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Theorem refsum2cnlem1 37358
Description: This is the core Lemma for refsum2cn 37359: 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 4492 . . . . . . . . 9  |-  F/_ k
( k  e.  {
1 ,  2 } 
|->  if ( k  =  1 ,  F ,  G ) )
42, 3nfcxfr 2590 . . . . . . . 8  |-  F/_ k A
5 nfcv 2592 . . . . . . . 8  |-  F/_ k
1
64, 5nffv 5872 . . . . . . 7  |-  F/_ k
( A `  1
)
7 nfcv 2592 . . . . . . 7  |-  F/_ k
x
86, 7nffv 5872 . . . . . 6  |-  F/_ k
( ( A ` 
1 ) `  x
)
98a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F/_ k
( ( A ` 
1 ) `  x
) )
10 nfcv 2592 . . . . . . . 8  |-  F/_ k
2
114, 10nffv 5872 . . . . . . 7  |-  F/_ k
( A `  2
)
1211, 7nffv 5872 . . . . . 6  |-  F/_ k
( ( A ` 
2 ) `  x
)
1312a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F/_ k
( ( A ` 
2 ) `  x
) )
14 1cnd 9659 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  1  e.  CC )
15 2cnd 10682 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  2  e.  CC )
16 1ex 9638 . . . . . . . . . . 11  |-  1  e.  _V
1716prid1 4080 . . . . . . . . . 10  |-  1  e.  { 1 ,  2 }
18 refsum2cnlem1.8 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
19 refsum2cnlem1.9 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
2018, 19ifcld 3924 . . . . . . . . . 10  |-  ( ph  ->  if ( 1  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
21 eqeq1 2455 . . . . . . . . . . . 12  |-  ( k  =  1  ->  (
k  =  1  <->  1  =  1 ) )
2221ifbid 3903 . . . . . . . . . . 11  |-  ( k  =  1  ->  if ( k  =  1 ,  F ,  G
)  =  if ( 1  =  1 ,  F ,  G ) )
2322, 2fvmptg 5946 . . . . . . . . . 10  |-  ( ( 1  e.  { 1 ,  2 }  /\  if ( 1  =  1 ,  F ,  G
)  e.  ( J  Cn  K ) )  ->  ( A ` 
1 )  =  if ( 1  =  1 ,  F ,  G
) )
2417, 20, 23sylancr 669 . . . . . . . . 9  |-  ( ph  ->  ( A `  1
)  =  if ( 1  =  1 ,  F ,  G ) )
25 eqid 2451 . . . . . . . . . 10  |-  1  =  1
2625iftruei 3888 . . . . . . . . 9  |-  if ( 1  =  1 ,  F ,  G )  =  F
2724, 26syl6eq 2501 . . . . . . . 8  |-  ( ph  ->  ( A `  1
)  =  F )
2827adantr 467 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( A `  1 )  =  F )
2928fveq1d 5867 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  1
) `  x )  =  ( F `  x ) )
30 eqid 2451 . . . . . . . . . . 11  |-  U. J  =  U. J
31 eqid 2451 . . . . . . . . . . 11  |-  U. K  =  U. K
3230, 31cnf 20262 . . . . . . . . . 10  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
3318, 32syl 17 . . . . . . . . 9  |-  ( ph  ->  F : U. J --> U. K )
34 refsum2cnlem1.7 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  (TopOn `  X ) )
35 toponuni 19942 . . . . . . . . . . . 12  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
3634, 35syl 17 . . . . . . . . . . 11  |-  ( ph  ->  X  =  U. J
)
3736eqcomd 2457 . . . . . . . . . 10  |-  ( ph  ->  U. J  =  X )
38 refsum2cnlem1.6 . . . . . . . . . . . . 13  |-  K  =  ( topGen `  ran  (,) )
3938unieqi 4207 . . . . . . . . . . . 12  |-  U. K  =  U. ( topGen `  ran  (,) )
40 uniretop 21783 . . . . . . . . . . . 12  |-  RR  =  U. ( topGen `  ran  (,) )
4139, 40eqtr4i 2476 . . . . . . . . . . 11  |-  U. K  =  RR
4241a1i 11 . . . . . . . . . 10  |-  ( ph  ->  U. K  =  RR )
4337, 42feq23d 5723 . . . . . . . . 9  |-  ( ph  ->  ( F : U. J
--> U. K  <->  F : X
--> RR ) )
4433, 43mpbid 214 . . . . . . . 8  |-  ( ph  ->  F : X --> RR )
4544anim1i 572 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( F : X --> RR  /\  x  e.  X )
)
46 ffvelrn 6020 . . . . . . 7  |-  ( ( F : X --> RR  /\  x  e.  X )  ->  ( F `  x
)  e.  RR )
47 recn 9629 . . . . . . 7  |-  ( ( F `  x )  e.  RR  ->  ( F `  x )  e.  CC )
4845, 46, 473syl 18 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  CC )
4929, 48eqeltrd 2529 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  1
) `  x )  e.  CC )
50 2ex 10681 . . . . . . . . . . 11  |-  2  e.  _V
5150prid2 4081 . . . . . . . . . 10  |-  2  e.  { 1 ,  2 }
5218, 19ifcld 3924 . . . . . . . . . 10  |-  ( ph  ->  if ( 2  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
53 eqeq1 2455 . . . . . . . . . . . 12  |-  ( k  =  2  ->  (
k  =  1  <->  2  =  1 ) )
5453ifbid 3903 . . . . . . . . . . 11  |-  ( k  =  2  ->  if ( k  =  1 ,  F ,  G
)  =  if ( 2  =  1 ,  F ,  G ) )
5554, 2fvmptg 5946 . . . . . . . . . 10  |-  ( ( 2  e.  { 1 ,  2 }  /\  if ( 2  =  1 ,  F ,  G
)  e.  ( J  Cn  K ) )  ->  ( A ` 
2 )  =  if ( 2  =  1 ,  F ,  G
) )
5651, 52, 55sylancr 669 . . . . . . . . 9  |-  ( ph  ->  ( A `  2
)  =  if ( 2  =  1 ,  F ,  G ) )
57 1ne2 10822 . . . . . . . . . . 11  |-  1  =/=  2
5857nesymi 2681 . . . . . . . . . 10  |-  -.  2  =  1
5958iffalsei 3891 . . . . . . . . 9  |-  if ( 2  =  1 ,  F ,  G )  =  G
6056, 59syl6eq 2501 . . . . . . . 8  |-  ( ph  ->  ( A `  2
)  =  G )
6160adantr 467 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( A `  2 )  =  G )
6261fveq1d 5867 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  2
) `  x )  =  ( G `  x ) )
6330, 31cnf 20262 . . . . . . . . . 10  |-  ( G  e.  ( J  Cn  K )  ->  G : U. J --> U. K
)
6419, 63syl 17 . . . . . . . . 9  |-  ( ph  ->  G : U. J --> U. K )
6537, 42feq23d 5723 . . . . . . . . 9  |-  ( ph  ->  ( G : U. J
--> U. K  <->  G : X
--> RR ) )
6664, 65mpbid 214 . . . . . . . 8  |-  ( ph  ->  G : X --> RR )
6766anim1i 572 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( G : X --> RR  /\  x  e.  X )
)
68 ffvelrn 6020 . . . . . . 7  |-  ( ( G : X --> RR  /\  x  e.  X )  ->  ( G `  x
)  e.  RR )
69 recn 9629 . . . . . . 7  |-  ( ( G `  x )  e.  RR  ->  ( G `  x )  e.  CC )
7067, 68, 693syl 18 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  CC )
7162, 70eqeltrd 2529 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  2
) `  x )  e.  CC )
7257a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  1  =/=  2 )
73 fveq2 5865 . . . . . . 7  |-  ( k  =  1  ->  ( A `  k )  =  ( A ` 
1 ) )
7473fveq1d 5867 . . . . . 6  |-  ( k  =  1  ->  (
( A `  k
) `  x )  =  ( ( A `
 1 ) `  x ) )
7574adantl 468 . . . . 5  |-  ( ( ( ph  /\  x  e.  X )  /\  k  =  1 )  -> 
( ( A `  k ) `  x
)  =  ( ( A `  1 ) `
 x ) )
76 fveq2 5865 . . . . . . 7  |-  ( k  =  2  ->  ( A `  k )  =  ( A ` 
2 ) )
7776fveq1d 5867 . . . . . 6  |-  ( k  =  2  ->  (
( A `  k
) `  x )  =  ( ( A `
 2 ) `  x ) )
7877adantl 468 . . . . 5  |-  ( ( ( ph  /\  x  e.  X )  /\  k  =  2 )  -> 
( ( A `  k ) `  x
)  =  ( ( A `  2 ) `
 x ) )
799, 13, 14, 15, 49, 71, 72, 75, 78sumpair 37356 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  sum_ k  e.  { 1 ,  2 }  ( ( A `
 k ) `  x )  =  ( ( ( A ` 
1 ) `  x
)  +  ( ( A `  2 ) `
 x ) ) )
8029, 62oveq12d 6308 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( A ` 
1 ) `  x
)  +  ( ( A `  2 ) `
 x ) )  =  ( ( F `
 x )  +  ( G `  x
) ) )
8179, 80eqtrd 2485 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  sum_ k  e.  { 1 ,  2 }  ( ( A `
 k ) `  x )  =  ( ( F `  x
)  +  ( G `
 x ) ) )
821, 81mpteq2da 4488 . 2  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  {
1 ,  2 }  ( ( A `  k ) `  x
) )  =  ( x  e.  X  |->  ( ( F `  x
)  +  ( G `
 x ) ) ) )
83 prfi 7846 . . . 4  |-  { 1 ,  2 }  e.  Fin
8483a1i 11 . . 3  |-  ( ph  ->  { 1 ,  2 }  e.  Fin )
85 eqid 2451 . . . . . . . . . 10  |-  X  =  X
8685ax-gen 1669 . . . . . . . . 9  |-  A. x  X  =  X
87 refsum2cnlem1.1 . . . . . . . . . . . 12  |-  F/_ x A
88 nfcv 2592 . . . . . . . . . . . 12  |-  F/_ x
k
8987, 88nffv 5872 . . . . . . . . . . 11  |-  F/_ x
( A `  k
)
90 refsum2cnlem1.2 . . . . . . . . . . 11  |-  F/_ x F
9189, 90nfeq 2603 . . . . . . . . . 10  |-  F/ x
( A `  k
)  =  F
92 fveq1 5864 . . . . . . . . . . 11  |-  ( ( A `  k )  =  F  ->  (
( A `  k
) `  x )  =  ( F `  x ) )
9392a1d 26 . . . . . . . . . 10  |-  ( ( A `  k )  =  F  ->  (
x  e.  X  -> 
( ( A `  k ) `  x
)  =  ( F `
 x ) ) )
9491, 93ralrimi 2788 . . . . . . . . 9  |-  ( ( A `  k )  =  F  ->  A. x  e.  X  ( ( A `  k ) `  x )  =  ( F `  x ) )
95 mpteq12f 4479 . . . . . . . . 9  |-  ( ( A. x  X  =  X  /\  A. x  e.  X  ( ( A `  k ) `  x )  =  ( F `  x ) )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  ( x  e.  X  |->  ( F `  x ) ) )
9686, 94, 95sylancr 669 . . . . . . . 8  |-  ( ( A `  k )  =  F  ->  (
x  e.  X  |->  ( ( A `  k
) `  x )
)  =  ( x  e.  X  |->  ( F `
 x ) ) )
9796adantl 468 . . . . . . 7  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  ( x  e.  X  |->  ( F `  x ) ) )
98 retopon 21784 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
9938, 98eqeltri 2525 . . . . . . . . . . . 12  |-  K  e.  (TopOn `  RR )
10099a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  K  e.  (TopOn `  RR ) )
101 cnf2 20265 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )  /\  F  e.  ( J  Cn  K ) )  ->  F : X --> RR )
10234, 100, 18, 101syl3anc 1268 . . . . . . . . . 10  |-  ( ph  ->  F : X --> RR )
103 ffn 5728 . . . . . . . . . 10  |-  ( F : X --> RR  ->  F  Fn  X )
104102, 103syl 17 . . . . . . . . 9  |-  ( ph  ->  F  Fn  X )
10590dffn5f 5920 . . . . . . . . 9  |-  ( F  Fn  X  <->  F  =  ( x  e.  X  |->  ( F `  x
) ) )
106104, 105sylib 200 . . . . . . . 8  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
107106adantr 467 . . . . . . 7  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  F  =  ( x  e.  X  |->  ( F `  x
) ) )
10897, 107eqtr4d 2488 . . . . . 6  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  F )
10918adantr 467 . . . . . 6  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  F  e.  ( J  Cn  K
) )
110108, 109eqeltrd 2529 . . . . 5  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  e.  ( J  Cn  K ) )
111110adantlr 721 . . . 4  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  ( A `  k )  =  F )  ->  (
x  e.  X  |->  ( ( A `  k
) `  x )
)  e.  ( J  Cn  K ) )
112 refsum2cnlem1.3 . . . . . . . . . . 11  |-  F/_ x G
11389, 112nfeq 2603 . . . . . . . . . 10  |-  F/ x
( A `  k
)  =  G
114 fveq1 5864 . . . . . . . . . . 11  |-  ( ( A `  k )  =  G  ->  (
( A `  k
) `  x )  =  ( G `  x ) )
115114a1d 26 . . . . . . . . . 10  |-  ( ( A `  k )  =  G  ->  (
x  e.  X  -> 
( ( A `  k ) `  x
)  =  ( G `
 x ) ) )
116113, 115ralrimi 2788 . . . . . . . . 9  |-  ( ( A `  k )  =  G  ->  A. x  e.  X  ( ( A `  k ) `  x )  =  ( G `  x ) )
117 mpteq12f 4479 . . . . . . . . 9  |-  ( ( A. x  X  =  X  /\  A. x  e.  X  ( ( A `  k ) `  x )  =  ( G `  x ) )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  ( x  e.  X  |->  ( G `  x ) ) )
11886, 116, 117sylancr 669 . . . . . . . 8  |-  ( ( A `  k )  =  G  ->  (
x  e.  X  |->  ( ( A `  k
) `  x )
)  =  ( x  e.  X  |->  ( G `
 x ) ) )
119118adantl 468 . . . . . . 7  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  ( x  e.  X  |->  ( G `  x ) ) )
120 cnf2 20265 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )  /\  G  e.  ( J  Cn  K ) )  ->  G : X --> RR )
12134, 100, 19, 120syl3anc 1268 . . . . . . . . . 10  |-  ( ph  ->  G : X --> RR )
122 ffn 5728 . . . . . . . . . 10  |-  ( G : X --> RR  ->  G  Fn  X )
123121, 122syl 17 . . . . . . . . 9  |-  ( ph  ->  G  Fn  X )
124112dffn5f 5920 . . . . . . . . 9  |-  ( G  Fn  X  <->  G  =  ( x  e.  X  |->  ( G `  x
) ) )
125123, 124sylib 200 . . . . . . . 8  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
126125adantr 467 . . . . . . 7  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  G  =  ( x  e.  X  |->  ( G `  x
) ) )
127119, 126eqtr4d 2488 . . . . . 6  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  G )
12819adantr 467 . . . . . 6  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  G  e.  ( J  Cn  K
) )
129127, 128eqeltrd 2529 . . . . 5  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  e.  ( J  Cn  K ) )
130129adantlr 721 . . . 4  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  ( A `  k )  =  G )  ->  (
x  e.  X  |->  ( ( A `  k
) `  x )
)  e.  ( J  Cn  K ) )
131 simpr 463 . . . . . . . 8  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  k  e.  { 1 ,  2 } )
13218, 19ifcld 3924 . . . . . . . . 9  |-  ( ph  ->  if ( k  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
133132adantr 467 . . . . . . . 8  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  if (
k  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
1342fvmpt2 5957 . . . . . . . 8  |-  ( ( k  e.  { 1 ,  2 }  /\  if ( k  =  1 ,  F ,  G
)  e.  ( J  Cn  K ) )  ->  ( A `  k )  =  if ( k  =  1 ,  F ,  G
) )
135131, 133, 134syl2anc 667 . . . . . . 7  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  ( A `  k )  =  if ( k  =  1 ,  F ,  G
) )
136 iftrue 3887 . . . . . . 7  |-  ( k  =  1  ->  if ( k  =  1 ,  F ,  G
)  =  F )
137135, 136sylan9eq 2505 . . . . . 6  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  1 )  -> 
( A `  k
)  =  F )
138137orcd 394 . . . . 5  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  1 )  -> 
( ( A `  k )  =  F  \/  ( A `  k )  =  G ) )
139135adantr 467 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  -> 
( A `  k
)  =  if ( k  =  1 ,  F ,  G ) )
140 neeq2 2687 . . . . . . . . . . . 12  |-  ( k  =  2  ->  (
1  =/=  k  <->  1  =/=  2 ) )
14157, 140mpbiri 237 . . . . . . . . . . 11  |-  ( k  =  2  ->  1  =/=  k )
142141necomd 2679 . . . . . . . . . 10  |-  ( k  =  2  ->  k  =/=  1 )
143142neneqd 2629 . . . . . . . . 9  |-  ( k  =  2  ->  -.  k  =  1 )
144143adantl 468 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  ->  -.  k  =  1
)
145144iffalsed 3892 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  ->  if ( k  =  1 ,  F ,  G
)  =  G )
146139, 145eqtrd 2485 . . . . . 6  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  -> 
( A `  k
)  =  G )
147146olcd 395 . . . . 5  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  -> 
( ( A `  k )  =  F  \/  ( A `  k )  =  G ) )
148 elpri 3985 . . . . . 6  |-  ( k  e.  { 1 ,  2 }  ->  (
k  =  1  \/  k  =  2 ) )
149148adantl 468 . . . . 5  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  ( k  =  1  \/  k  =  2 ) )
150138, 147, 149mpjaodan 795 . . . 4  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  ( ( A `  k )  =  F  \/  ( A `  k )  =  G ) )
151111, 130, 150mpjaodan 795 . . 3  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  e.  ( J  Cn  K ) )
1521, 38, 34, 84, 151refsumcn 37351 . 2  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  {
1 ,  2 }  ( ( A `  k ) `  x
) )  e.  ( J  Cn  K ) )
15382, 152eqeltrrd 2530 1  |-  ( ph  ->  ( x  e.  X  |->  ( ( F `  x )  +  ( G `  x ) ) )  e.  ( J  Cn  K ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 370    /\ wa 371   A.wal 1442    = wceq 1444   F/wnf 1667    e. wcel 1887   F/_wnfc 2579    =/= wne 2622   A.wral 2737   ifcif 3881   {cpr 3970   U.cuni 4198    |-> cmpt 4461   ran crn 4835    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   Fincfn 7569   CCcc 9537   RRcr 9538   1c1 9540    + caddc 9542   2c2 10659   (,)cioo 11635   sum_csu 13752   topGenctg 15336  TopOnctopon 19918    Cn ccn 20240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-er 7363  df-map 7474  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cn 20243  df-cnp 20244  df-tx 20577  df-hmeo 20770  df-xms 21335  df-ms 21336  df-tms 21337
This theorem is referenced by:  refsum2cn  37359
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