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Theorem refsum2cnlem1 27708
Description: This is the core Lemma for refsum2cn 27709: 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 4109 . . . . . . . . 9  |-  F/_ k
( k  e.  {
1 ,  2 } 
|->  if ( k  =  1 ,  F ,  G ) )
42, 3nfcxfr 2416 . . . . . . . 8  |-  F/_ k A
5 nfcv 2419 . . . . . . . 8  |-  F/_ k
1
64, 5nffv 5532 . . . . . . 7  |-  F/_ k
( A `  1
)
7 nfcv 2419 . . . . . . 7  |-  F/_ k
x
86, 7nffv 5532 . . . . . 6  |-  F/_ k
( ( A ` 
1 ) `  x
)
98a1i 10 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F/_ k
( ( A ` 
1 ) `  x
) )
10 nfcv 2419 . . . . . . . 8  |-  F/_ k
2
114, 10nffv 5532 . . . . . . 7  |-  F/_ k
( A `  2
)
1211, 7nffv 5532 . . . . . 6  |-  F/_ k
( ( A ` 
2 ) `  x
)
1312a1i 10 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F/_ k
( ( A ` 
2 ) `  x
) )
14 ax-1cn 8795 . . . . . 6  |-  1  e.  CC
1514a1i 10 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  1  e.  CC )
16 2cn 9816 . . . . . 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 2283 . . . . . . . . . . . 12  |-  U. J  =  U. J
20 eqid 2283 . . . . . . . . . . . 12  |-  U. K  =  U. K
2119, 20cnf 16976 . . . . . . . . . . 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 16665 . . . . . . . . . . . . . 14  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
2523, 24syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  X  =  U. J
)
2625eqcomd 2288 . . . . . . . . . . . 12  |-  ( ph  ->  U. J  =  X )
27 refsum2cnlem1.6 . . . . . . . . . . . . . . 15  |-  K  =  ( topGen `  ran  (,) )
2827unieqi 3837 . . . . . . . . . . . . . 14  |-  U. K  =  U. ( topGen `  ran  (,) )
29 uniretop 18271 . . . . . . . . . . . . . 14  |-  RR  =  U. ( topGen `  ran  (,) )
3028, 29eqtr4i 2306 . . . . . . . . . . . . 13  |-  U. K  =  RR
3130a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  U. K  =  RR )
3226, 31feq23d 5386 . . . . . . . . . . 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 5663 . . . . . . 7  |-  ( ( F : X --> RR  /\  x  e.  X )  ->  ( F `  x
)  e.  RR )
39 recn 8827 . . . . . . 7  |-  ( ( F `  x )  e.  RR  ->  ( F `  x )  e.  CC )
4037, 38, 393syl 18 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  CC )
41 1ex 8833 . . . . . . . . . . . . . 14  |-  1  e.  _V
4241prid1 3734 . . . . . . . . . . . . 13  |-  1  e.  { 1 ,  2 }
4342a1i 10 . . . . . . . . . . . 12  |-  ( ph  ->  1  e.  { 1 ,  2 } )
44 eqid 2283 . . . . . . . . . . . . . 14  |-  1  =  1
45 iftrue 3571 . . . . . . . . . . . . . 14  |-  ( 1  =  1  ->  if ( 1  =  1 ,  F ,  G
)  =  F )
4644, 45ax-mp 8 . . . . . . . . . . . . 13  |-  if ( 1  =  1 ,  F ,  G )  =  F
4746, 18syl5eqel 2367 . . . . . . . . . . . 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 2419 . . . . . . . . . . . 12  |-  F/_ k if ( 1  =  1 ,  F ,  G
)
50 eqeq1 2289 . . . . . . . . . . . . 13  |-  ( k  =  1  ->  (
k  =  1  <->  1  =  1 ) )
5150ifbid 3583 . . . . . . . . . . . 12  |-  ( k  =  1  ->  if ( k  =  1 ,  F ,  G
)  =  if ( 1  =  1 ,  F ,  G ) )
525, 49, 51, 2fvmptf 5616 . . . . . . . . . . 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 2331 . . . . . . . . 9  |-  ( ph  ->  ( A `  1
)  =  F )
5554adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  ( A `  1 )  =  F )
5655fveq1d 5527 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  1
) `  x )  =  ( F `  x ) )
5756eleq1d 2349 . . . . . 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 2796 . . . . . . . . . . . . . 14  |-  ( 2  e.  CC  ->  2  e.  _V )
6016, 59ax-mp 8 . . . . . . . . . . . . 13  |-  2  e.  _V
6160prid2 3735 . . . . . . . . . . . 12  |-  2  e.  { 1 ,  2 }
6261a1i 10 . . . . . . . . . . 11  |-  ( ph  ->  2  e.  { 1 ,  2 } )
63 1ne2 9931 . . . . . . . . . . . . . . 15  |-  1  =/=  2
6463necomi 2528 . . . . . . . . . . . . . 14  |-  2  =/=  1
65 df-ne 2448 . . . . . . . . . . . . . 14  |-  ( 2  =/=  1  <->  -.  2  =  1 )
6664, 65mpbi 199 . . . . . . . . . . . . 13  |-  -.  2  =  1
67 iffalse 3572 . . . . . . . . . . . . 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 2367 . . . . . . . . . . 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 2419 . . . . . . . . . . 11  |-  F/_ k if ( 2  =  1 ,  F ,  G
)
73 eqeq1 2289 . . . . . . . . . . . 12  |-  ( k  =  2  ->  (
k  =  1  <->  2  =  1 ) )
7473ifbid 3583 . . . . . . . . . . 11  |-  ( k  =  2  ->  if ( k  =  1 ,  F ,  G
)  =  if ( 2  =  1 ,  F ,  G ) )
7510, 72, 74, 2fvmptf 5616 . . . . . . . . . 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 2331 . . . . . . . 8  |-  ( ph  ->  ( A `  2
)  =  G )
7877adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( A `  2 )  =  G )
7978fveq1d 5527 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  2
) `  x )  =  ( G `  x ) )
8019, 20cnf 16976 . . . . . . . . . . 11  |-  ( G  e.  ( J  Cn  K )  ->  G : U. J --> U. K
)
8169, 80syl 15 . . . . . . . . . 10  |-  ( ph  ->  G : U. J --> U. K )
8226, 31feq23d 5386 . . . . . . . . . . 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 5663 . . . . . . 7  |-  ( ( G : X --> RR  /\  x  e.  X )  ->  ( G `  x
)  e.  RR )
88 recn 8827 . . . . . . 7  |-  ( ( G `  x )  e.  RR  ->  ( G `  x )  e.  CC )
8986, 87, 883syl 18 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  CC )
9079, 89eqeltrd 2357 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  2
) `  x )  e.  CC )
9163a1i 10 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  1  =/=  2 )
92 fveq2 5525 . . . . . . 7  |-  ( k  =  1  ->  ( A `  k )  =  ( A ` 
1 ) )
9392fveq1d 5527 . . . . . 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 5525 . . . . . . 7  |-  ( k  =  2  ->  ( A `  k )  =  ( A ` 
2 ) )
9695fveq1d 5527 . . . . . 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 27706 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  sum_ k  e.  { 1 ,  2 }  ( ( A `
 k ) `  x )  =  ( ( ( A ` 
1 ) `  x
)  +  ( ( A `  2 ) `
 x ) ) )
9956, 79oveq12d 5876 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( A ` 
1 ) `  x
)  +  ( ( A `  2 ) `
 x ) )  =  ( ( F `
 x )  +  ( G `  x
) ) )
10098, 99eqtrd 2315 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  sum_ k  e.  { 1 ,  2 }  ( ( A `
 k ) `  x )  =  ( ( F `  x
)  +  ( G `
 x ) ) )
1011, 100mpteq2da 4105 . 2  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  {
1 ,  2 }  ( ( A `  k ) `  x
) )  =  ( x  e.  X  |->  ( ( F `  x
)  +  ( G `
 x ) ) ) )
102 prfi 7131 . . . 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 2283 . . . . . . . . . . . . 13  |-  X  =  X
106105ax-gen 1533 . . . . . . . . . . . 12  |-  A. x  X  =  X
107106a1i 10 . . . . . . . . . . 11  |-  ( ( A `  k )  =  F  ->  A. x  X  =  X )
108 refsum2cnlem1.1 . . . . . . . . . . . . . 14  |-  F/_ x A
109 nfcv 2419 . . . . . . . . . . . . . 14  |-  F/_ x
k
110108, 109nffv 5532 . . . . . . . . . . . . 13  |-  F/_ x
( A `  k
)
111 refsum2cnlem1.2 . . . . . . . . . . . . 13  |-  F/_ x F
112110, 111nfeq 2426 . . . . . . . . . . . 12  |-  F/ x
( A `  k
)  =  F
113 fveq1 5524 . . . . . . . . . . . . 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 2624 . . . . . . . . . . 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 4096 . . . . . . . . . 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 18272 . . . . . . . . . . . . . . 15  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
12127, 120eqeltri 2353 . . . . . . . . . . . . . 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 16979 . . . . . . . . . . . 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 5389 . . . . . . . . . . 11  |-  ( F : X --> RR  ->  F  Fn  X )
127125, 126syl 15 . . . . . . . . . 10  |-  ( ph  ->  F  Fn  X )
128111dffn5f 5577 . . . . . . . . . 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 2318 . . . . . . 7  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  F )
132131eleq1d 2349 . . . . . 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 2426 . . . . . . . . . . . 12  |-  F/ x
( A `  k
)  =  G
139 fveq1 5524 . . . . . . . . . . . . 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 2624 . . . . . . . . . . 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 4096 . . . . . . . . . 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 16979 . . . . . . . . . . . 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 5389 . . . . . . . . . . 11  |-  ( G : X --> RR  ->  G  Fn  X )
150148, 149syl 15 . . . . . . . . . 10  |-  ( ph  ->  G  Fn  X )
151137dffn5f 5577 . . . . . . . . . 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 2318 . . . . . . 7  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  G )
155154eleq1d 2349 . . . . . 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 3601 . . . . . . . . . . . 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 5608 . . . . . . . . 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 3571 . . . . . . . 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 2315 . . . . . 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 2455 . . . . . . . . . . . 12  |-  ( k  =  2  ->  (
1  =/=  k  <->  1  =/=  2 ) )
17363, 172mpbiri 224 . . . . . . . . . . 11  |-  ( k  =  2  ->  1  =/=  k )
174173necomd 2529 . . . . . . . . . 10  |-  ( k  =  2  ->  k  =/=  1 )
175 df-ne 2448 . . . . . . . . . 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 3572 . . . . . . . 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 2315 . . . . . 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 3660 . . . . . 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 27701 . 2  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  {
1 ,  2 }  ( ( A `  k ) `  x
) )  e.  ( J  Cn  K ) )
187101, 186eqeltrrd 2358 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 1527   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406    =/= wne 2446   A.wral 2543   _Vcvv 2788   ifcif 3565   {cpr 3641   U.cuni 3827    e. cmpt 4077   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   RRcr 8736   1c1 8738    + caddc 8740   2c2 9795   (,)cioo 10656   sum_csu 12158   topGenctg 13342  TopOnctopon 16632    Cn ccn 16954
This theorem is referenced by:  refsum2cn  27709
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-cnp 16958  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887
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