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Theorem refsum2cnlem1 30809
Description: This is the core Lemma for refsum2cn 30810: 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 4529 . . . . . . . . 9  |-  F/_ k
( k  e.  {
1 ,  2 } 
|->  if ( k  =  1 ,  F ,  G ) )
42, 3nfcxfr 2620 . . . . . . . 8  |-  F/_ k A
5 nfcv 2622 . . . . . . . 8  |-  F/_ k
1
64, 5nffv 5864 . . . . . . 7  |-  F/_ k
( A `  1
)
7 nfcv 2622 . . . . . . 7  |-  F/_ k
x
86, 7nffv 5864 . . . . . 6  |-  F/_ k
( ( A ` 
1 ) `  x
)
98a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F/_ k
( ( A ` 
1 ) `  x
) )
10 nfcv 2622 . . . . . . . 8  |-  F/_ k
2
114, 10nffv 5864 . . . . . . 7  |-  F/_ k
( A `  2
)
1211, 7nffv 5864 . . . . . 6  |-  F/_ k
( ( A ` 
2 ) `  x
)
1312a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F/_ k
( ( A ` 
2 ) `  x
) )
14 ax-1cn 9539 . . . . . 6  |-  1  e.  CC
1514a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  1  e.  CC )
16 2cnd 10597 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  2  e.  CC )
17 1ex 9580 . . . . . . . . . . 11  |-  1  e.  _V
1817prid1 4128 . . . . . . . . . 10  |-  1  e.  { 1 ,  2 }
19 eqid 2460 . . . . . . . . . . . 12  |-  1  =  1
2019iftruei 3939 . . . . . . . . . . 11  |-  if ( 1  =  1 ,  F ,  G )  =  F
21 refsum2cnlem1.8 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
2220, 21syl5eqel 2552 . . . . . . . . . 10  |-  ( ph  ->  if ( 1  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
23 eqeq1 2464 . . . . . . . . . . . 12  |-  ( k  =  1  ->  (
k  =  1  <->  1  =  1 ) )
2423ifbid 3954 . . . . . . . . . . 11  |-  ( k  =  1  ->  if ( k  =  1 ,  F ,  G
)  =  if ( 1  =  1 ,  F ,  G ) )
2524, 2fvmptg 5939 . . . . . . . . . 10  |-  ( ( 1  e.  { 1 ,  2 }  /\  if ( 1  =  1 ,  F ,  G
)  e.  ( J  Cn  K ) )  ->  ( A ` 
1 )  =  if ( 1  =  1 ,  F ,  G
) )
2618, 22, 25sylancr 663 . . . . . . . . 9  |-  ( ph  ->  ( A `  1
)  =  if ( 1  =  1 ,  F ,  G ) )
2726, 20syl6eq 2517 . . . . . . . 8  |-  ( ph  ->  ( A `  1
)  =  F )
2827adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( A `  1 )  =  F )
2928fveq1d 5859 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  1
) `  x )  =  ( F `  x ) )
30 eqid 2460 . . . . . . . . . . 11  |-  U. J  =  U. J
31 eqid 2460 . . . . . . . . . . 11  |-  U. K  =  U. K
3230, 31cnf 19506 . . . . . . . . . 10  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> U. K
)
3321, 32syl 16 . . . . . . . . 9  |-  ( ph  ->  F : U. J --> U. K )
34 refsum2cnlem1.7 . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  (TopOn `  X ) )
35 toponuni 19188 . . . . . . . . . . . 12  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
3634, 35syl 16 . . . . . . . . . . 11  |-  ( ph  ->  X  =  U. J
)
3736eqcomd 2468 . . . . . . . . . 10  |-  ( ph  ->  U. J  =  X )
38 refsum2cnlem1.6 . . . . . . . . . . . . 13  |-  K  =  ( topGen `  ran  (,) )
3938unieqi 4247 . . . . . . . . . . . 12  |-  U. K  =  U. ( topGen `  ran  (,) )
40 uniretop 20997 . . . . . . . . . . . 12  |-  RR  =  U. ( topGen `  ran  (,) )
4139, 40eqtr4i 2492 . . . . . . . . . . 11  |-  U. K  =  RR
4241a1i 11 . . . . . . . . . 10  |-  ( ph  ->  U. K  =  RR )
4337, 42feq23d 5717 . . . . . . . . 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 9571 . . . . . . 7  |-  ( ( F `  x )  e.  RR  ->  ( F `  x )  e.  CC )
4845, 46, 473syl 20 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  CC )
4929, 48eqeltrd 2548 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  1
) `  x )  e.  CC )
50 2ex 10596 . . . . . . . . . . 11  |-  2  e.  _V
5150prid2 4129 . . . . . . . . . 10  |-  2  e.  { 1 ,  2 }
52 1ne2 10737 . . . . . . . . . . . . . 14  |-  1  =/=  2
5352necomi 2730 . . . . . . . . . . . . 13  |-  2  =/=  1
54 df-ne 2657 . . . . . . . . . . . . 13  |-  ( 2  =/=  1  <->  -.  2  =  1 )
5553, 54mpbi 208 . . . . . . . . . . . 12  |-  -.  2  =  1
5655iffalsei 3942 . . . . . . . . . . 11  |-  if ( 2  =  1 ,  F ,  G )  =  G
57 refsum2cnlem1.9 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
5856, 57syl5eqel 2552 . . . . . . . . . 10  |-  ( ph  ->  if ( 2  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
59 eqeq1 2464 . . . . . . . . . . . 12  |-  ( k  =  2  ->  (
k  =  1  <->  2  =  1 ) )
6059ifbid 3954 . . . . . . . . . . 11  |-  ( k  =  2  ->  if ( k  =  1 ,  F ,  G
)  =  if ( 2  =  1 ,  F ,  G ) )
6160, 2fvmptg 5939 . . . . . . . . . 10  |-  ( ( 2  e.  { 1 ,  2 }  /\  if ( 2  =  1 ,  F ,  G
)  e.  ( J  Cn  K ) )  ->  ( A ` 
2 )  =  if ( 2  =  1 ,  F ,  G
) )
6251, 58, 61sylancr 663 . . . . . . . . 9  |-  ( ph  ->  ( A `  2
)  =  if ( 2  =  1 ,  F ,  G ) )
6362, 56syl6eq 2517 . . . . . . . 8  |-  ( ph  ->  ( A `  2
)  =  G )
6463adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( A `  2 )  =  G )
6564fveq1d 5859 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  2
) `  x )  =  ( G `  x ) )
6630, 31cnf 19506 . . . . . . . . . 10  |-  ( G  e.  ( J  Cn  K )  ->  G : U. J --> U. K
)
6757, 66syl 16 . . . . . . . . 9  |-  ( ph  ->  G : U. J --> U. K )
6837, 42feq23d 5717 . . . . . . . . 9  |-  ( ph  ->  ( G : U. J
--> U. K  <->  G : X
--> RR ) )
6967, 68mpbid 210 . . . . . . . 8  |-  ( ph  ->  G : X --> RR )
7069anim1i 568 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( G : X --> RR  /\  x  e.  X )
)
71 ffvelrn 6010 . . . . . . 7  |-  ( ( G : X --> RR  /\  x  e.  X )  ->  ( G `  x
)  e.  RR )
72 recn 9571 . . . . . . 7  |-  ( ( G `  x )  e.  RR  ->  ( G `  x )  e.  CC )
7370, 71, 723syl 20 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  CC )
7465, 73eqeltrd 2548 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  2
) `  x )  e.  CC )
7552a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  1  =/=  2 )
76 fveq2 5857 . . . . . . 7  |-  ( k  =  1  ->  ( A `  k )  =  ( A ` 
1 ) )
7776fveq1d 5859 . . . . . 6  |-  ( k  =  1  ->  (
( A `  k
) `  x )  =  ( ( A `
 1 ) `  x ) )
7877adantl 466 . . . . 5  |-  ( ( ( ph  /\  x  e.  X )  /\  k  =  1 )  -> 
( ( A `  k ) `  x
)  =  ( ( A `  1 ) `
 x ) )
79 fveq2 5857 . . . . . . 7  |-  ( k  =  2  ->  ( A `  k )  =  ( A ` 
2 ) )
8079fveq1d 5859 . . . . . 6  |-  ( k  =  2  ->  (
( A `  k
) `  x )  =  ( ( A `
 2 ) `  x ) )
8180adantl 466 . . . . 5  |-  ( ( ( ph  /\  x  e.  X )  /\  k  =  2 )  -> 
( ( A `  k ) `  x
)  =  ( ( A `  2 ) `
 x ) )
829, 13, 15, 16, 49, 74, 75, 78, 81sumpair 30807 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  sum_ k  e.  { 1 ,  2 }  ( ( A `
 k ) `  x )  =  ( ( ( A ` 
1 ) `  x
)  +  ( ( A `  2 ) `
 x ) ) )
8329, 65oveq12d 6293 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( A ` 
1 ) `  x
)  +  ( ( A `  2 ) `
 x ) )  =  ( ( F `
 x )  +  ( G `  x
) ) )
8482, 83eqtrd 2501 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  sum_ k  e.  { 1 ,  2 }  ( ( A `
 k ) `  x )  =  ( ( F `  x
)  +  ( G `
 x ) ) )
851, 84mpteq2da 4525 . 2  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  {
1 ,  2 }  ( ( A `  k ) `  x
) )  =  ( x  e.  X  |->  ( ( F `  x
)  +  ( G `
 x ) ) ) )
86 prfi 7784 . . . 4  |-  { 1 ,  2 }  e.  Fin
8786a1i 11 . . 3  |-  ( ph  ->  { 1 ,  2 }  e.  Fin )
88 eqid 2460 . . . . . . . . . 10  |-  X  =  X
8988ax-gen 1596 . . . . . . . . 9  |-  A. x  X  =  X
90 refsum2cnlem1.1 . . . . . . . . . . . 12  |-  F/_ x A
91 nfcv 2622 . . . . . . . . . . . 12  |-  F/_ x
k
9290, 91nffv 5864 . . . . . . . . . . 11  |-  F/_ x
( A `  k
)
93 refsum2cnlem1.2 . . . . . . . . . . 11  |-  F/_ x F
9492, 93nfeq 2633 . . . . . . . . . 10  |-  F/ x
( A `  k
)  =  F
95 fveq1 5856 . . . . . . . . . . 11  |-  ( ( A `  k )  =  F  ->  (
( A `  k
) `  x )  =  ( F `  x ) )
9695a1d 25 . . . . . . . . . 10  |-  ( ( A `  k )  =  F  ->  (
x  e.  X  -> 
( ( A `  k ) `  x
)  =  ( F `
 x ) ) )
9794, 96ralrimi 2857 . . . . . . . . 9  |-  ( ( A `  k )  =  F  ->  A. x  e.  X  ( ( A `  k ) `  x )  =  ( F `  x ) )
98 mpteq12f 4516 . . . . . . . . 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 ) ) )
9989, 97, 98sylancr 663 . . . . . . . 8  |-  ( ( A `  k )  =  F  ->  (
x  e.  X  |->  ( ( A `  k
) `  x )
)  =  ( x  e.  X  |->  ( F `
 x ) ) )
10099adantl 466 . . . . . . 7  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  ( x  e.  X  |->  ( F `  x ) ) )
101 retopon 20998 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
10238, 101eqeltri 2544 . . . . . . . . . . . 12  |-  K  e.  (TopOn `  RR )
103102a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  K  e.  (TopOn `  RR ) )
104 cnf2 19509 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )  /\  F  e.  ( J  Cn  K ) )  ->  F : X --> RR )
10534, 103, 21, 104syl3anc 1223 . . . . . . . . . 10  |-  ( ph  ->  F : X --> RR )
106 ffn 5722 . . . . . . . . . 10  |-  ( F : X --> RR  ->  F  Fn  X )
107105, 106syl 16 . . . . . . . . 9  |-  ( ph  ->  F  Fn  X )
10893dffn5f 5913 . . . . . . . . 9  |-  ( F  Fn  X  <->  F  =  ( x  e.  X  |->  ( F `  x
) ) )
109107, 108sylib 196 . . . . . . . 8  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
110109adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  F  =  ( x  e.  X  |->  ( F `  x
) ) )
111100, 110eqtr4d 2504 . . . . . 6  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  F )
11221adantr 465 . . . . . 6  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  F  e.  ( J  Cn  K
) )
113111, 112eqeltrd 2548 . . . . 5  |-  ( (
ph  /\  ( A `  k )  =  F )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  e.  ( J  Cn  K ) )
114113adantlr 714 . . . 4  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  ( A `  k )  =  F )  ->  (
x  e.  X  |->  ( ( A `  k
) `  x )
)  e.  ( J  Cn  K ) )
115 refsum2cnlem1.3 . . . . . . . . . . 11  |-  F/_ x G
11692, 115nfeq 2633 . . . . . . . . . 10  |-  F/ x
( A `  k
)  =  G
117 fveq1 5856 . . . . . . . . . . 11  |-  ( ( A `  k )  =  G  ->  (
( A `  k
) `  x )  =  ( G `  x ) )
118117a1d 25 . . . . . . . . . 10  |-  ( ( A `  k )  =  G  ->  (
x  e.  X  -> 
( ( A `  k ) `  x
)  =  ( G `
 x ) ) )
119116, 118ralrimi 2857 . . . . . . . . 9  |-  ( ( A `  k )  =  G  ->  A. x  e.  X  ( ( A `  k ) `  x )  =  ( G `  x ) )
120 mpteq12f 4516 . . . . . . . . 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 ) ) )
12189, 119, 120sylancr 663 . . . . . . . 8  |-  ( ( A `  k )  =  G  ->  (
x  e.  X  |->  ( ( A `  k
) `  x )
)  =  ( x  e.  X  |->  ( G `
 x ) ) )
122121adantl 466 . . . . . . 7  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  ( x  e.  X  |->  ( G `  x ) ) )
123 cnf2 19509 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )  /\  G  e.  ( J  Cn  K ) )  ->  G : X --> RR )
12434, 103, 57, 123syl3anc 1223 . . . . . . . . . 10  |-  ( ph  ->  G : X --> RR )
125 ffn 5722 . . . . . . . . . 10  |-  ( G : X --> RR  ->  G  Fn  X )
126124, 125syl 16 . . . . . . . . 9  |-  ( ph  ->  G  Fn  X )
127115dffn5f 5913 . . . . . . . . 9  |-  ( G  Fn  X  <->  G  =  ( x  e.  X  |->  ( G `  x
) ) )
128126, 127sylib 196 . . . . . . . 8  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
129128adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  G  =  ( x  e.  X  |->  ( G `  x
) ) )
130122, 129eqtr4d 2504 . . . . . 6  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  =  G )
13157adantr 465 . . . . . 6  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  G  e.  ( J  Cn  K
) )
132130, 131eqeltrd 2548 . . . . 5  |-  ( (
ph  /\  ( A `  k )  =  G )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  e.  ( J  Cn  K ) )
133132adantlr 714 . . . 4  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  ( A `  k )  =  G )  ->  (
x  e.  X  |->  ( ( A `  k
) `  x )
)  e.  ( J  Cn  K ) )
134 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  k  e.  { 1 ,  2 } )
135 ifcl 3974 . . . . . . . . . 10  |-  ( ( F  e.  ( J  Cn  K )  /\  G  e.  ( J  Cn  K ) )  ->  if ( k  =  1 ,  F ,  G
)  e.  ( J  Cn  K ) )
13621, 57, 135syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  if ( k  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
137136adantr 465 . . . . . . . 8  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  if (
k  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
1382fvmpt2 5948 . . . . . . . 8  |-  ( ( k  e.  { 1 ,  2 }  /\  if ( k  =  1 ,  F ,  G
)  e.  ( J  Cn  K ) )  ->  ( A `  k )  =  if ( k  =  1 ,  F ,  G
) )
139134, 137, 138syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  ( A `  k )  =  if ( k  =  1 ,  F ,  G
) )
140 iftrue 3938 . . . . . . 7  |-  ( k  =  1  ->  if ( k  =  1 ,  F ,  G
)  =  F )
141139, 140sylan9eq 2521 . . . . . 6  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  1 )  -> 
( A `  k
)  =  F )
142141orcd 392 . . . . 5  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  1 )  -> 
( ( A `  k )  =  F  \/  ( A `  k )  =  G ) )
143139adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  -> 
( A `  k
)  =  if ( k  =  1 ,  F ,  G ) )
144 neeq2 2743 . . . . . . . . . . . 12  |-  ( k  =  2  ->  (
1  =/=  k  <->  1  =/=  2 ) )
14552, 144mpbiri 233 . . . . . . . . . . 11  |-  ( k  =  2  ->  1  =/=  k )
146145necomd 2731 . . . . . . . . . 10  |-  ( k  =  2  ->  k  =/=  1 )
147146neneqd 2662 . . . . . . . . 9  |-  ( k  =  2  ->  -.  k  =  1 )
148147adantl 466 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  ->  -.  k  =  1
)
149 iffalse 3941 . . . . . . . 8  |-  ( -.  k  =  1  ->  if ( k  =  1 ,  F ,  G
)  =  G )
150148, 149syl 16 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  ->  if ( k  =  1 ,  F ,  G
)  =  G )
151143, 150eqtrd 2501 . . . . . 6  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  -> 
( A `  k
)  =  G )
152151olcd 393 . . . . 5  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  -> 
( ( A `  k )  =  F  \/  ( A `  k )  =  G ) )
153 elpri 4040 . . . . . 6  |-  ( k  e.  { 1 ,  2 }  ->  (
k  =  1  \/  k  =  2 ) )
154153adantl 466 . . . . 5  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  ( k  =  1  \/  k  =  2 ) )
155142, 152, 154mpjaodan 784 . . . 4  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  ( ( A `  k )  =  F  \/  ( A `  k )  =  G ) )
156114, 133, 155mpjaodan 784 . . 3  |-  ( (
ph  /\  k  e.  { 1 ,  2 } )  ->  ( x  e.  X  |->  ( ( A `  k ) `
 x ) )  e.  ( J  Cn  K ) )
1571, 38, 34, 87, 156refsumcn 30802 . 2  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  {
1 ,  2 }  ( ( A `  k ) `  x
) )  e.  ( J  Cn  K ) )
15885, 157eqeltrrd 2549 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 1372    = wceq 1374   F/wnf 1594    e. wcel 1762   F/_wnfc 2608    =/= wne 2655   A.wral 2807   ifcif 3932   {cpr 4022   U.cuni 4238    |-> cmpt 4498   ran crn 4993    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275   Fincfn 7506   CCcc 9479   RRcr 9480   1c1 9482    + caddc 9484   2c2 10574   (,)cioo 11518   sum_csu 13457   topGenctg 14682  TopOnctopon 19155    Cn ccn 19484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-icc 11525  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-sum 13458  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cn 19487  df-cnp 19488  df-tx 19791  df-hmeo 19984  df-xms 20551  df-ms 20552  df-tms 20553
This theorem is referenced by:  refsum2cn  30810
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