Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  refsum2cnlem1 Unicode version

Theorem refsum2cnlem1 27811
Description: This is the core Lemma for refsum2cn 27812: the sum of two continuous real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
refsum2cnlem1.1  |-  F/_ x A
refsum2cnlem1.2  |-  F/_ x F
refsum2cnlem1.3  |-  F/_ x G
refsum2cnlem1.4  |-  F/ x ph
refsum2cnlem1.5  |-  A  =  ( k  e.  {
1 ,  2 } 
|->  if ( k  =  1 ,  F ,  G ) )
refsum2cnlem1.6  |-  K  =  ( topGen `  ran  (,) )
refsum2cnlem1.7  |-  ( ph  ->  J  e.  (TopOn `  X ) )
refsum2cnlem1.8  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
refsum2cnlem1.9  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
Assertion
Ref Expression
refsum2cnlem1  |-  ( ph  ->  ( x  e.  X  |->  ( ( F `  x )  +  ( G `  x ) ) )  e.  ( J  Cn  K ) )
Distinct variable groups:    x, k, J    k, F    k, G    k, K, x    k, X, x    ph, k
Allowed substitution hints:    ph( x)    A( x, k)    F( x)    G( x)

Proof of Theorem refsum2cnlem1
StepHypRef Expression
1 refsum2cnlem1.4 . . 3  |-  F/ x ph
2 refsum2cnlem1.5 . . . . . . . . 9  |-  A  =  ( k  e.  {
1 ,  2 } 
|->  if ( k  =  1 ,  F ,  G ) )
3 nfmpt1 4125 . . . . . . . . 9  |-  F/_ k
( k  e.  {
1 ,  2 } 
|->  if ( k  =  1 ,  F ,  G ) )
42, 3nfcxfr 2429 . . . . . . . 8  |-  F/_ k A
5 nfcv 2432 . . . . . . . 8  |-  F/_ k
1
64, 5nffv 5548 . . . . . . 7  |-  F/_ k
( A `  1
)
7 nfcv 2432 . . . . . . 7  |-  F/_ k
x
86, 7nffv 5548 . . . . . 6  |-  F/_ k
( ( A ` 
1 ) `  x
)
98a1i 10 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F/_ k
( ( A ` 
1 ) `  x
) )
10 nfcv 2432 . . . . . . . 8  |-  F/_ k
2
114, 10nffv 5548 . . . . . . 7  |-  F/_ k
( A `  2
)
1211, 7nffv 5548 . . . . . 6  |-  F/_ k
( ( A ` 
2 ) `  x
)
1312a1i 10 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F/_ k
( ( A ` 
2 ) `  x
) )
14 ax-1cn 8811 . . . . . 6  |-  1  e.  CC
1514a1i 10 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  1  e.  CC )
16 2cn 9832 . . . . . 6  |-  2  e.  CC
1716a1i 10 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  2  e.  CC )
18 refsum2cnlem1.8 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
19 eqid 2296 . . . . . . . . . . . 12  |-  U. J  =  U. J
20 eqid 2296 . . . . . . . . . . . 12  |-  U. K  =  U. K
2119, 20cnf 16992 . . . . . . . . . . 11  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
2218, 21syl 15 . . . . . . . . . 10  |-  ( ph  ->  F : U. J --> U. K )
23 refsum2cnlem1.7 . . . . . . . . . . . . . 14  |-  ( ph  ->  J  e.  (TopOn `  X ) )
24 toponuni 16681 . . . . . . . . . . . . . 14  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
2523, 24syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  X  =  U. J
)
2625eqcomd 2301 . . . . . . . . . . . 12  |-  ( ph  ->  U. J  =  X )
27 refsum2cnlem1.6 . . . . . . . . . . . . . . 15  |-  K  =  ( topGen `  ran  (,) )
2827unieqi 3853 . . . . . . . . . . . . . 14  |-  U. K  =  U. ( topGen `  ran  (,) )
29 uniretop 18287 . . . . . . . . . . . . . 14  |-  RR  =  U. ( topGen `  ran  (,) )
3028, 29eqtr4i 2319 . . . . . . . . . . . . 13  |-  U. K  =  RR
3130a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  U. K  =  RR )
3226, 31feq23d 5402 . . . . . . . . . . 11  |-  ( ph  ->  ( F : U. J
--> U. K  <->  F : X
--> RR ) )
3332biimpd 198 . . . . . . . . . 10  |-  ( ph  ->  ( F : U. J
--> U. K  ->  F : X --> RR ) )
3422, 33mpd 14 . . . . . . . . 9  |-  ( ph  ->  F : X --> RR )
3534adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  F : X --> RR )
36 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
3735, 36jca 518 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( F : X --> RR  /\  x  e.  X )
)
38 ffvelrn 5679 . . . . . . 7  |-  ( ( F : X --> RR  /\  x  e.  X )  ->  ( F `  x
)  e.  RR )
39 recn 8843 . . . . . . 7  |-  ( ( F `  x )  e.  RR  ->  ( F `  x )  e.  CC )
4037, 38, 393syl 18 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  CC )
41 1ex 8849 . . . . . . . . . . . . . 14  |-  1  e.  _V
4241prid1 3747 . . . . . . . . . . . . 13  |-  1  e.  { 1 ,  2 }
4342a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  { 1 ,  2 } )
44 eqid 2296 . . . . . . . . . . . . . 14  |-  1  =  1
45 iftrue 3584 . . . . . . . . . . . . . 14  |-  ( 1  =  1  ->  if ( 1  =  1 ,  F ,  G
)  =  F )
4644, 45ax-mp 8 . . . . . . . . . . . . 13  |-  if ( 1  =  1 ,  F ,  G )  =  F
4746, 18syl5eqel 2380 . . . . . . . . . . . 12  |-  ( ph  ->  if ( 1  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
4843, 47jca 518 . . . . . . . . . . 11  |-  ( ph  ->  ( 1  e.  {
1 ,  2 }  /\  if ( 1  =  1 ,  F ,  G )  e.  ( J  Cn  K ) ) )
49 nfcv 2432 . . . . . . . . . . . 12  |-  F/_ k if ( 1  =  1 ,  F ,  G
)
50 eqeq1 2302 . . . . . . . . . . . . 13  |-  ( k  =  1  ->  (
k  =  1  <->  1  =  1 ) )
5150ifbid 3596 . . . . . . . . . . . 12  |-  ( k  =  1  ->  if ( k  =  1 ,  F ,  G
)  =  if ( 1  =  1 ,  F ,  G ) )
525, 49, 51, 2fvmptf 5632 . . . . . . . . . . 11  |-  ( ( 1  e.  { 1 ,  2 }  /\  if ( 1  =  1 ,  F ,  G
)  e.  ( J  Cn  K ) )  ->  ( A ` 
1 )  =  if ( 1  =  1 ,  F ,  G
) )
5348, 52syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( A `  1
)  =  if ( 1  =  1 ,  F ,  G ) )
5453, 46syl6eq 2344 . . . . . . . . 9  |-  ( ph  ->  ( A `  1
)  =  F )
5554adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  ( A `  1 )  =  F )
5655fveq1d 5543 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  1
) `  x )  =  ( F `  x ) )
5756eleq1d 2362 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( A ` 
1 ) `  x
)  e.  CC  <->  ( F `  x )  e.  CC ) )
5840, 57mpbird 223 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  1
) `  x )  e.  CC )
59 elex 2809 . . . . . . . . . . . . . 14  |-  ( 2  e.  CC  ->  2  e.  _V )
6016, 59ax-mp 8 . . . . . . . . . . . . 13  |-  2  e.  _V
6160prid2 3748 . . . . . . . . . . . 12  |-  2  e.  { 1 ,  2 }
6261a1i 10 . . . . . . . . . . 11  |-  ( ph  ->  2  e.  { 1 ,  2 } )
63 1ne2 9947 . . . . . . . . . . . . . . 15  |-  1  =/=  2
6463necomi 2541 . . . . . . . . . . . . . 14  |-  2  =/=  1
65 df-ne 2461 . . . . . . . . . . . . . 14  |-  ( 2  =/=  1  <->  -.  2  =  1 )
6664, 65mpbi 199 . . . . . . . . . . . . 13  |-  -.  2  =  1
67 iffalse 3585 . . . . . . . . . . . . 13  |-  ( -.  2  =  1  ->  if ( 2  =  1 ,  F ,  G
)  =  G )
6866, 67ax-mp 8 . . . . . . . . . . . 12  |-  if ( 2  =  1 ,  F ,  G )  =  G
69 refsum2cnlem1.9 . . . . . . . . . . . 12  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
7068, 69syl5eqel 2380 . . . . . . . . . . 11  |-  ( ph  ->  if ( 2  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
7162, 70jca 518 . . . . . . . . . 10  |-  ( ph  ->  ( 2  e.  {
1 ,  2 }  /\  if ( 2  =  1 ,  F ,  G )  e.  ( J  Cn  K ) ) )
72 nfcv 2432 . . . . . . . . . . 11  |-  F/_ k if ( 2  =  1 ,  F ,  G
)
73 eqeq1 2302 . . . . . . . . . . . 12  |-  ( k  =  2  ->  (
k  =  1  <->  2  =  1 ) )
7473ifbid 3596 . . . . . . . . . . 11  |-  ( k  =  2  ->  if ( k  =  1 ,  F ,  G
)  =  if ( 2  =  1 ,  F ,  G ) )
7510, 72, 74, 2fvmptf 5632 . . . . . . . . . 10  |-  ( ( 2  e.  { 1 ,  2 }  /\  if ( 2  =  1 ,  F ,  G
)  e.  ( J  Cn  K ) )  ->  ( A ` 
2 )  =  if ( 2  =  1 ,  F ,  G
) )
7671, 75syl 15 . . . . . . . . 9  |-  ( ph  ->  ( A `  2
)  =  if ( 2  =  1 ,  F ,  G ) )
7776, 68syl6eq 2344 . . . . . . . 8  |-  ( ph  ->  ( A `  2
)  =  G )
7877adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( A `  2 )  =  G )
7978fveq1d 5543 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  2
) `  x )  =  ( G `  x ) )
8019, 20cnf 16992 . . . . . . . . . . 11  |-  ( G  e.  ( J  Cn  K )  ->  G : U. J --> U. K
)
8169, 80syl 15 . . . . . . . . . 10  |-  ( ph  ->  G : U. J --> U. K )
8226, 31feq23d 5402 . . . . . . . . . . 11  |-  ( ph  ->  ( G : U. J
--> U. K  <->  G : X
--> RR ) )
8382biimpd 198 . . . . . . . . . 10  |-  ( ph  ->  ( G : U. J
--> U. K  ->  G : X --> RR ) )
8481, 83mpd 14 . . . . . . . . 9  |-  ( ph  ->  G : X --> RR )
8584adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  G : X --> RR )
8685, 36jca 518 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( G : X --> RR  /\  x  e.  X )
)
87 ffvelrn 5679 . . . . . . 7  |-  ( ( G : X --> RR  /\  x  e.  X )  ->  ( G `  x
)  e.  RR )
88 recn 8843 . . . . . . 7  |-  ( ( G `  x )  e.  RR  ->  ( G `  x )  e.  CC )
8986, 87, 883syl 18 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  CC )
9079, 89eqeltrd 2370 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  2
) `  x )  e.  CC )
9163a1i 10 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  1  =/=  2 )
92 fveq2 5541 . . . . . . 7  |-  ( k  =  1  ->  ( A `  k )  =  ( A ` 
1 ) )
9392fveq1d 5543 . . . . . 6  |-  ( k  =  1  ->  (
( A `  k
) `  x )  =  ( ( A `
 1 ) `  x ) )
9493adantl 452 . . . . 5  |-  ( ( ( ph  /\  x  e.  X )  /\  k  =  1 )  -> 
( ( A `  k ) `  x
)  =  ( ( A `  1 ) `
 x ) )
95 fveq2 5541 . . . . . . 7  |-  ( k  =  2  ->  ( A `  k )  =  ( A ` 
2 ) )
9695fveq1d 5543 . . . . . 6  |-  ( k  =  2  ->  (
( A `  k
) `  x )  =  ( ( A `
 2 ) `  x ) )
9796adantl 452 . . . . 5  |-  ( ( ( ph  /\  x  e.  X )  /\  k  =  2 )  -> 
( ( A `  k ) `  x
)  =  ( ( A `  2 ) `
 x ) )
989, 13, 15, 17, 58, 90, 91, 94, 97sumpair 27809 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  sum_ k  e.  { 1 ,  2 }  ( ( A `
 k ) `  x )  =  ( ( ( A ` 
1 ) `  x
)  +  ( ( A `  2 ) `
 x ) ) )
9956, 79oveq12d 5892 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( A ` 
1 ) `  x
)  +  ( ( A `  2 ) `
 x ) )  =  ( ( F `
 x )  +  ( G `  x
) ) )
10098, 99eqtrd 2328 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  sum_ k  e.  { 1 ,  2 }  ( ( A `
 k ) `  x )  =  ( ( F `  x
)  +  ( G `
 x ) ) )
1011, 100mpteq2da 4121 . 2  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  {
1 ,  2 }  ( ( A `  k ) `  x
) )  =  ( x  e.  X  |->  ( ( F `  x
)  +  ( G `
 x ) ) ) )
102 prfi 7147 . . . 4  |-  { 1 ,  2 }  e.  Fin
103102a1i 10 . . 3  |-  ( ph  ->  { 1 ,  2 }  e.  Fin )
10418adantr 451 . . . . . 6  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  F  e.  ( J  Cn  K
) )
105 eqid 2296 . . . . . . . . . . . . 13  |-  X  =  X
106105ax-gen 1536 . . . . . . . . . . . 12  |-  A. x  X  =  X
107106a1i 10 . . . . . . . . . . 11  |-  ( ( A `  k )  =  F  ->  A. x  X  =  X )
108 refsum2cnlem1.1 . . . . . . . . . . . . . 14  |-  F/_ x A
109 nfcv 2432 . . . . . . . . . . . . . 14  |-  F/_ x
k
110108, 109nffv 5548 . . . . . . . . . . . . 13  |-  F/_ x
( A `  k
)
111 refsum2cnlem1.2 . . . . . . . . . . . . 13  |-  F/_ x F
112110, 111nfeq 2439 . . . . . . . . . . . 12  |-  F/ x
( A `  k
)  =  F
113 fveq1 5540 . . . . . . . . . . . . 13  |-  ( ( A `  k )  =  F  ->  (
( A `  k
) `  x )  =  ( F `  x ) )
114113a1d 22 . . . . . . . . . . . 12  |-  ( ( A `  k )  =  F  ->  (
x  e.  X  -> 
( ( A `  k ) `  x
)  =  ( F `
 x ) ) )
115112, 114ralrimi 2637 . . . . . . . . . . 11  |-  ( ( A `  k )  =  F  ->  A. x  e.  X  ( ( A `  k ) `  x )  =  ( F `  x ) )
116107, 115jca 518 . . . . . . . . . 10  |-  ( ( A `  k )  =  F  ->  ( A. x  X  =  X  /\  A. x  e.  X  ( ( A `
 k ) `  x )  =  ( F `  x ) ) )
117 mpteq12f 4112 . . . . . . . . . 10  |-  ( ( A. x  X  =  X  /\  A. x  e.  X  ( ( A `  k ) `  x )  =  ( F `  x ) )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  ( x  e.  X  |->  ( F `  x ) ) )
118116, 117syl 15 . . . . . . . . 9  |-  ( ( A `  k )  =  F  ->  (
x  e.  X  |->  ( ( A `  k
) `  x )
)  =  ( x  e.  X  |->  ( F `
 x ) ) )
119118adantl 452 . . . . . . . 8  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  ( x  e.  X  |->  ( F `  x ) ) )
120 retopon 18288 . . . . . . . . . . . . . . 15  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
12127, 120eqeltri 2366 . . . . . . . . . . . . . 14  |-  K  e.  (TopOn `  RR )
122121a1i 10 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  (TopOn `  RR ) )
12323, 122, 183jca 1132 . . . . . . . . . . . 12  |-  ( ph  ->  ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )  /\  F  e.  ( J  Cn  K ) ) )
124 cnf2 16995 . . . . . . . . . . . 12  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )  /\  F  e.  ( J  Cn  K ) )  ->  F : X --> RR )
125123, 124syl 15 . . . . . . . . . . 11  |-  ( ph  ->  F : X --> RR )
126 ffn 5405 . . . . . . . . . . 11  |-  ( F : X --> RR  ->  F  Fn  X )
127125, 126syl 15 . . . . . . . . . 10  |-  ( ph  ->  F  Fn  X )
128111dffn5f 5593 . . . . . . . . . 10  |-  ( F  Fn  X  <->  F  =  ( x  e.  X  |->  ( F `  x
) ) )
129127, 128sylib 188 . . . . . . . . 9  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
130129adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  F  =  ( x  e.  X  |->  ( F `  x
) ) )
131119, 130eqtr4d 2331 . . . . . . 7  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  F )
132131eleq1d 2362 . . . . . 6  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  ( (
x  e.  X  |->  ( ( A `  k
) `  x )
)  e.  ( J  Cn  K )  <->  F  e.  ( J  Cn  K
) ) )
133104, 132mpbird 223 . . . . 5  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  e.  ( J  Cn  K ) )
134133adantlr 695 . . . 4  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  ( A `  k )  =  F )  ->  (
x  e.  X  |->  ( ( A `  k
) `  x )
)  e.  ( J  Cn  K ) )
13569adantr 451 . . . . . 6  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  G  e.  ( J  Cn  K
) )
136106a1i 10 . . . . . . . . . . 11  |-  ( ( A `  k )  =  G  ->  A. x  X  =  X )
137 refsum2cnlem1.3 . . . . . . . . . . . . 13  |-  F/_ x G
138110, 137nfeq 2439 . . . . . . . . . . . 12  |-  F/ x
( A `  k
)  =  G
139 fveq1 5540 . . . . . . . . . . . . 13  |-  ( ( A `  k )  =  G  ->  (
( A `  k
) `  x )  =  ( G `  x ) )
140139a1d 22 . . . . . . . . . . . 12  |-  ( ( A `  k )  =  G  ->  (
x  e.  X  -> 
( ( A `  k ) `  x
)  =  ( G `
 x ) ) )
141138, 140ralrimi 2637 . . . . . . . . . . 11  |-  ( ( A `  k )  =  G  ->  A. x  e.  X  ( ( A `  k ) `  x )  =  ( G `  x ) )
142136, 141jca 518 . . . . . . . . . 10  |-  ( ( A `  k )  =  G  ->  ( A. x  X  =  X  /\  A. x  e.  X  ( ( A `
 k ) `  x )  =  ( G `  x ) ) )
143 mpteq12f 4112 . . . . . . . . . 10  |-  ( ( A. x  X  =  X  /\  A. x  e.  X  ( ( A `  k ) `  x )  =  ( G `  x ) )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  ( x  e.  X  |->  ( G `  x ) ) )
144142, 143syl 15 . . . . . . . . 9  |-  ( ( A `  k )  =  G  ->  (
x  e.  X  |->  ( ( A `  k
) `  x )
)  =  ( x  e.  X  |->  ( G `
 x ) ) )
145144adantl 452 . . . . . . . 8  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  ( x  e.  X  |->  ( G `  x ) ) )
14623, 122, 693jca 1132 . . . . . . . . . . . 12  |-  ( ph  ->  ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )  /\  G  e.  ( J  Cn  K ) ) )
147 cnf2 16995 . . . . . . . . . . . 12  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )  /\  G  e.  ( J  Cn  K ) )  ->  G : X --> RR )
148146, 147syl 15 . . . . . . . . . . 11  |-  ( ph  ->  G : X --> RR )
149 ffn 5405 . . . . . . . . . . 11  |-  ( G : X --> RR  ->  G  Fn  X )
150148, 149syl 15 . . . . . . . . . 10  |-  ( ph  ->  G  Fn  X )
151137dffn5f 5593 . . . . . . . . . 10  |-  ( G  Fn  X  <->  G  =  ( x  e.  X  |->  ( G `  x
) ) )
152150, 151sylib 188 . . . . . . . . 9  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
153152adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  G  =  ( x  e.  X  |->  ( G `  x
) ) )
154145, 153eqtr4d 2331 . . . . . . 7  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  G )
155154eleq1d 2362 . . . . . 6  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  ( (
x  e.  X  |->  ( ( A `  k
) `  x )
)  e.  ( J  Cn  K )  <->  G  e.  ( J  Cn  K
) ) )
156135, 155mpbird 223 . . . . 5  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  e.  ( J  Cn  K ) )
157156adantlr 695 . . . 4  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  ( A `  k )  =  G )  ->  (
x  e.  X  |->  ( ( A `  k
) `  x )
)  e.  ( J  Cn  K ) )
158 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  k  e.  { 1 ,  2 } )
15918, 69jca 518 . . . . . . . . . . . 12  |-  ( ph  ->  ( F  e.  ( J  Cn  K )  /\  G  e.  ( J  Cn  K ) ) )
160 ifcl 3614 . . . . . . . . . . . 12  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( J  Cn  K ) )  ->  if ( k  =  1 ,  F ,  G
)  e.  ( J  Cn  K ) )
161159, 160syl 15 . . . . . . . . . . 11  |-  ( ph  ->  if ( k  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
162161adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  if (
k  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
163158, 162jca 518 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  ( k  e.  { 1 ,  2 }  /\  if ( k  =  1 ,  F ,  G )  e.  ( J  Cn  K ) ) )
1642fvmpt2 5624 . . . . . . . . 9  |-  ( ( k  e.  { 1 ,  2 }  /\  if ( k  =  1 ,  F ,  G
)  e.  ( J  Cn  K ) )  ->  ( A `  k )  =  if ( k  =  1 ,  F ,  G
) )
165163, 164syl 15 . . . . . . . 8  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  ( A `  k )  =  if ( k  =  1 ,  F ,  G
) )
166165adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  1 )  -> 
( A `  k
)  =  if ( k  =  1 ,  F ,  G ) )
167 iftrue 3584 . . . . . . . 8  |-  ( k  =  1  ->  if ( k  =  1 ,  F ,  G
)  =  F )
168167adantl 452 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  1 )  ->  if ( k  =  1 ,  F ,  G
)  =  F )
169166, 168eqtrd 2328 . . . . . 6  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  1 )  -> 
( A `  k
)  =  F )
170169orcd 381 . . . . 5  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  1 )  -> 
( ( A `  k )  =  F  \/  ( A `  k )  =  G ) )
171165adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  -> 
( A `  k
)  =  if ( k  =  1 ,  F ,  G ) )
172 neeq2 2468 . . . . . . . . . . . 12  |-  ( k  =  2  ->  (
1  =/=  k  <->  1  =/=  2 ) )
17363, 172mpbiri 224 . . . . . . . . . . 11  |-  ( k  =  2  ->  1  =/=  k )
174173necomd 2542 . . . . . . . . . 10  |-  ( k  =  2  ->  k  =/=  1 )
175 df-ne 2461 . . . . . . . . . 10  |-  ( k  =/=  1  <->  -.  k  =  1 )
176174, 175sylib 188 . . . . . . . . 9  |-  ( k  =  2  ->  -.  k  =  1 )
177176adantl 452 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  ->  -.  k  =  1
)
178 iffalse 3585 . . . . . . . 8  |-  ( -.  k  =  1  ->  if ( k  =  1 ,  F ,  G
)  =  G )
179177, 178syl 15 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  ->  if ( k  =  1 ,  F ,  G
)  =  G )
180171, 179eqtrd 2328 . . . . . 6  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  -> 
( A `  k
)  =  G )
181180olcd 382 . . . . 5  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  -> 
( ( A `  k )  =  F  \/  ( A `  k )  =  G ) )
182 elpri 3673 . . . . . 6  |-  ( k  e.  { 1 ,  2 }  ->  (
k  =  1  \/  k  =  2 ) )
183182adantl 452 . . . . 5  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  ( k  =  1  \/  k  =  2 ) )
184170, 181, 183mpjaodan 761 . . . 4  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  ( ( A `  k )  =  F  \/  ( A `  k )  =  G ) )
185134, 157, 184mpjaodan 761 . . 3  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  e.  ( J  Cn  K ) )
1861, 27, 23, 103, 185refsumcn 27804 . 2  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  {
1 ,  2 }  ( ( A `  k ) `  x
) )  e.  ( J  Cn  K ) )
187101, 186eqeltrrd 2371 1  |-  ( ph  ->  ( x  e.  X  |->  ( ( F `  x )  +  ( G `  x ) ) )  e.  ( J  Cn  K ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934   A.wal 1530   F/wnf 1534    = wceq 1632    e. wcel 1696   F/_wnfc 2419    =/= wne 2459   A.wral 2556   _Vcvv 2801   ifcif 3578   {cpr 3654   U.cuni 3843    e. cmpt 4093   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   CCcc 8751   RRcr 8752   1c1 8754    + caddc 8756   2c2 9811   (,)cioo 10672   sum_csu 12174   topGenctg 13358  TopOnctopon 16648    Cn ccn 16970
This theorem is referenced by:  refsum2cn  27812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cn 16973  df-cnp 16974  df-tx 17273  df-hmeo 17462  df-xms 17901  df-ms 17902  df-tms 17903
  Copyright terms: Public domain W3C validator