Step |
Hyp |
Ref |
Expression |
1 |
|
ccfldextrr |
|- CCfld /FldExt RRfld |
2 |
|
extdgval |
|- ( CCfld /FldExt RRfld -> ( CCfld [:] RRfld ) = ( dim ` ( ( subringAlg ` CCfld ) ` ( Base ` RRfld ) ) ) ) |
3 |
1 2
|
ax-mp |
|- ( CCfld [:] RRfld ) = ( dim ` ( ( subringAlg ` CCfld ) ` ( Base ` RRfld ) ) ) |
4 |
|
rebase |
|- RR = ( Base ` RRfld ) |
5 |
4
|
fveq2i |
|- ( ( subringAlg ` CCfld ) ` RR ) = ( ( subringAlg ` CCfld ) ` ( Base ` RRfld ) ) |
6 |
5
|
fveq2i |
|- ( dim ` ( ( subringAlg ` CCfld ) ` RR ) ) = ( dim ` ( ( subringAlg ` CCfld ) ` ( Base ` RRfld ) ) ) |
7 |
|
ccfldsrarelvec |
|- ( ( subringAlg ` CCfld ) ` RR ) e. LVec |
8 |
|
df-pr |
|- { 1 , _i } = ( { 1 } u. { _i } ) |
9 |
|
eqid |
|- ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) = ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) |
10 |
|
eqidd |
|- ( T. -> ( ( subringAlg ` CCfld ) ` RR ) = ( ( subringAlg ` CCfld ) ` RR ) ) |
11 |
|
cnfld0 |
|- 0 = ( 0g ` CCfld ) |
12 |
11
|
a1i |
|- ( T. -> 0 = ( 0g ` CCfld ) ) |
13 |
|
ax-resscn |
|- RR C_ CC |
14 |
|
cnfldbas |
|- CC = ( Base ` CCfld ) |
15 |
13 14
|
sseqtri |
|- RR C_ ( Base ` CCfld ) |
16 |
15
|
a1i |
|- ( T. -> RR C_ ( Base ` CCfld ) ) |
17 |
10 12 16
|
sralmod0 |
|- ( T. -> 0 = ( 0g ` ( ( subringAlg ` CCfld ) ` RR ) ) ) |
18 |
17
|
mptru |
|- 0 = ( 0g ` ( ( subringAlg ` CCfld ) ` RR ) ) |
19 |
7
|
a1i |
|- ( T. -> ( ( subringAlg ` CCfld ) ` RR ) e. LVec ) |
20 |
|
ax-1cn |
|- 1 e. CC |
21 |
|
ax-1ne0 |
|- 1 =/= 0 |
22 |
10 16
|
srabase |
|- ( T. -> ( Base ` CCfld ) = ( Base ` ( ( subringAlg ` CCfld ) ` RR ) ) ) |
23 |
22
|
mptru |
|- ( Base ` CCfld ) = ( Base ` ( ( subringAlg ` CCfld ) ` RR ) ) |
24 |
14 23
|
eqtri |
|- CC = ( Base ` ( ( subringAlg ` CCfld ) ` RR ) ) |
25 |
24 18
|
lindssn |
|- ( ( ( ( subringAlg ` CCfld ) ` RR ) e. LVec /\ 1 e. CC /\ 1 =/= 0 ) -> { 1 } e. ( LIndS ` ( ( subringAlg ` CCfld ) ` RR ) ) ) |
26 |
7 20 21 25
|
mp3an |
|- { 1 } e. ( LIndS ` ( ( subringAlg ` CCfld ) ` RR ) ) |
27 |
26
|
a1i |
|- ( T. -> { 1 } e. ( LIndS ` ( ( subringAlg ` CCfld ) ` RR ) ) ) |
28 |
|
ax-icn |
|- _i e. CC |
29 |
|
ine0 |
|- _i =/= 0 |
30 |
24 18
|
lindssn |
|- ( ( ( ( subringAlg ` CCfld ) ` RR ) e. LVec /\ _i e. CC /\ _i =/= 0 ) -> { _i } e. ( LIndS ` ( ( subringAlg ` CCfld ) ` RR ) ) ) |
31 |
7 28 29 30
|
mp3an |
|- { _i } e. ( LIndS ` ( ( subringAlg ` CCfld ) ` RR ) ) |
32 |
31
|
a1i |
|- ( T. -> { _i } e. ( LIndS ` ( ( subringAlg ` CCfld ) ` RR ) ) ) |
33 |
|
lveclmod |
|- ( ( ( subringAlg ` CCfld ) ` RR ) e. LVec -> ( ( subringAlg ` CCfld ) ` RR ) e. LMod ) |
34 |
7 33
|
ax-mp |
|- ( ( subringAlg ` CCfld ) ` RR ) e. LMod |
35 |
|
df-refld |
|- RRfld = ( CCfld |`s RR ) |
36 |
10 16
|
srasca |
|- ( T. -> ( CCfld |`s RR ) = ( Scalar ` ( ( subringAlg ` CCfld ) ` RR ) ) ) |
37 |
36
|
mptru |
|- ( CCfld |`s RR ) = ( Scalar ` ( ( subringAlg ` CCfld ) ` RR ) ) |
38 |
35 37
|
eqtri |
|- RRfld = ( Scalar ` ( ( subringAlg ` CCfld ) ` RR ) ) |
39 |
|
cnfldmul |
|- x. = ( .r ` CCfld ) |
40 |
10 16
|
sravsca |
|- ( T. -> ( .r ` CCfld ) = ( .s ` ( ( subringAlg ` CCfld ) ` RR ) ) ) |
41 |
40
|
mptru |
|- ( .r ` CCfld ) = ( .s ` ( ( subringAlg ` CCfld ) ` RR ) ) |
42 |
39 41
|
eqtri |
|- x. = ( .s ` ( ( subringAlg ` CCfld ) ` RR ) ) |
43 |
38 4 24 42 9
|
lspsnel |
|- ( ( ( ( subringAlg ` CCfld ) ` RR ) e. LMod /\ 1 e. CC ) -> ( z e. ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { 1 } ) <-> E. x e. RR z = ( x x. 1 ) ) ) |
44 |
34 20 43
|
mp2an |
|- ( z e. ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { 1 } ) <-> E. x e. RR z = ( x x. 1 ) ) |
45 |
38 4 24 42 9
|
lspsnel |
|- ( ( ( ( subringAlg ` CCfld ) ` RR ) e. LMod /\ _i e. CC ) -> ( z e. ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { _i } ) <-> E. y e. RR z = ( y x. _i ) ) ) |
46 |
34 28 45
|
mp2an |
|- ( z e. ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { _i } ) <-> E. y e. RR z = ( y x. _i ) ) |
47 |
44 46
|
anbi12i |
|- ( ( z e. ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { 1 } ) /\ z e. ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { _i } ) ) <-> ( E. x e. RR z = ( x x. 1 ) /\ E. y e. RR z = ( y x. _i ) ) ) |
48 |
|
reeanv |
|- ( E. x e. RR E. y e. RR ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) <-> ( E. x e. RR z = ( x x. 1 ) /\ E. y e. RR z = ( y x. _i ) ) ) |
49 |
|
simprl |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> z = ( x x. 1 ) ) |
50 |
|
simpll |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> x e. RR ) |
51 |
50
|
recnd |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> x e. CC ) |
52 |
51
|
mulid1d |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> ( x x. 1 ) = x ) |
53 |
49 52
|
eqtrd |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> z = x ) |
54 |
53
|
negeqd |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> -u z = -u x ) |
55 |
|
simprr |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> z = ( y x. _i ) ) |
56 |
|
simplr |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> y e. RR ) |
57 |
56
|
recnd |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> y e. CC ) |
58 |
28
|
a1i |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> _i e. CC ) |
59 |
57 58
|
mulcomd |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> ( y x. _i ) = ( _i x. y ) ) |
60 |
55 59
|
eqtrd |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> z = ( _i x. y ) ) |
61 |
54 60
|
oveq12d |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> ( -u z + z ) = ( -u x + ( _i x. y ) ) ) |
62 |
53 51
|
eqeltrd |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> z e. CC ) |
63 |
62
|
subidd |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> ( z - z ) = 0 ) |
64 |
63
|
negeqd |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> -u ( z - z ) = -u 0 ) |
65 |
62 62
|
negsubdid |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> -u ( z - z ) = ( -u z + z ) ) |
66 |
|
neg0 |
|- -u 0 = 0 |
67 |
66
|
a1i |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> -u 0 = 0 ) |
68 |
64 65 67
|
3eqtr3d |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> ( -u z + z ) = 0 ) |
69 |
61 68
|
eqtr3d |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> ( -u x + ( _i x. y ) ) = 0 ) |
70 |
50
|
renegcld |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> -u x e. RR ) |
71 |
|
creq0 |
|- ( ( -u x e. RR /\ y e. RR ) -> ( ( -u x = 0 /\ y = 0 ) <-> ( -u x + ( _i x. y ) ) = 0 ) ) |
72 |
70 56 71
|
syl2anc |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> ( ( -u x = 0 /\ y = 0 ) <-> ( -u x + ( _i x. y ) ) = 0 ) ) |
73 |
69 72
|
mpbird |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> ( -u x = 0 /\ y = 0 ) ) |
74 |
73
|
simpld |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> -u x = 0 ) |
75 |
51 74
|
negcon1ad |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> -u 0 = x ) |
76 |
53 75 67
|
3eqtr2d |
|- ( ( ( x e. RR /\ y e. RR ) /\ ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) -> z = 0 ) |
77 |
76
|
ex |
|- ( ( x e. RR /\ y e. RR ) -> ( ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) -> z = 0 ) ) |
78 |
77
|
rexlimivv |
|- ( E. x e. RR E. y e. RR ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) -> z = 0 ) |
79 |
|
0red |
|- ( z = 0 -> 0 e. RR ) |
80 |
|
simpr |
|- ( ( z = 0 /\ x = 0 ) -> x = 0 ) |
81 |
80
|
oveq1d |
|- ( ( z = 0 /\ x = 0 ) -> ( x x. 1 ) = ( 0 x. 1 ) ) |
82 |
81
|
eqeq2d |
|- ( ( z = 0 /\ x = 0 ) -> ( z = ( x x. 1 ) <-> z = ( 0 x. 1 ) ) ) |
83 |
82
|
anbi1d |
|- ( ( z = 0 /\ x = 0 ) -> ( ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) <-> ( z = ( 0 x. 1 ) /\ z = ( y x. _i ) ) ) ) |
84 |
83
|
rexbidv |
|- ( ( z = 0 /\ x = 0 ) -> ( E. y e. RR ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) <-> E. y e. RR ( z = ( 0 x. 1 ) /\ z = ( y x. _i ) ) ) ) |
85 |
|
simpr |
|- ( ( z = 0 /\ y = 0 ) -> y = 0 ) |
86 |
85
|
oveq1d |
|- ( ( z = 0 /\ y = 0 ) -> ( y x. _i ) = ( 0 x. _i ) ) |
87 |
86
|
eqeq2d |
|- ( ( z = 0 /\ y = 0 ) -> ( z = ( y x. _i ) <-> z = ( 0 x. _i ) ) ) |
88 |
87
|
anbi2d |
|- ( ( z = 0 /\ y = 0 ) -> ( ( z = ( 0 x. 1 ) /\ z = ( y x. _i ) ) <-> ( z = ( 0 x. 1 ) /\ z = ( 0 x. _i ) ) ) ) |
89 |
20
|
mul02i |
|- ( 0 x. 1 ) = 0 |
90 |
89
|
eqeq2i |
|- ( z = ( 0 x. 1 ) <-> z = 0 ) |
91 |
90
|
biimpri |
|- ( z = 0 -> z = ( 0 x. 1 ) ) |
92 |
28
|
mul02i |
|- ( 0 x. _i ) = 0 |
93 |
92
|
eqeq2i |
|- ( z = ( 0 x. _i ) <-> z = 0 ) |
94 |
93
|
biimpri |
|- ( z = 0 -> z = ( 0 x. _i ) ) |
95 |
91 94
|
jca |
|- ( z = 0 -> ( z = ( 0 x. 1 ) /\ z = ( 0 x. _i ) ) ) |
96 |
79 88 95
|
rspcedvd |
|- ( z = 0 -> E. y e. RR ( z = ( 0 x. 1 ) /\ z = ( y x. _i ) ) ) |
97 |
79 84 96
|
rspcedvd |
|- ( z = 0 -> E. x e. RR E. y e. RR ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) ) |
98 |
78 97
|
impbii |
|- ( E. x e. RR E. y e. RR ( z = ( x x. 1 ) /\ z = ( y x. _i ) ) <-> z = 0 ) |
99 |
47 48 98
|
3bitr2i |
|- ( ( z e. ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { 1 } ) /\ z e. ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { _i } ) ) <-> z = 0 ) |
100 |
|
elin |
|- ( z e. ( ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { 1 } ) i^i ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { _i } ) ) <-> ( z e. ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { 1 } ) /\ z e. ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { _i } ) ) ) |
101 |
|
velsn |
|- ( z e. { 0 } <-> z = 0 ) |
102 |
99 100 101
|
3bitr4i |
|- ( z e. ( ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { 1 } ) i^i ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { _i } ) ) <-> z e. { 0 } ) |
103 |
102
|
eqriv |
|- ( ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { 1 } ) i^i ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { _i } ) ) = { 0 } |
104 |
103
|
a1i |
|- ( T. -> ( ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { 1 } ) i^i ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { _i } ) ) = { 0 } ) |
105 |
9 18 19 27 32 104
|
lindsun |
|- ( T. -> ( { 1 } u. { _i } ) e. ( LIndS ` ( ( subringAlg ` CCfld ) ` RR ) ) ) |
106 |
105
|
mptru |
|- ( { 1 } u. { _i } ) e. ( LIndS ` ( ( subringAlg ` CCfld ) ` RR ) ) |
107 |
8 106
|
eqeltri |
|- { 1 , _i } e. ( LIndS ` ( ( subringAlg ` CCfld ) ` RR ) ) |
108 |
|
cnfldadd |
|- + = ( +g ` CCfld ) |
109 |
10 16
|
sraaddg |
|- ( T. -> ( +g ` CCfld ) = ( +g ` ( ( subringAlg ` CCfld ) ` RR ) ) ) |
110 |
109
|
mptru |
|- ( +g ` CCfld ) = ( +g ` ( ( subringAlg ` CCfld ) ` RR ) ) |
111 |
108 110
|
eqtri |
|- + = ( +g ` ( ( subringAlg ` CCfld ) ` RR ) ) |
112 |
34
|
a1i |
|- ( T. -> ( ( subringAlg ` CCfld ) ` RR ) e. LMod ) |
113 |
|
1cnd |
|- ( T. -> 1 e. CC ) |
114 |
28
|
a1i |
|- ( T. -> _i e. CC ) |
115 |
24 111 38 4 42 9 112 113 114
|
lspprel |
|- ( T. -> ( z e. ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { 1 , _i } ) <-> E. x e. RR E. y e. RR z = ( ( x x. 1 ) + ( y x. _i ) ) ) ) |
116 |
115
|
mptru |
|- ( z e. ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { 1 , _i } ) <-> E. x e. RR E. y e. RR z = ( ( x x. 1 ) + ( y x. _i ) ) ) |
117 |
|
simpl |
|- ( ( x e. RR /\ y e. RR ) -> x e. RR ) |
118 |
117
|
recnd |
|- ( ( x e. RR /\ y e. RR ) -> x e. CC ) |
119 |
|
1cnd |
|- ( ( x e. RR /\ y e. RR ) -> 1 e. CC ) |
120 |
118 119
|
mulcld |
|- ( ( x e. RR /\ y e. RR ) -> ( x x. 1 ) e. CC ) |
121 |
|
simpr |
|- ( ( x e. RR /\ y e. RR ) -> y e. RR ) |
122 |
121
|
recnd |
|- ( ( x e. RR /\ y e. RR ) -> y e. CC ) |
123 |
28
|
a1i |
|- ( ( x e. RR /\ y e. RR ) -> _i e. CC ) |
124 |
122 123
|
mulcld |
|- ( ( x e. RR /\ y e. RR ) -> ( y x. _i ) e. CC ) |
125 |
120 124
|
addcld |
|- ( ( x e. RR /\ y e. RR ) -> ( ( x x. 1 ) + ( y x. _i ) ) e. CC ) |
126 |
|
eleq1 |
|- ( z = ( ( x x. 1 ) + ( y x. _i ) ) -> ( z e. CC <-> ( ( x x. 1 ) + ( y x. _i ) ) e. CC ) ) |
127 |
125 126
|
syl5ibrcom |
|- ( ( x e. RR /\ y e. RR ) -> ( z = ( ( x x. 1 ) + ( y x. _i ) ) -> z e. CC ) ) |
128 |
127
|
rexlimivv |
|- ( E. x e. RR E. y e. RR z = ( ( x x. 1 ) + ( y x. _i ) ) -> z e. CC ) |
129 |
|
recl |
|- ( z e. CC -> ( Re ` z ) e. RR ) |
130 |
|
simpr |
|- ( ( z e. CC /\ x = ( Re ` z ) ) -> x = ( Re ` z ) ) |
131 |
130
|
oveq1d |
|- ( ( z e. CC /\ x = ( Re ` z ) ) -> ( x x. 1 ) = ( ( Re ` z ) x. 1 ) ) |
132 |
131
|
oveq1d |
|- ( ( z e. CC /\ x = ( Re ` z ) ) -> ( ( x x. 1 ) + ( y x. _i ) ) = ( ( ( Re ` z ) x. 1 ) + ( y x. _i ) ) ) |
133 |
132
|
eqeq2d |
|- ( ( z e. CC /\ x = ( Re ` z ) ) -> ( z = ( ( x x. 1 ) + ( y x. _i ) ) <-> z = ( ( ( Re ` z ) x. 1 ) + ( y x. _i ) ) ) ) |
134 |
133
|
rexbidv |
|- ( ( z e. CC /\ x = ( Re ` z ) ) -> ( E. y e. RR z = ( ( x x. 1 ) + ( y x. _i ) ) <-> E. y e. RR z = ( ( ( Re ` z ) x. 1 ) + ( y x. _i ) ) ) ) |
135 |
|
imcl |
|- ( z e. CC -> ( Im ` z ) e. RR ) |
136 |
|
simpr |
|- ( ( z e. CC /\ y = ( Im ` z ) ) -> y = ( Im ` z ) ) |
137 |
136
|
oveq1d |
|- ( ( z e. CC /\ y = ( Im ` z ) ) -> ( y x. _i ) = ( ( Im ` z ) x. _i ) ) |
138 |
137
|
oveq2d |
|- ( ( z e. CC /\ y = ( Im ` z ) ) -> ( ( ( Re ` z ) x. 1 ) + ( y x. _i ) ) = ( ( ( Re ` z ) x. 1 ) + ( ( Im ` z ) x. _i ) ) ) |
139 |
138
|
eqeq2d |
|- ( ( z e. CC /\ y = ( Im ` z ) ) -> ( z = ( ( ( Re ` z ) x. 1 ) + ( y x. _i ) ) <-> z = ( ( ( Re ` z ) x. 1 ) + ( ( Im ` z ) x. _i ) ) ) ) |
140 |
|
replim |
|- ( z e. CC -> z = ( ( Re ` z ) + ( _i x. ( Im ` z ) ) ) ) |
141 |
129
|
recnd |
|- ( z e. CC -> ( Re ` z ) e. CC ) |
142 |
141
|
mulid1d |
|- ( z e. CC -> ( ( Re ` z ) x. 1 ) = ( Re ` z ) ) |
143 |
135
|
recnd |
|- ( z e. CC -> ( Im ` z ) e. CC ) |
144 |
28
|
a1i |
|- ( z e. CC -> _i e. CC ) |
145 |
143 144
|
mulcomd |
|- ( z e. CC -> ( ( Im ` z ) x. _i ) = ( _i x. ( Im ` z ) ) ) |
146 |
142 145
|
oveq12d |
|- ( z e. CC -> ( ( ( Re ` z ) x. 1 ) + ( ( Im ` z ) x. _i ) ) = ( ( Re ` z ) + ( _i x. ( Im ` z ) ) ) ) |
147 |
140 146
|
eqtr4d |
|- ( z e. CC -> z = ( ( ( Re ` z ) x. 1 ) + ( ( Im ` z ) x. _i ) ) ) |
148 |
135 139 147
|
rspcedvd |
|- ( z e. CC -> E. y e. RR z = ( ( ( Re ` z ) x. 1 ) + ( y x. _i ) ) ) |
149 |
129 134 148
|
rspcedvd |
|- ( z e. CC -> E. x e. RR E. y e. RR z = ( ( x x. 1 ) + ( y x. _i ) ) ) |
150 |
128 149
|
impbii |
|- ( E. x e. RR E. y e. RR z = ( ( x x. 1 ) + ( y x. _i ) ) <-> z e. CC ) |
151 |
116 150
|
bitri |
|- ( z e. ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { 1 , _i } ) <-> z e. CC ) |
152 |
151
|
eqriv |
|- ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { 1 , _i } ) = CC |
153 |
|
eqid |
|- ( LBasis ` ( ( subringAlg ` CCfld ) ` RR ) ) = ( LBasis ` ( ( subringAlg ` CCfld ) ` RR ) ) |
154 |
24 153 9
|
islbs4 |
|- ( { 1 , _i } e. ( LBasis ` ( ( subringAlg ` CCfld ) ` RR ) ) <-> ( { 1 , _i } e. ( LIndS ` ( ( subringAlg ` CCfld ) ` RR ) ) /\ ( ( LSpan ` ( ( subringAlg ` CCfld ) ` RR ) ) ` { 1 , _i } ) = CC ) ) |
155 |
107 152 154
|
mpbir2an |
|- { 1 , _i } e. ( LBasis ` ( ( subringAlg ` CCfld ) ` RR ) ) |
156 |
153
|
dimval |
|- ( ( ( ( subringAlg ` CCfld ) ` RR ) e. LVec /\ { 1 , _i } e. ( LBasis ` ( ( subringAlg ` CCfld ) ` RR ) ) ) -> ( dim ` ( ( subringAlg ` CCfld ) ` RR ) ) = ( # ` { 1 , _i } ) ) |
157 |
7 155 156
|
mp2an |
|- ( dim ` ( ( subringAlg ` CCfld ) ` RR ) ) = ( # ` { 1 , _i } ) |
158 |
|
1nei |
|- 1 =/= _i |
159 |
|
hashprg |
|- ( ( 1 e. CC /\ _i e. CC ) -> ( 1 =/= _i <-> ( # ` { 1 , _i } ) = 2 ) ) |
160 |
20 28 159
|
mp2an |
|- ( 1 =/= _i <-> ( # ` { 1 , _i } ) = 2 ) |
161 |
158 160
|
mpbi |
|- ( # ` { 1 , _i } ) = 2 |
162 |
157 161
|
eqtri |
|- ( dim ` ( ( subringAlg ` CCfld ) ` RR ) ) = 2 |
163 |
3 6 162
|
3eqtr2i |
|- ( CCfld [:] RRfld ) = 2 |