Step |
Hyp |
Ref |
Expression |
1 |
|
i1ff |
|- ( F e. dom S.1 -> F : RR --> RR ) |
2 |
1
|
ffvelrnda |
|- ( ( F e. dom S.1 /\ x e. RR ) -> ( F ` x ) e. RR ) |
3 |
|
i1ff |
|- ( G e. dom S.1 -> G : RR --> RR ) |
4 |
3
|
ffvelrnda |
|- ( ( G e. dom S.1 /\ x e. RR ) -> ( G ` x ) e. RR ) |
5 |
|
absreim |
|- ( ( ( F ` x ) e. RR /\ ( G ` x ) e. RR ) -> ( abs ` ( ( F ` x ) + ( _i x. ( G ` x ) ) ) ) = ( sqrt ` ( ( ( F ` x ) ^ 2 ) + ( ( G ` x ) ^ 2 ) ) ) ) |
6 |
2 4 5
|
syl2an |
|- ( ( ( F e. dom S.1 /\ x e. RR ) /\ ( G e. dom S.1 /\ x e. RR ) ) -> ( abs ` ( ( F ` x ) + ( _i x. ( G ` x ) ) ) ) = ( sqrt ` ( ( ( F ` x ) ^ 2 ) + ( ( G ` x ) ^ 2 ) ) ) ) |
7 |
6
|
anandirs |
|- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( abs ` ( ( F ` x ) + ( _i x. ( G ` x ) ) ) ) = ( sqrt ` ( ( ( F ` x ) ^ 2 ) + ( ( G ` x ) ^ 2 ) ) ) ) |
8 |
2
|
recnd |
|- ( ( F e. dom S.1 /\ x e. RR ) -> ( F ` x ) e. CC ) |
9 |
8
|
sqvald |
|- ( ( F e. dom S.1 /\ x e. RR ) -> ( ( F ` x ) ^ 2 ) = ( ( F ` x ) x. ( F ` x ) ) ) |
10 |
4
|
recnd |
|- ( ( G e. dom S.1 /\ x e. RR ) -> ( G ` x ) e. CC ) |
11 |
10
|
sqvald |
|- ( ( G e. dom S.1 /\ x e. RR ) -> ( ( G ` x ) ^ 2 ) = ( ( G ` x ) x. ( G ` x ) ) ) |
12 |
9 11
|
oveqan12d |
|- ( ( ( F e. dom S.1 /\ x e. RR ) /\ ( G e. dom S.1 /\ x e. RR ) ) -> ( ( ( F ` x ) ^ 2 ) + ( ( G ` x ) ^ 2 ) ) = ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) ) |
13 |
12
|
anandirs |
|- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( ( ( F ` x ) ^ 2 ) + ( ( G ` x ) ^ 2 ) ) = ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) ) |
14 |
13
|
fveq2d |
|- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( sqrt ` ( ( ( F ` x ) ^ 2 ) + ( ( G ` x ) ^ 2 ) ) ) = ( sqrt ` ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) ) ) |
15 |
7 14
|
eqtrd |
|- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( abs ` ( ( F ` x ) + ( _i x. ( G ` x ) ) ) ) = ( sqrt ` ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) ) ) |
16 |
15
|
mpteq2dva |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( x e. RR |-> ( abs ` ( ( F ` x ) + ( _i x. ( G ` x ) ) ) ) ) = ( x e. RR |-> ( sqrt ` ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) ) ) ) |
17 |
|
ax-icn |
|- _i e. CC |
18 |
|
mulcl |
|- ( ( _i e. CC /\ ( G ` x ) e. CC ) -> ( _i x. ( G ` x ) ) e. CC ) |
19 |
17 10 18
|
sylancr |
|- ( ( G e. dom S.1 /\ x e. RR ) -> ( _i x. ( G ` x ) ) e. CC ) |
20 |
|
addcl |
|- ( ( ( F ` x ) e. CC /\ ( _i x. ( G ` x ) ) e. CC ) -> ( ( F ` x ) + ( _i x. ( G ` x ) ) ) e. CC ) |
21 |
8 19 20
|
syl2an |
|- ( ( ( F e. dom S.1 /\ x e. RR ) /\ ( G e. dom S.1 /\ x e. RR ) ) -> ( ( F ` x ) + ( _i x. ( G ` x ) ) ) e. CC ) |
22 |
21
|
anandirs |
|- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( ( F ` x ) + ( _i x. ( G ` x ) ) ) e. CC ) |
23 |
|
reex |
|- RR e. _V |
24 |
23
|
a1i |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> RR e. _V ) |
25 |
2
|
adantlr |
|- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( F ` x ) e. RR ) |
26 |
|
ovexd |
|- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( _i x. ( G ` x ) ) e. _V ) |
27 |
1
|
feqmptd |
|- ( F e. dom S.1 -> F = ( x e. RR |-> ( F ` x ) ) ) |
28 |
27
|
adantr |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> F = ( x e. RR |-> ( F ` x ) ) ) |
29 |
23
|
a1i |
|- ( G e. dom S.1 -> RR e. _V ) |
30 |
17
|
a1i |
|- ( ( G e. dom S.1 /\ x e. RR ) -> _i e. CC ) |
31 |
|
fconstmpt |
|- ( RR X. { _i } ) = ( x e. RR |-> _i ) |
32 |
31
|
a1i |
|- ( G e. dom S.1 -> ( RR X. { _i } ) = ( x e. RR |-> _i ) ) |
33 |
3
|
feqmptd |
|- ( G e. dom S.1 -> G = ( x e. RR |-> ( G ` x ) ) ) |
34 |
29 30 4 32 33
|
offval2 |
|- ( G e. dom S.1 -> ( ( RR X. { _i } ) oF x. G ) = ( x e. RR |-> ( _i x. ( G ` x ) ) ) ) |
35 |
34
|
adantl |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( ( RR X. { _i } ) oF x. G ) = ( x e. RR |-> ( _i x. ( G ` x ) ) ) ) |
36 |
24 25 26 28 35
|
offval2 |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( F oF + ( ( RR X. { _i } ) oF x. G ) ) = ( x e. RR |-> ( ( F ` x ) + ( _i x. ( G ` x ) ) ) ) ) |
37 |
|
absf |
|- abs : CC --> RR |
38 |
37
|
a1i |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> abs : CC --> RR ) |
39 |
38
|
feqmptd |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> abs = ( y e. CC |-> ( abs ` y ) ) ) |
40 |
|
fveq2 |
|- ( y = ( ( F ` x ) + ( _i x. ( G ` x ) ) ) -> ( abs ` y ) = ( abs ` ( ( F ` x ) + ( _i x. ( G ` x ) ) ) ) ) |
41 |
22 36 39 40
|
fmptco |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( abs o. ( F oF + ( ( RR X. { _i } ) oF x. G ) ) ) = ( x e. RR |-> ( abs ` ( ( F ` x ) + ( _i x. ( G ` x ) ) ) ) ) ) |
42 |
8 8
|
mulcld |
|- ( ( F e. dom S.1 /\ x e. RR ) -> ( ( F ` x ) x. ( F ` x ) ) e. CC ) |
43 |
10 10
|
mulcld |
|- ( ( G e. dom S.1 /\ x e. RR ) -> ( ( G ` x ) x. ( G ` x ) ) e. CC ) |
44 |
|
addcl |
|- ( ( ( ( F ` x ) x. ( F ` x ) ) e. CC /\ ( ( G ` x ) x. ( G ` x ) ) e. CC ) -> ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) e. CC ) |
45 |
42 43 44
|
syl2an |
|- ( ( ( F e. dom S.1 /\ x e. RR ) /\ ( G e. dom S.1 /\ x e. RR ) ) -> ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) e. CC ) |
46 |
45
|
anandirs |
|- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) e. CC ) |
47 |
42
|
adantlr |
|- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( ( F ` x ) x. ( F ` x ) ) e. CC ) |
48 |
43
|
adantll |
|- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. RR ) -> ( ( G ` x ) x. ( G ` x ) ) e. CC ) |
49 |
23
|
a1i |
|- ( F e. dom S.1 -> RR e. _V ) |
50 |
49 2 2 27 27
|
offval2 |
|- ( F e. dom S.1 -> ( F oF x. F ) = ( x e. RR |-> ( ( F ` x ) x. ( F ` x ) ) ) ) |
51 |
50
|
adantr |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( F oF x. F ) = ( x e. RR |-> ( ( F ` x ) x. ( F ` x ) ) ) ) |
52 |
29 4 4 33 33
|
offval2 |
|- ( G e. dom S.1 -> ( G oF x. G ) = ( x e. RR |-> ( ( G ` x ) x. ( G ` x ) ) ) ) |
53 |
52
|
adantl |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( G oF x. G ) = ( x e. RR |-> ( ( G ` x ) x. ( G ` x ) ) ) ) |
54 |
24 47 48 51 53
|
offval2 |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( ( F oF x. F ) oF + ( G oF x. G ) ) = ( x e. RR |-> ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) ) ) |
55 |
|
sqrtf |
|- sqrt : CC --> CC |
56 |
55
|
a1i |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> sqrt : CC --> CC ) |
57 |
56
|
feqmptd |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> sqrt = ( y e. CC |-> ( sqrt ` y ) ) ) |
58 |
|
fveq2 |
|- ( y = ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) -> ( sqrt ` y ) = ( sqrt ` ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) ) ) |
59 |
46 54 57 58
|
fmptco |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) = ( x e. RR |-> ( sqrt ` ( ( ( F ` x ) x. ( F ` x ) ) + ( ( G ` x ) x. ( G ` x ) ) ) ) ) ) |
60 |
16 41 59
|
3eqtr4d |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( abs o. ( F oF + ( ( RR X. { _i } ) oF x. G ) ) ) = ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) ) |
61 |
|
elrege0 |
|- ( x e. ( 0 [,) +oo ) <-> ( x e. RR /\ 0 <_ x ) ) |
62 |
|
resqrtcl |
|- ( ( x e. RR /\ 0 <_ x ) -> ( sqrt ` x ) e. RR ) |
63 |
61 62
|
sylbi |
|- ( x e. ( 0 [,) +oo ) -> ( sqrt ` x ) e. RR ) |
64 |
63
|
adantl |
|- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. ( 0 [,) +oo ) ) -> ( sqrt ` x ) e. RR ) |
65 |
|
id |
|- ( sqrt : CC --> CC -> sqrt : CC --> CC ) |
66 |
65
|
feqmptd |
|- ( sqrt : CC --> CC -> sqrt = ( x e. CC |-> ( sqrt ` x ) ) ) |
67 |
55 66
|
ax-mp |
|- sqrt = ( x e. CC |-> ( sqrt ` x ) ) |
68 |
67
|
reseq1i |
|- ( sqrt |` ( 0 [,) +oo ) ) = ( ( x e. CC |-> ( sqrt ` x ) ) |` ( 0 [,) +oo ) ) |
69 |
|
rge0ssre |
|- ( 0 [,) +oo ) C_ RR |
70 |
|
ax-resscn |
|- RR C_ CC |
71 |
69 70
|
sstri |
|- ( 0 [,) +oo ) C_ CC |
72 |
|
resmpt |
|- ( ( 0 [,) +oo ) C_ CC -> ( ( x e. CC |-> ( sqrt ` x ) ) |` ( 0 [,) +oo ) ) = ( x e. ( 0 [,) +oo ) |-> ( sqrt ` x ) ) ) |
73 |
71 72
|
ax-mp |
|- ( ( x e. CC |-> ( sqrt ` x ) ) |` ( 0 [,) +oo ) ) = ( x e. ( 0 [,) +oo ) |-> ( sqrt ` x ) ) |
74 |
68 73
|
eqtri |
|- ( sqrt |` ( 0 [,) +oo ) ) = ( x e. ( 0 [,) +oo ) |-> ( sqrt ` x ) ) |
75 |
64 74
|
fmptd |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( sqrt |` ( 0 [,) +oo ) ) : ( 0 [,) +oo ) --> RR ) |
76 |
|
ge0addcl |
|- ( ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) -> ( x + y ) e. ( 0 [,) +oo ) ) |
77 |
76
|
adantl |
|- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ ( x e. ( 0 [,) +oo ) /\ y e. ( 0 [,) +oo ) ) ) -> ( x + y ) e. ( 0 [,) +oo ) ) |
78 |
|
oveq12 |
|- ( ( z = F /\ z = F ) -> ( z oF x. z ) = ( F oF x. F ) ) |
79 |
78
|
anidms |
|- ( z = F -> ( z oF x. z ) = ( F oF x. F ) ) |
80 |
79
|
feq1d |
|- ( z = F -> ( ( z oF x. z ) : RR --> ( 0 [,) +oo ) <-> ( F oF x. F ) : RR --> ( 0 [,) +oo ) ) ) |
81 |
|
i1ff |
|- ( z e. dom S.1 -> z : RR --> RR ) |
82 |
81
|
ffvelrnda |
|- ( ( z e. dom S.1 /\ x e. RR ) -> ( z ` x ) e. RR ) |
83 |
82 82
|
remulcld |
|- ( ( z e. dom S.1 /\ x e. RR ) -> ( ( z ` x ) x. ( z ` x ) ) e. RR ) |
84 |
82
|
msqge0d |
|- ( ( z e. dom S.1 /\ x e. RR ) -> 0 <_ ( ( z ` x ) x. ( z ` x ) ) ) |
85 |
|
elrege0 |
|- ( ( ( z ` x ) x. ( z ` x ) ) e. ( 0 [,) +oo ) <-> ( ( ( z ` x ) x. ( z ` x ) ) e. RR /\ 0 <_ ( ( z ` x ) x. ( z ` x ) ) ) ) |
86 |
83 84 85
|
sylanbrc |
|- ( ( z e. dom S.1 /\ x e. RR ) -> ( ( z ` x ) x. ( z ` x ) ) e. ( 0 [,) +oo ) ) |
87 |
86
|
fmpttd |
|- ( z e. dom S.1 -> ( x e. RR |-> ( ( z ` x ) x. ( z ` x ) ) ) : RR --> ( 0 [,) +oo ) ) |
88 |
23
|
a1i |
|- ( z e. dom S.1 -> RR e. _V ) |
89 |
81
|
feqmptd |
|- ( z e. dom S.1 -> z = ( x e. RR |-> ( z ` x ) ) ) |
90 |
88 82 82 89 89
|
offval2 |
|- ( z e. dom S.1 -> ( z oF x. z ) = ( x e. RR |-> ( ( z ` x ) x. ( z ` x ) ) ) ) |
91 |
90
|
feq1d |
|- ( z e. dom S.1 -> ( ( z oF x. z ) : RR --> ( 0 [,) +oo ) <-> ( x e. RR |-> ( ( z ` x ) x. ( z ` x ) ) ) : RR --> ( 0 [,) +oo ) ) ) |
92 |
87 91
|
mpbird |
|- ( z e. dom S.1 -> ( z oF x. z ) : RR --> ( 0 [,) +oo ) ) |
93 |
80 92
|
vtoclga |
|- ( F e. dom S.1 -> ( F oF x. F ) : RR --> ( 0 [,) +oo ) ) |
94 |
93
|
adantr |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( F oF x. F ) : RR --> ( 0 [,) +oo ) ) |
95 |
|
oveq12 |
|- ( ( z = G /\ z = G ) -> ( z oF x. z ) = ( G oF x. G ) ) |
96 |
95
|
anidms |
|- ( z = G -> ( z oF x. z ) = ( G oF x. G ) ) |
97 |
96
|
feq1d |
|- ( z = G -> ( ( z oF x. z ) : RR --> ( 0 [,) +oo ) <-> ( G oF x. G ) : RR --> ( 0 [,) +oo ) ) ) |
98 |
97 92
|
vtoclga |
|- ( G e. dom S.1 -> ( G oF x. G ) : RR --> ( 0 [,) +oo ) ) |
99 |
98
|
adantl |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( G oF x. G ) : RR --> ( 0 [,) +oo ) ) |
100 |
|
inidm |
|- ( RR i^i RR ) = RR |
101 |
77 94 99 24 24 100
|
off |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( ( F oF x. F ) oF + ( G oF x. G ) ) : RR --> ( 0 [,) +oo ) ) |
102 |
|
fco2 |
|- ( ( ( sqrt |` ( 0 [,) +oo ) ) : ( 0 [,) +oo ) --> RR /\ ( ( F oF x. F ) oF + ( G oF x. G ) ) : RR --> ( 0 [,) +oo ) ) -> ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) : RR --> RR ) |
103 |
75 101 102
|
syl2anc |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) : RR --> RR ) |
104 |
|
rnco |
|- ran ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) = ran ( sqrt |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) |
105 |
|
ffn |
|- ( sqrt : CC --> CC -> sqrt Fn CC ) |
106 |
55 105
|
ax-mp |
|- sqrt Fn CC |
107 |
|
readdcl |
|- ( ( x e. RR /\ y e. RR ) -> ( x + y ) e. RR ) |
108 |
107
|
adantl |
|- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ ( x e. RR /\ y e. RR ) ) -> ( x + y ) e. RR ) |
109 |
|
remulcl |
|- ( ( x e. RR /\ y e. RR ) -> ( x x. y ) e. RR ) |
110 |
109
|
adantl |
|- ( ( F e. dom S.1 /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. RR ) |
111 |
110 1 1 49 49 100
|
off |
|- ( F e. dom S.1 -> ( F oF x. F ) : RR --> RR ) |
112 |
111
|
adantr |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( F oF x. F ) : RR --> RR ) |
113 |
109
|
adantl |
|- ( ( G e. dom S.1 /\ ( x e. RR /\ y e. RR ) ) -> ( x x. y ) e. RR ) |
114 |
113 3 3 29 29 100
|
off |
|- ( G e. dom S.1 -> ( G oF x. G ) : RR --> RR ) |
115 |
114
|
adantl |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( G oF x. G ) : RR --> RR ) |
116 |
108 112 115 24 24 100
|
off |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( ( F oF x. F ) oF + ( G oF x. G ) ) : RR --> RR ) |
117 |
116
|
frnd |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ran ( ( F oF x. F ) oF + ( G oF x. G ) ) C_ RR ) |
118 |
117 70
|
sstrdi |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ran ( ( F oF x. F ) oF + ( G oF x. G ) ) C_ CC ) |
119 |
|
fnssres |
|- ( ( sqrt Fn CC /\ ran ( ( F oF x. F ) oF + ( G oF x. G ) ) C_ CC ) -> ( sqrt |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) Fn ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) |
120 |
106 118 119
|
sylancr |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( sqrt |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) Fn ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) |
121 |
|
id |
|- ( F e. dom S.1 -> F e. dom S.1 ) |
122 |
121 121
|
i1fmul |
|- ( F e. dom S.1 -> ( F oF x. F ) e. dom S.1 ) |
123 |
122
|
adantr |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( F oF x. F ) e. dom S.1 ) |
124 |
|
id |
|- ( G e. dom S.1 -> G e. dom S.1 ) |
125 |
124 124
|
i1fmul |
|- ( G e. dom S.1 -> ( G oF x. G ) e. dom S.1 ) |
126 |
125
|
adantl |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( G oF x. G ) e. dom S.1 ) |
127 |
123 126
|
i1fadd |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( ( F oF x. F ) oF + ( G oF x. G ) ) e. dom S.1 ) |
128 |
|
i1frn |
|- ( ( ( F oF x. F ) oF + ( G oF x. G ) ) e. dom S.1 -> ran ( ( F oF x. F ) oF + ( G oF x. G ) ) e. Fin ) |
129 |
127 128
|
syl |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ran ( ( F oF x. F ) oF + ( G oF x. G ) ) e. Fin ) |
130 |
|
fnfi |
|- ( ( ( sqrt |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) Fn ran ( ( F oF x. F ) oF + ( G oF x. G ) ) /\ ran ( ( F oF x. F ) oF + ( G oF x. G ) ) e. Fin ) -> ( sqrt |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) e. Fin ) |
131 |
120 129 130
|
syl2anc |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( sqrt |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) e. Fin ) |
132 |
|
rnfi |
|- ( ( sqrt |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) e. Fin -> ran ( sqrt |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) e. Fin ) |
133 |
131 132
|
syl |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ran ( sqrt |` ran ( ( F oF x. F ) oF + ( G oF x. G ) ) ) e. Fin ) |
134 |
104 133
|
eqeltrid |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ran ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) e. Fin ) |
135 |
|
cnvco |
|- `' ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) = ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) o. `' sqrt ) |
136 |
135
|
imaeq1i |
|- ( `' ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) " { x } ) = ( ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) o. `' sqrt ) " { x } ) |
137 |
|
imaco |
|- ( ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) o. `' sqrt ) " { x } ) = ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) " ( `' sqrt " { x } ) ) |
138 |
136 137
|
eqtri |
|- ( `' ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) " { x } ) = ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) " ( `' sqrt " { x } ) ) |
139 |
|
i1fima |
|- ( ( ( F oF x. F ) oF + ( G oF x. G ) ) e. dom S.1 -> ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) " ( `' sqrt " { x } ) ) e. dom vol ) |
140 |
127 139
|
syl |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) " ( `' sqrt " { x } ) ) e. dom vol ) |
141 |
138 140
|
eqeltrid |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( `' ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) " { x } ) e. dom vol ) |
142 |
141
|
adantr |
|- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. ( ran ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) \ { 0 } ) ) -> ( `' ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) " { x } ) e. dom vol ) |
143 |
138
|
fveq2i |
|- ( vol ` ( `' ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) " { x } ) ) = ( vol ` ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) " ( `' sqrt " { x } ) ) ) |
144 |
|
eldifsni |
|- ( x e. ( ran ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) \ { 0 } ) -> x =/= 0 ) |
145 |
|
c0ex |
|- 0 e. _V |
146 |
145
|
elsn |
|- ( 0 e. { x } <-> 0 = x ) |
147 |
|
eqcom |
|- ( 0 = x <-> x = 0 ) |
148 |
146 147
|
bitri |
|- ( 0 e. { x } <-> x = 0 ) |
149 |
148
|
necon3bbii |
|- ( -. 0 e. { x } <-> x =/= 0 ) |
150 |
|
sqrt0 |
|- ( sqrt ` 0 ) = 0 |
151 |
150
|
eleq1i |
|- ( ( sqrt ` 0 ) e. { x } <-> 0 e. { x } ) |
152 |
149 151
|
xchnxbir |
|- ( -. ( sqrt ` 0 ) e. { x } <-> x =/= 0 ) |
153 |
144 152
|
sylibr |
|- ( x e. ( ran ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) \ { 0 } ) -> -. ( sqrt ` 0 ) e. { x } ) |
154 |
153
|
olcd |
|- ( x e. ( ran ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) \ { 0 } ) -> ( -. 0 e. CC \/ -. ( sqrt ` 0 ) e. { x } ) ) |
155 |
|
ianor |
|- ( -. ( 0 e. CC /\ ( sqrt ` 0 ) e. { x } ) <-> ( -. 0 e. CC \/ -. ( sqrt ` 0 ) e. { x } ) ) |
156 |
|
elpreima |
|- ( sqrt Fn CC -> ( 0 e. ( `' sqrt " { x } ) <-> ( 0 e. CC /\ ( sqrt ` 0 ) e. { x } ) ) ) |
157 |
55 105 156
|
mp2b |
|- ( 0 e. ( `' sqrt " { x } ) <-> ( 0 e. CC /\ ( sqrt ` 0 ) e. { x } ) ) |
158 |
155 157
|
xchnxbir |
|- ( -. 0 e. ( `' sqrt " { x } ) <-> ( -. 0 e. CC \/ -. ( sqrt ` 0 ) e. { x } ) ) |
159 |
154 158
|
sylibr |
|- ( x e. ( ran ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) \ { 0 } ) -> -. 0 e. ( `' sqrt " { x } ) ) |
160 |
|
i1fima2 |
|- ( ( ( ( F oF x. F ) oF + ( G oF x. G ) ) e. dom S.1 /\ -. 0 e. ( `' sqrt " { x } ) ) -> ( vol ` ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) " ( `' sqrt " { x } ) ) ) e. RR ) |
161 |
127 159 160
|
syl2an |
|- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. ( ran ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) \ { 0 } ) ) -> ( vol ` ( `' ( ( F oF x. F ) oF + ( G oF x. G ) ) " ( `' sqrt " { x } ) ) ) e. RR ) |
162 |
143 161
|
eqeltrid |
|- ( ( ( F e. dom S.1 /\ G e. dom S.1 ) /\ x e. ( ran ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) \ { 0 } ) ) -> ( vol ` ( `' ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) " { x } ) ) e. RR ) |
163 |
103 134 142 162
|
i1fd |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( sqrt o. ( ( F oF x. F ) oF + ( G oF x. G ) ) ) e. dom S.1 ) |
164 |
60 163
|
eqeltrd |
|- ( ( F e. dom S.1 /\ G e. dom S.1 ) -> ( abs o. ( F oF + ( ( RR X. { _i } ) oF x. G ) ) ) e. dom S.1 ) |