Step |
Hyp |
Ref |
Expression |
1 |
|
cnre2csqima.1 |
|- F = ( x e. RR , y e. RR |-> ( x + ( _i x. y ) ) ) |
2 |
|
ioossre |
|- ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) C_ RR |
3 |
|
ioossre |
|- ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) C_ RR |
4 |
|
xpinpreima2 |
|- ( ( ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) C_ RR /\ ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) C_ RR ) -> ( ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) X. ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) = ( ( `' ( 1st |` ( RR X. RR ) ) " ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) ) ) |
5 |
4
|
eleq2d |
|- ( ( ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) C_ RR /\ ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) C_ RR ) -> ( Y e. ( ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) X. ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) <-> Y e. ( ( `' ( 1st |` ( RR X. RR ) ) " ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) ) ) ) |
6 |
2 3 5
|
mp2an |
|- ( Y e. ( ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) X. ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) <-> Y e. ( ( `' ( 1st |` ( RR X. RR ) ) " ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) ) ) |
7 |
|
elin |
|- ( Y e. ( ( `' ( 1st |` ( RR X. RR ) ) " ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) ) <-> ( Y e. ( `' ( 1st |` ( RR X. RR ) ) " ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) ) /\ Y e. ( `' ( 2nd |` ( RR X. RR ) ) " ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) ) ) |
8 |
|
simpl |
|- ( ( x e. RR /\ y e. RR ) -> x e. RR ) |
9 |
8
|
recnd |
|- ( ( x e. RR /\ y e. RR ) -> x e. CC ) |
10 |
|
ax-icn |
|- _i e. CC |
11 |
10
|
a1i |
|- ( ( x e. RR /\ y e. RR ) -> _i e. CC ) |
12 |
|
simpr |
|- ( ( x e. RR /\ y e. RR ) -> y e. RR ) |
13 |
12
|
recnd |
|- ( ( x e. RR /\ y e. RR ) -> y e. CC ) |
14 |
11 13
|
mulcld |
|- ( ( x e. RR /\ y e. RR ) -> ( _i x. y ) e. CC ) |
15 |
9 14
|
addcld |
|- ( ( x e. RR /\ y e. RR ) -> ( x + ( _i x. y ) ) e. CC ) |
16 |
|
reval |
|- ( ( x + ( _i x. y ) ) e. CC -> ( Re ` ( x + ( _i x. y ) ) ) = ( ( ( x + ( _i x. y ) ) + ( * ` ( x + ( _i x. y ) ) ) ) / 2 ) ) |
17 |
15 16
|
syl |
|- ( ( x e. RR /\ y e. RR ) -> ( Re ` ( x + ( _i x. y ) ) ) = ( ( ( x + ( _i x. y ) ) + ( * ` ( x + ( _i x. y ) ) ) ) / 2 ) ) |
18 |
|
crre |
|- ( ( x e. RR /\ y e. RR ) -> ( Re ` ( x + ( _i x. y ) ) ) = x ) |
19 |
17 18
|
eqtr3d |
|- ( ( x e. RR /\ y e. RR ) -> ( ( ( x + ( _i x. y ) ) + ( * ` ( x + ( _i x. y ) ) ) ) / 2 ) = x ) |
20 |
19
|
mpoeq3ia |
|- ( x e. RR , y e. RR |-> ( ( ( x + ( _i x. y ) ) + ( * ` ( x + ( _i x. y ) ) ) ) / 2 ) ) = ( x e. RR , y e. RR |-> x ) |
21 |
15
|
adantl |
|- ( ( T. /\ ( x e. RR /\ y e. RR ) ) -> ( x + ( _i x. y ) ) e. CC ) |
22 |
1
|
a1i |
|- ( T. -> F = ( x e. RR , y e. RR |-> ( x + ( _i x. y ) ) ) ) |
23 |
|
df-re |
|- Re = ( z e. CC |-> ( ( z + ( * ` z ) ) / 2 ) ) |
24 |
23
|
a1i |
|- ( T. -> Re = ( z e. CC |-> ( ( z + ( * ` z ) ) / 2 ) ) ) |
25 |
|
id |
|- ( z = ( x + ( _i x. y ) ) -> z = ( x + ( _i x. y ) ) ) |
26 |
|
fveq2 |
|- ( z = ( x + ( _i x. y ) ) -> ( * ` z ) = ( * ` ( x + ( _i x. y ) ) ) ) |
27 |
25 26
|
oveq12d |
|- ( z = ( x + ( _i x. y ) ) -> ( z + ( * ` z ) ) = ( ( x + ( _i x. y ) ) + ( * ` ( x + ( _i x. y ) ) ) ) ) |
28 |
27
|
oveq1d |
|- ( z = ( x + ( _i x. y ) ) -> ( ( z + ( * ` z ) ) / 2 ) = ( ( ( x + ( _i x. y ) ) + ( * ` ( x + ( _i x. y ) ) ) ) / 2 ) ) |
29 |
21 22 24 28
|
fmpoco |
|- ( T. -> ( Re o. F ) = ( x e. RR , y e. RR |-> ( ( ( x + ( _i x. y ) ) + ( * ` ( x + ( _i x. y ) ) ) ) / 2 ) ) ) |
30 |
29
|
mptru |
|- ( Re o. F ) = ( x e. RR , y e. RR |-> ( ( ( x + ( _i x. y ) ) + ( * ` ( x + ( _i x. y ) ) ) ) / 2 ) ) |
31 |
|
df1stres |
|- ( 1st |` ( RR X. RR ) ) = ( x e. RR , y e. RR |-> x ) |
32 |
20 30 31
|
3eqtr4ri |
|- ( 1st |` ( RR X. RR ) ) = ( Re o. F ) |
33 |
15
|
rgen2 |
|- A. x e. RR A. y e. RR ( x + ( _i x. y ) ) e. CC |
34 |
1
|
fnmpo |
|- ( A. x e. RR A. y e. RR ( x + ( _i x. y ) ) e. CC -> F Fn ( RR X. RR ) ) |
35 |
33 34
|
ax-mp |
|- F Fn ( RR X. RR ) |
36 |
|
fo1st |
|- 1st : _V -onto-> _V |
37 |
|
fofn |
|- ( 1st : _V -onto-> _V -> 1st Fn _V ) |
38 |
36 37
|
ax-mp |
|- 1st Fn _V |
39 |
|
xp1st |
|- ( z e. ( RR X. RR ) -> ( 1st ` z ) e. RR ) |
40 |
1
|
rnmpo |
|- ran F = { z | E. x e. RR E. y e. RR z = ( x + ( _i x. y ) ) } |
41 |
|
simpr |
|- ( ( ( x e. RR /\ y e. RR ) /\ z = ( x + ( _i x. y ) ) ) -> z = ( x + ( _i x. y ) ) ) |
42 |
15
|
adantr |
|- ( ( ( x e. RR /\ y e. RR ) /\ z = ( x + ( _i x. y ) ) ) -> ( x + ( _i x. y ) ) e. CC ) |
43 |
41 42
|
eqeltrd |
|- ( ( ( x e. RR /\ y e. RR ) /\ z = ( x + ( _i x. y ) ) ) -> z e. CC ) |
44 |
43
|
ex |
|- ( ( x e. RR /\ y e. RR ) -> ( z = ( x + ( _i x. y ) ) -> z e. CC ) ) |
45 |
44
|
rexlimivv |
|- ( E. x e. RR E. y e. RR z = ( x + ( _i x. y ) ) -> z e. CC ) |
46 |
45
|
abssi |
|- { z | E. x e. RR E. y e. RR z = ( x + ( _i x. y ) ) } C_ CC |
47 |
40 46
|
eqsstri |
|- ran F C_ CC |
48 |
|
simpl |
|- ( ( z e. ran F /\ u e. ran F ) -> z e. ran F ) |
49 |
47 48
|
sseldi |
|- ( ( z e. ran F /\ u e. ran F ) -> z e. CC ) |
50 |
|
simpr |
|- ( ( z e. ran F /\ u e. ran F ) -> u e. ran F ) |
51 |
47 50
|
sseldi |
|- ( ( z e. ran F /\ u e. ran F ) -> u e. CC ) |
52 |
49 51
|
resubd |
|- ( ( z e. ran F /\ u e. ran F ) -> ( Re ` ( z - u ) ) = ( ( Re ` z ) - ( Re ` u ) ) ) |
53 |
32 35 38 39 52
|
cnre2csqlem |
|- ( ( X e. ( RR X. RR ) /\ Y e. ( RR X. RR ) /\ D e. RR+ ) -> ( Y e. ( `' ( 1st |` ( RR X. RR ) ) " ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) ) -> ( abs ` ( Re ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D ) ) |
54 |
|
imval |
|- ( ( x + ( _i x. y ) ) e. CC -> ( Im ` ( x + ( _i x. y ) ) ) = ( Re ` ( ( x + ( _i x. y ) ) / _i ) ) ) |
55 |
15 54
|
syl |
|- ( ( x e. RR /\ y e. RR ) -> ( Im ` ( x + ( _i x. y ) ) ) = ( Re ` ( ( x + ( _i x. y ) ) / _i ) ) ) |
56 |
|
crim |
|- ( ( x e. RR /\ y e. RR ) -> ( Im ` ( x + ( _i x. y ) ) ) = y ) |
57 |
55 56
|
eqtr3d |
|- ( ( x e. RR /\ y e. RR ) -> ( Re ` ( ( x + ( _i x. y ) ) / _i ) ) = y ) |
58 |
57
|
mpoeq3ia |
|- ( x e. RR , y e. RR |-> ( Re ` ( ( x + ( _i x. y ) ) / _i ) ) ) = ( x e. RR , y e. RR |-> y ) |
59 |
|
df-im |
|- Im = ( z e. CC |-> ( Re ` ( z / _i ) ) ) |
60 |
59
|
a1i |
|- ( T. -> Im = ( z e. CC |-> ( Re ` ( z / _i ) ) ) ) |
61 |
|
fvoveq1 |
|- ( z = ( x + ( _i x. y ) ) -> ( Re ` ( z / _i ) ) = ( Re ` ( ( x + ( _i x. y ) ) / _i ) ) ) |
62 |
21 22 60 61
|
fmpoco |
|- ( T. -> ( Im o. F ) = ( x e. RR , y e. RR |-> ( Re ` ( ( x + ( _i x. y ) ) / _i ) ) ) ) |
63 |
62
|
mptru |
|- ( Im o. F ) = ( x e. RR , y e. RR |-> ( Re ` ( ( x + ( _i x. y ) ) / _i ) ) ) |
64 |
|
df2ndres |
|- ( 2nd |` ( RR X. RR ) ) = ( x e. RR , y e. RR |-> y ) |
65 |
58 63 64
|
3eqtr4ri |
|- ( 2nd |` ( RR X. RR ) ) = ( Im o. F ) |
66 |
|
fo2nd |
|- 2nd : _V -onto-> _V |
67 |
|
fofn |
|- ( 2nd : _V -onto-> _V -> 2nd Fn _V ) |
68 |
66 67
|
ax-mp |
|- 2nd Fn _V |
69 |
|
xp2nd |
|- ( z e. ( RR X. RR ) -> ( 2nd ` z ) e. RR ) |
70 |
49 51
|
imsubd |
|- ( ( z e. ran F /\ u e. ran F ) -> ( Im ` ( z - u ) ) = ( ( Im ` z ) - ( Im ` u ) ) ) |
71 |
65 35 68 69 70
|
cnre2csqlem |
|- ( ( X e. ( RR X. RR ) /\ Y e. ( RR X. RR ) /\ D e. RR+ ) -> ( Y e. ( `' ( 2nd |` ( RR X. RR ) ) " ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) -> ( abs ` ( Im ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D ) ) |
72 |
53 71
|
anim12d |
|- ( ( X e. ( RR X. RR ) /\ Y e. ( RR X. RR ) /\ D e. RR+ ) -> ( ( Y e. ( `' ( 1st |` ( RR X. RR ) ) " ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) ) /\ Y e. ( `' ( 2nd |` ( RR X. RR ) ) " ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) ) -> ( ( abs ` ( Re ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D /\ ( abs ` ( Im ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D ) ) ) |
73 |
7 72
|
syl5bi |
|- ( ( X e. ( RR X. RR ) /\ Y e. ( RR X. RR ) /\ D e. RR+ ) -> ( Y e. ( ( `' ( 1st |` ( RR X. RR ) ) " ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) ) i^i ( `' ( 2nd |` ( RR X. RR ) ) " ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) ) -> ( ( abs ` ( Re ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D /\ ( abs ` ( Im ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D ) ) ) |
74 |
6 73
|
syl5bi |
|- ( ( X e. ( RR X. RR ) /\ Y e. ( RR X. RR ) /\ D e. RR+ ) -> ( Y e. ( ( ( ( 1st ` X ) - D ) (,) ( ( 1st ` X ) + D ) ) X. ( ( ( 2nd ` X ) - D ) (,) ( ( 2nd ` X ) + D ) ) ) -> ( ( abs ` ( Re ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D /\ ( abs ` ( Im ` ( ( F ` Y ) - ( F ` X ) ) ) ) < D ) ) ) |