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Theorem refsum2cnlem1 29897
Description: This is the core Lemma for refsum2cn 29898: 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 4479 . . . . . . . . 9  |-  F/_ k
( k  e.  {
1 ,  2 } 
|->  if ( k  =  1 ,  F ,  G ) )
42, 3nfcxfr 2611 . . . . . . . 8  |-  F/_ k A
5 nfcv 2613 . . . . . . . 8  |-  F/_ k
1
64, 5nffv 5796 . . . . . . 7  |-  F/_ k
( A `  1
)
7 nfcv 2613 . . . . . . 7  |-  F/_ k
x
86, 7nffv 5796 . . . . . 6  |-  F/_ k
( ( A ` 
1 ) `  x
)
98a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F/_ k
( ( A ` 
1 ) `  x
) )
10 nfcv 2613 . . . . . . . 8  |-  F/_ k
2
114, 10nffv 5796 . . . . . . 7  |-  F/_ k
( A `  2
)
1211, 7nffv 5796 . . . . . 6  |-  F/_ k
( ( A ` 
2 ) `  x
)
1312a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F/_ k
( ( A ` 
2 ) `  x
) )
14 ax-1cn 9441 . . . . . 6  |-  1  e.  CC
1514a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  1  e.  CC )
16 2cnd 10495 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  2  e.  CC )
17 1ex 9482 . . . . . . . . . . 11  |-  1  e.  _V
1817prid1 4081 . . . . . . . . . 10  |-  1  e.  { 1 ,  2 }
19 eqid 2451 . . . . . . . . . . . 12  |-  1  =  1
2019iftruei 3896 . . . . . . . . . . 11  |-  if ( 1  =  1 ,  F ,  G )  =  F
21 refsum2cnlem1.8 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
2220, 21syl5eqel 2543 . . . . . . . . . 10  |-  ( ph  ->  if ( 1  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
23 eqeq1 2455 . . . . . . . . . . . 12  |-  ( k  =  1  ->  (
k  =  1  <->  1  =  1 ) )
2423ifbid 3909 . . . . . . . . . . 11  |-  ( k  =  1  ->  if ( k  =  1 ,  F ,  G
)  =  if ( 1  =  1 ,  F ,  G ) )
2524, 2fvmptg 5871 . . . . . . . . . 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 2508 . . . . . . . 8  |-  ( ph  ->  ( A `  1
)  =  F )
2827adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( A `  1 )  =  F )
2928fveq1d 5791 . . . . . 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 18966 . . . . . . . . . 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 18648 . . . . . . . . . . . 12  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
3634, 35syl 16 . . . . . . . . . . 11  |-  ( ph  ->  X  =  U. J
)
3736eqcomd 2459 . . . . . . . . . 10  |-  ( ph  ->  U. J  =  X )
38 refsum2cnlem1.6 . . . . . . . . . . . . 13  |-  K  =  ( topGen `  ran  (,) )
3938unieqi 4198 . . . . . . . . . . . 12  |-  U. K  =  U. ( topGen `  ran  (,) )
40 uniretop 20457 . . . . . . . . . . . 12  |-  RR  =  U. ( topGen `  ran  (,) )
4139, 40eqtr4i 2483 . . . . . . . . . . 11  |-  U. K  =  RR
4241a1i 11 . . . . . . . . . 10  |-  ( ph  ->  U. K  =  RR )
4337, 42feq23d 5652 . . . . . . . . 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 5940 . . . . . . 7  |-  ( ( F : X --> RR  /\  x  e.  X )  ->  ( F `  x
)  e.  RR )
47 recn 9473 . . . . . . 7  |-  ( ( F `  x )  e.  RR  ->  ( F `  x )  e.  CC )
4845, 46, 473syl 20 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  CC )
4929, 48eqeltrd 2539 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  1
) `  x )  e.  CC )
50 2ex 10494 . . . . . . . . . . 11  |-  2  e.  _V
5150prid2 4082 . . . . . . . . . 10  |-  2  e.  { 1 ,  2 }
52 1ne2 10635 . . . . . . . . . . . . . 14  |-  1  =/=  2
5352necomi 2718 . . . . . . . . . . . . 13  |-  2  =/=  1
54 df-ne 2646 . . . . . . . . . . . . 13  |-  ( 2  =/=  1  <->  -.  2  =  1 )
5553, 54mpbi 208 . . . . . . . . . . . 12  |-  -.  2  =  1
5655iffalsei 3898 . . . . . . . . . . 11  |-  if ( 2  =  1 ,  F ,  G )  =  G
57 refsum2cnlem1.9 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
5856, 57syl5eqel 2543 . . . . . . . . . 10  |-  ( ph  ->  if ( 2  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
59 eqeq1 2455 . . . . . . . . . . . 12  |-  ( k  =  2  ->  (
k  =  1  <->  2  =  1 ) )
6059ifbid 3909 . . . . . . . . . . 11  |-  ( k  =  2  ->  if ( k  =  1 ,  F ,  G
)  =  if ( 2  =  1 ,  F ,  G ) )
6160, 2fvmptg 5871 . . . . . . . . . 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 2508 . . . . . . . 8  |-  ( ph  ->  ( A `  2
)  =  G )
6463adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( A `  2 )  =  G )
6564fveq1d 5791 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  2
) `  x )  =  ( G `  x ) )
6630, 31cnf 18966 . . . . . . . . . 10  |-  ( G  e.  ( J  Cn  K )  ->  G : U. J --> U. K
)
6757, 66syl 16 . . . . . . . . 9  |-  ( ph  ->  G : U. J --> U. K )
6837, 42feq23d 5652 . . . . . . . . 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 5940 . . . . . . 7  |-  ( ( G : X --> RR  /\  x  e.  X )  ->  ( G `  x
)  e.  RR )
72 recn 9473 . . . . . . 7  |-  ( ( G `  x )  e.  RR  ->  ( G `  x )  e.  CC )
7370, 71, 723syl 20 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  CC )
7465, 73eqeltrd 2539 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  2
) `  x )  e.  CC )
7552a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  1  =/=  2 )
76 fveq2 5789 . . . . . . 7  |-  ( k  =  1  ->  ( A `  k )  =  ( A ` 
1 ) )
7776fveq1d 5791 . . . . . 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 5789 . . . . . . 7  |-  ( k  =  2  ->  ( A `  k )  =  ( A ` 
2 ) )
8079fveq1d 5791 . . . . . 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 29895 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  sum_ k  e.  { 1 ,  2 }  ( ( A `
 k ) `  x )  =  ( ( ( A ` 
1 ) `  x
)  +  ( ( A `  2 ) `
 x ) ) )
8329, 65oveq12d 6208 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( A ` 
1 ) `  x
)  +  ( ( A `  2 ) `
 x ) )  =  ( ( F `
 x )  +  ( G `  x
) ) )
8482, 83eqtrd 2492 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  sum_ k  e.  { 1 ,  2 }  ( ( A `
 k ) `  x )  =  ( ( F `  x
)  +  ( G `
 x ) ) )
851, 84mpteq2da 4475 . 2  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  {
1 ,  2 }  ( ( A `  k ) `  x
) )  =  ( x  e.  X  |->  ( ( F `  x
)  +  ( G `
 x ) ) ) )
86 prfi 7687 . . . 4  |-  { 1 ,  2 }  e.  Fin
8786a1i 11 . . 3  |-  ( ph  ->  { 1 ,  2 }  e.  Fin )
88 eqid 2451 . . . . . . . . . 10  |-  X  =  X
8988ax-gen 1592 . . . . . . . . 9  |-  A. x  X  =  X
90 refsum2cnlem1.1 . . . . . . . . . . . 12  |-  F/_ x A
91 nfcv 2613 . . . . . . . . . . . 12  |-  F/_ x
k
9290, 91nffv 5796 . . . . . . . . . . 11  |-  F/_ x
( A `  k
)
93 refsum2cnlem1.2 . . . . . . . . . . 11  |-  F/_ x F
9492, 93nfeq 2623 . . . . . . . . . 10  |-  F/ x
( A `  k
)  =  F
95 fveq1 5788 . . . . . . . . . . 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 2815 . . . . . . . . 9  |-  ( ( A `  k )  =  F  ->  A. x  e.  X  ( ( A `  k ) `  x )  =  ( F `  x ) )
98 mpteq12f 4466 . . . . . . . . 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 20458 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
10238, 101eqeltri 2535 . . . . . . . . . . . 12  |-  K  e.  (TopOn `  RR )
103102a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  K  e.  (TopOn `  RR ) )
104 cnf2 18969 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )  /\  F  e.  ( J  Cn  K ) )  ->  F : X --> RR )
10534, 103, 21, 104syl3anc 1219 . . . . . . . . . 10  |-  ( ph  ->  F : X --> RR )
106 ffn 5657 . . . . . . . . . 10  |-  ( F : X --> RR  ->  F  Fn  X )
107105, 106syl 16 . . . . . . . . 9  |-  ( ph  ->  F  Fn  X )
10893dffn5f 5845 . . . . . . . . 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 2495 . . . . . 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 2539 . . . . 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 2623 . . . . . . . . . 10  |-  F/ x
( A `  k
)  =  G
117 fveq1 5788 . . . . . . . . . . 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 2815 . . . . . . . . 9  |-  ( ( A `  k )  =  G  ->  A. x  e.  X  ( ( A `  k ) `  x )  =  ( G `  x ) )
120 mpteq12f 4466 . . . . . . . . 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 18969 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )  /\  G  e.  ( J  Cn  K ) )  ->  G : X --> RR )
12434, 103, 57, 123syl3anc 1219 . . . . . . . . . 10  |-  ( ph  ->  G : X --> RR )
125 ffn 5657 . . . . . . . . . 10  |-  ( G : X --> RR  ->  G  Fn  X )
126124, 125syl 16 . . . . . . . . 9  |-  ( ph  ->  G  Fn  X )
127115dffn5f 5845 . . . . . . . . 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 2495 . . . . . 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 2539 . . . . 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 3929 . . . . . . . . . 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 5880 . . . . . . . 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 3895 . . . . . . 7  |-  ( k  =  1  ->  if ( k  =  1 ,  F ,  G
)  =  F )
141139, 140sylan9eq 2512 . . . . . 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 2731 . . . . . . . . . . . 12  |-  ( k  =  2  ->  (
1  =/=  k  <->  1  =/=  2 ) )
14552, 144mpbiri 233 . . . . . . . . . . 11  |-  ( k  =  2  ->  1  =/=  k )
146145necomd 2719 . . . . . . . . . 10  |-  ( k  =  2  ->  k  =/=  1 )
147146neneqd 2651 . . . . . . . . 9  |-  ( k  =  2  ->  -.  k  =  1 )
148147adantl 466 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  ->  -.  k  =  1
)
149 iffalse 3897 . . . . . . . 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 2492 . . . . . 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 3995 . . . . . 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 29890 . 2  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  {
1 ,  2 }  ( ( A `  k ) `  x
) )  e.  ( J  Cn  K ) )
15885, 157eqeltrrd 2540 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 1368    = wceq 1370   F/wnf 1590    e. wcel 1758   F/_wnfc 2599    =/= wne 2644   A.wral 2795   ifcif 3889   {cpr 3977   U.cuni 4189    |-> cmpt 4448   ran crn 4939    Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190   Fincfn 7410   CCcc 9381   RRcr 9382   1c1 9384    + caddc 9386   2c2 10472   (,)cioo 11401   sum_csu 13265   topGenctg 14478  TopOnctopon 18615    Cn ccn 18944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461  ax-addf 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-supp 6791  df-recs 6932  df-rdg 6966  df-1o 7020  df-2o 7021  df-oadd 7024  df-er 7201  df-map 7316  df-ixp 7364  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-fsupp 7722  df-fi 7762  df-sup 7792  df-oi 7825  df-card 8210  df-cda 8438  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-q 11055  df-rp 11093  df-xneg 11190  df-xadd 11191  df-xmul 11192  df-ioo 11405  df-icc 11408  df-fz 11539  df-fzo 11650  df-seq 11908  df-exp 11967  df-hash 12205  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-clim 13068  df-sum 13266  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-starv 14355  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-unif 14363  df-hom 14364  df-cco 14365  df-rest 14463  df-topn 14464  df-0g 14482  df-gsum 14483  df-topgen 14484  df-pt 14485  df-prds 14488  df-xrs 14542  df-qtop 14547  df-imas 14548  df-xps 14550  df-mre 14626  df-mrc 14627  df-acs 14629  df-mnd 15517  df-submnd 15567  df-mulg 15650  df-cntz 15937  df-cmn 16383  df-psmet 17918  df-xmet 17919  df-met 17920  df-bl 17921  df-mopn 17922  df-cnfld 17928  df-top 18619  df-bases 18621  df-topon 18622  df-topsp 18623  df-cn 18947  df-cnp 18948  df-tx 19251  df-hmeo 19444  df-xms 20011  df-ms 20012  df-tms 20013
This theorem is referenced by:  refsum2cn  29898
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