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Theorem refsum2cnlem1 31359
Description: This is the core Lemma for refsum2cn 31360: 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 4522 . . . . . . . . 9  |-  F/_ k
( k  e.  {
1 ,  2 } 
|->  if ( k  =  1 ,  F ,  G ) )
42, 3nfcxfr 2601 . . . . . . . 8  |-  F/_ k A
5 nfcv 2603 . . . . . . . 8  |-  F/_ k
1
64, 5nffv 5859 . . . . . . 7  |-  F/_ k
( A `  1
)
7 nfcv 2603 . . . . . . 7  |-  F/_ k
x
86, 7nffv 5859 . . . . . 6  |-  F/_ k
( ( A ` 
1 ) `  x
)
98a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F/_ k
( ( A ` 
1 ) `  x
) )
10 nfcv 2603 . . . . . . . 8  |-  F/_ k
2
114, 10nffv 5859 . . . . . . 7  |-  F/_ k
( A `  2
)
1211, 7nffv 5859 . . . . . 6  |-  F/_ k
( ( A ` 
2 ) `  x
)
1312a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F/_ k
( ( A ` 
2 ) `  x
) )
14 1cnd 9610 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  1  e.  CC )
15 2cnd 10609 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  2  e.  CC )
16 1ex 9589 . . . . . . . . . . 11  |-  1  e.  _V
1716prid1 4119 . . . . . . . . . 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 3965 . . . . . . . . . 10  |-  ( ph  ->  if ( 1  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
21 eqeq1 2445 . . . . . . . . . . . 12  |-  ( k  =  1  ->  (
k  =  1  <->  1  =  1 ) )
2221ifbid 3944 . . . . . . . . . . 11  |-  ( k  =  1  ->  if ( k  =  1 ,  F ,  G
)  =  if ( 1  =  1 ,  F ,  G ) )
2322, 2fvmptg 5935 . . . . . . . . . 10  |-  ( ( 1  e.  { 1 ,  2 }  /\  if ( 1  =  1 ,  F ,  G
)  e.  ( J  Cn  K ) )  ->  ( A ` 
1 )  =  if ( 1  =  1 ,  F ,  G
) )
2417, 20, 23sylancr 663 . . . . . . . . 9  |-  ( ph  ->  ( A `  1
)  =  if ( 1  =  1 ,  F ,  G ) )
25 eqid 2441 . . . . . . . . . 10  |-  1  =  1
2625iftruei 3929 . . . . . . . . 9  |-  if ( 1  =  1 ,  F ,  G )  =  F
2724, 26syl6eq 2498 . . . . . . . 8  |-  ( ph  ->  ( A `  1
)  =  F )
2827adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( A `  1 )  =  F )
2928fveq1d 5854 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  1
) `  x )  =  ( F `  x ) )
30 eqid 2441 . . . . . . . . . . 11  |-  U. J  =  U. J
31 eqid 2441 . . . . . . . . . . 11  |-  U. K  =  U. K
3230, 31cnf 19613 . . . . . . . . . 10  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
3318, 32syl 16 . . . . . . . . 9  |-  ( ph  ->  F : U. J --> U. K )
34 refsum2cnlem1.7 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  (TopOn `  X ) )
35 toponuni 19295 . . . . . . . . . . . 12  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
3634, 35syl 16 . . . . . . . . . . 11  |-  ( ph  ->  X  =  U. J
)
3736eqcomd 2449 . . . . . . . . . 10  |-  ( ph  ->  U. J  =  X )
38 refsum2cnlem1.6 . . . . . . . . . . . . 13  |-  K  =  ( topGen `  ran  (,) )
3938unieqi 4239 . . . . . . . . . . . 12  |-  U. K  =  U. ( topGen `  ran  (,) )
40 uniretop 21135 . . . . . . . . . . . 12  |-  RR  =  U. ( topGen `  ran  (,) )
4139, 40eqtr4i 2473 . . . . . . . . . . 11  |-  U. K  =  RR
4241a1i 11 . . . . . . . . . 10  |-  ( ph  ->  U. K  =  RR )
4337, 42feq23d 5712 . . . . . . . . 9  |-  ( ph  ->  ( F : U. J
--> U. K  <->  F : X
--> RR ) )
4433, 43mpbid 210 . . . . . . . 8  |-  ( ph  ->  F : X --> RR )
4544anim1i 568 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( F : X --> RR  /\  x  e.  X )
)
46 ffvelrn 6010 . . . . . . 7  |-  ( ( F : X --> RR  /\  x  e.  X )  ->  ( F `  x
)  e.  RR )
47 recn 9580 . . . . . . 7  |-  ( ( F `  x )  e.  RR  ->  ( F `  x )  e.  CC )
4845, 46, 473syl 20 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  CC )
4929, 48eqeltrd 2529 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  1
) `  x )  e.  CC )
50 2ex 10608 . . . . . . . . . . 11  |-  2  e.  _V
5150prid2 4120 . . . . . . . . . 10  |-  2  e.  { 1 ,  2 }
5218, 19ifcld 3965 . . . . . . . . . 10  |-  ( ph  ->  if ( 2  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
53 eqeq1 2445 . . . . . . . . . . . 12  |-  ( k  =  2  ->  (
k  =  1  <->  2  =  1 ) )
5453ifbid 3944 . . . . . . . . . . 11  |-  ( k  =  2  ->  if ( k  =  1 ,  F ,  G
)  =  if ( 2  =  1 ,  F ,  G ) )
5554, 2fvmptg 5935 . . . . . . . . . 10  |-  ( ( 2  e.  { 1 ,  2 }  /\  if ( 2  =  1 ,  F ,  G
)  e.  ( J  Cn  K ) )  ->  ( A ` 
2 )  =  if ( 2  =  1 ,  F ,  G
) )
5651, 52, 55sylancr 663 . . . . . . . . 9  |-  ( ph  ->  ( A `  2
)  =  if ( 2  =  1 ,  F ,  G ) )
57 1ne2 10749 . . . . . . . . . . 11  |-  1  =/=  2
5857nesymi 2714 . . . . . . . . . 10  |-  -.  2  =  1
5958iffalsei 3932 . . . . . . . . 9  |-  if ( 2  =  1 ,  F ,  G )  =  G
6056, 59syl6eq 2498 . . . . . . . 8  |-  ( ph  ->  ( A `  2
)  =  G )
6160adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( A `  2 )  =  G )
6261fveq1d 5854 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  2
) `  x )  =  ( G `  x ) )
6330, 31cnf 19613 . . . . . . . . . 10  |-  ( G  e.  ( J  Cn  K )  ->  G : U. J --> U. K
)
6419, 63syl 16 . . . . . . . . 9  |-  ( ph  ->  G : U. J --> U. K )
6537, 42feq23d 5712 . . . . . . . . 9  |-  ( ph  ->  ( G : U. J
--> U. K  <->  G : X
--> RR ) )
6664, 65mpbid 210 . . . . . . . 8  |-  ( ph  ->  G : X --> RR )
6766anim1i 568 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( G : X --> RR  /\  x  e.  X )
)
68 ffvelrn 6010 . . . . . . 7  |-  ( ( G : X --> RR  /\  x  e.  X )  ->  ( G `  x
)  e.  RR )
69 recn 9580 . . . . . . 7  |-  ( ( G `  x )  e.  RR  ->  ( G `  x )  e.  CC )
7067, 68, 693syl 20 . . . . . 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 5852 . . . . . . 7  |-  ( k  =  1  ->  ( A `  k )  =  ( A ` 
1 ) )
7473fveq1d 5854 . . . . . 6  |-  ( k  =  1  ->  (
( A `  k
) `  x )  =  ( ( A `
 1 ) `  x ) )
7574adantl 466 . . . . 5  |-  ( ( ( ph  /\  x  e.  X )  /\  k  =  1 )  -> 
( ( A `  k ) `  x
)  =  ( ( A `  1 ) `
 x ) )
76 fveq2 5852 . . . . . . 7  |-  ( k  =  2  ->  ( A `  k )  =  ( A ` 
2 ) )
7776fveq1d 5854 . . . . . 6  |-  ( k  =  2  ->  (
( A `  k
) `  x )  =  ( ( A `
 2 ) `  x ) )
7877adantl 466 . . . . 5  |-  ( ( ( ph  /\  x  e.  X )  /\  k  =  2 )  -> 
( ( A `  k ) `  x
)  =  ( ( A `  2 ) `
 x ) )
799, 13, 14, 15, 49, 71, 72, 75, 78sumpair 31357 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  sum_ k  e.  { 1 ,  2 }  ( ( A `
 k ) `  x )  =  ( ( ( A ` 
1 ) `  x
)  +  ( ( A `  2 ) `
 x ) ) )
8029, 62oveq12d 6295 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( A ` 
1 ) `  x
)  +  ( ( A `  2 ) `
 x ) )  =  ( ( F `
 x )  +  ( G `  x
) ) )
8179, 80eqtrd 2482 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  sum_ k  e.  { 1 ,  2 }  ( ( A `
 k ) `  x )  =  ( ( F `  x
)  +  ( G `
 x ) ) )
821, 81mpteq2da 4518 . 2  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  {
1 ,  2 }  ( ( A `  k ) `  x
) )  =  ( x  e.  X  |->  ( ( F `  x
)  +  ( G `
 x ) ) ) )
83 prfi 7793 . . . 4  |-  { 1 ,  2 }  e.  Fin
8483a1i 11 . . 3  |-  ( ph  ->  { 1 ,  2 }  e.  Fin )
85 eqid 2441 . . . . . . . . . 10  |-  X  =  X
8685ax-gen 1603 . . . . . . . . 9  |-  A. x  X  =  X
87 refsum2cnlem1.1 . . . . . . . . . . . 12  |-  F/_ x A
88 nfcv 2603 . . . . . . . . . . . 12  |-  F/_ x
k
8987, 88nffv 5859 . . . . . . . . . . 11  |-  F/_ x
( A `  k
)
90 refsum2cnlem1.2 . . . . . . . . . . 11  |-  F/_ x F
9189, 90nfeq 2614 . . . . . . . . . 10  |-  F/ x
( A `  k
)  =  F
92 fveq1 5851 . . . . . . . . . . 11  |-  ( ( A `  k )  =  F  ->  (
( A `  k
) `  x )  =  ( F `  x ) )
9392a1d 25 . . . . . . . . . 10  |-  ( ( A `  k )  =  F  ->  (
x  e.  X  -> 
( ( A `  k ) `  x
)  =  ( F `
 x ) ) )
9491, 93ralrimi 2841 . . . . . . . . 9  |-  ( ( A `  k )  =  F  ->  A. x  e.  X  ( ( A `  k ) `  x )  =  ( F `  x ) )
95 mpteq12f 4509 . . . . . . . . 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 663 . . . . . . . 8  |-  ( ( A `  k )  =  F  ->  (
x  e.  X  |->  ( ( A `  k
) `  x )
)  =  ( x  e.  X  |->  ( F `
 x ) ) )
9796adantl 466 . . . . . . 7  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  ( x  e.  X  |->  ( F `  x ) ) )
98 retopon 21136 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
9938, 98eqeltri 2525 . . . . . . . . . . . 12  |-  K  e.  (TopOn `  RR )
10099a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  K  e.  (TopOn `  RR ) )
101 cnf2 19616 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )  /\  F  e.  ( J  Cn  K ) )  ->  F : X --> RR )
10234, 100, 18, 101syl3anc 1227 . . . . . . . . . 10  |-  ( ph  ->  F : X --> RR )
103 ffn 5717 . . . . . . . . . 10  |-  ( F : X --> RR  ->  F  Fn  X )
104102, 103syl 16 . . . . . . . . 9  |-  ( ph  ->  F  Fn  X )
10590dffn5f 5909 . . . . . . . . 9  |-  ( F  Fn  X  <->  F  =  ( x  e.  X  |->  ( F `  x
) ) )
106104, 105sylib 196 . . . . . . . 8  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
107106adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  F  =  ( x  e.  X  |->  ( F `  x
) ) )
10897, 107eqtr4d 2485 . . . . . 6  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  F )
10918adantr 465 . . . . . 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 714 . . . 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 2614 . . . . . . . . . 10  |-  F/ x
( A `  k
)  =  G
114 fveq1 5851 . . . . . . . . . . 11  |-  ( ( A `  k )  =  G  ->  (
( A `  k
) `  x )  =  ( G `  x ) )
115114a1d 25 . . . . . . . . . 10  |-  ( ( A `  k )  =  G  ->  (
x  e.  X  -> 
( ( A `  k ) `  x
)  =  ( G `
 x ) ) )
116113, 115ralrimi 2841 . . . . . . . . 9  |-  ( ( A `  k )  =  G  ->  A. x  e.  X  ( ( A `  k ) `  x )  =  ( G `  x ) )
117 mpteq12f 4509 . . . . . . . . 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 663 . . . . . . . 8  |-  ( ( A `  k )  =  G  ->  (
x  e.  X  |->  ( ( A `  k
) `  x )
)  =  ( x  e.  X  |->  ( G `
 x ) ) )
119118adantl 466 . . . . . . 7  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  ( x  e.  X  |->  ( G `  x ) ) )
120 cnf2 19616 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )  /\  G  e.  ( J  Cn  K ) )  ->  G : X --> RR )
12134, 100, 19, 120syl3anc 1227 . . . . . . . . . 10  |-  ( ph  ->  G : X --> RR )
122 ffn 5717 . . . . . . . . . 10  |-  ( G : X --> RR  ->  G  Fn  X )
123121, 122syl 16 . . . . . . . . 9  |-  ( ph  ->  G  Fn  X )
124112dffn5f 5909 . . . . . . . . 9  |-  ( G  Fn  X  <->  G  =  ( x  e.  X  |->  ( G `  x
) ) )
125123, 124sylib 196 . . . . . . . 8  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
126125adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  G  =  ( x  e.  X  |->  ( G `  x
) ) )
127119, 126eqtr4d 2485 . . . . . 6  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  G )
12819adantr 465 . . . . . 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 714 . . . 4  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  ( A `  k )  =  G )  ->  (
x  e.  X  |->  ( ( A `  k
) `  x )
)  e.  ( J  Cn  K ) )
131 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  k  e.  { 1 ,  2 } )
13218, 19ifcld 3965 . . . . . . . . 9  |-  ( ph  ->  if ( k  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
133132adantr 465 . . . . . . . 8  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  if (
k  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
1342fvmpt2 5944 . . . . . . . 8  |-  ( ( k  e.  { 1 ,  2 }  /\  if ( k  =  1 ,  F ,  G
)  e.  ( J  Cn  K ) )  ->  ( A `  k )  =  if ( k  =  1 ,  F ,  G
) )
135131, 133, 134syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  ( A `  k )  =  if ( k  =  1 ,  F ,  G
) )
136 iftrue 3928 . . . . . . 7  |-  ( k  =  1  ->  if ( k  =  1 ,  F ,  G
)  =  F )
137135, 136sylan9eq 2502 . . . . . 6  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  1 )  -> 
( A `  k
)  =  F )
138137orcd 392 . . . . 5  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  1 )  -> 
( ( A `  k )  =  F  \/  ( A `  k )  =  G ) )
139135adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  -> 
( A `  k
)  =  if ( k  =  1 ,  F ,  G ) )
140 neeq2 2724 . . . . . . . . . . . 12  |-  ( k  =  2  ->  (
1  =/=  k  <->  1  =/=  2 ) )
14157, 140mpbiri 233 . . . . . . . . . . 11  |-  ( k  =  2  ->  1  =/=  k )
142141necomd 2712 . . . . . . . . . 10  |-  ( k  =  2  ->  k  =/=  1 )
143142neneqd 2643 . . . . . . . . 9  |-  ( k  =  2  ->  -.  k  =  1 )
144143adantl 466 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  ->  -.  k  =  1
)
145144iffalsed 3933 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  ->  if ( k  =  1 ,  F ,  G
)  =  G )
146139, 145eqtrd 2482 . . . . . 6  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  -> 
( A `  k
)  =  G )
147146olcd 393 . . . . 5  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  -> 
( ( A `  k )  =  F  \/  ( A `  k )  =  G ) )
148 elpri 4030 . . . . . 6  |-  ( k  e.  { 1 ,  2 }  ->  (
k  =  1  \/  k  =  2 ) )
149148adantl 466 . . . . 5  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  ( k  =  1  \/  k  =  2 ) )
150138, 147, 149mpjaodan 784 . . . 4  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  ( ( A `  k )  =  F  \/  ( A `  k )  =  G ) )
151111, 130, 150mpjaodan 784 . . 3  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  e.  ( J  Cn  K ) )
1521, 38, 34, 84, 151refsumcn 31352 . 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 368    /\ wa 369   A.wal 1379    = wceq 1381   F/wnf 1601    e. wcel 1802   F/_wnfc 2589    =/= wne 2636   A.wral 2791   ifcif 3922   {cpr 4012   U.cuni 4230    |-> cmpt 4491   ran crn 4986    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277   Fincfn 7514   CCcc 9488   RRcr 9489   1c1 9491    + caddc 9493   2c2 10586   (,)cioo 11533   sum_csu 13482   topGenctg 14707  TopOnctopon 19262    Cn ccn 19591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-er 7309  df-map 7420  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fsupp 7828  df-fi 7869  df-sup 7899  df-oi 7933  df-card 8318  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-q 11187  df-rp 11225  df-xneg 11322  df-xadd 11323  df-xmul 11324  df-ioo 11537  df-icc 11540  df-fz 11677  df-fzo 11799  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-sum 13483  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-starv 14584  df-sca 14585  df-vsca 14586  df-ip 14587  df-tset 14588  df-ple 14589  df-ds 14591  df-unif 14592  df-hom 14593  df-cco 14594  df-rest 14692  df-topn 14693  df-0g 14711  df-gsum 14712  df-topgen 14713  df-pt 14714  df-prds 14717  df-xrs 14771  df-qtop 14776  df-imas 14777  df-xps 14779  df-mre 14855  df-mrc 14856  df-acs 14858  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-submnd 15836  df-mulg 15929  df-cntz 16224  df-cmn 16669  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cn 19594  df-cnp 19595  df-tx 19929  df-hmeo 20122  df-xms 20689  df-ms 20690  df-tms 20691
This theorem is referenced by:  refsum2cn  31360
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