) /\ ( u - u ) e. QQ ) ) |
60 |
55 56 59
|
3bitr4i |
|- ( u e. ( 0 [,] 1 ) <-> u .~ u ) |
61 |
2 23 48 60
|
iseri |
|- .~ Er ( 0 [,] 1 ) |