Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  refsum2cnlem1 Structured version   Unicode version

Theorem refsum2cnlem1 29730
Description: This is the core Lemma for refsum2cn 29731: 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 4376 . . . . . . . . 9  |-  F/_ k
( k  e.  {
1 ,  2 } 
|->  if ( k  =  1 ,  F ,  G ) )
42, 3nfcxfr 2571 . . . . . . . 8  |-  F/_ k A
5 nfcv 2574 . . . . . . . 8  |-  F/_ k
1
64, 5nffv 5693 . . . . . . 7  |-  F/_ k
( A `  1
)
7 nfcv 2574 . . . . . . 7  |-  F/_ k
x
86, 7nffv 5693 . . . . . 6  |-  F/_ k
( ( A ` 
1 ) `  x
)
98a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F/_ k
( ( A ` 
1 ) `  x
) )
10 nfcv 2574 . . . . . . . 8  |-  F/_ k
2
114, 10nffv 5693 . . . . . . 7  |-  F/_ k
( A `  2
)
1211, 7nffv 5693 . . . . . 6  |-  F/_ k
( ( A ` 
2 ) `  x
)
1312a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  F/_ k
( ( A ` 
2 ) `  x
) )
14 ax-1cn 9332 . . . . . 6  |-  1  e.  CC
1514a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  1  e.  CC )
16 2cnd 10386 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  2  e.  CC )
17 1ex 9373 . . . . . . . . . . 11  |-  1  e.  _V
1817prid1 3978 . . . . . . . . . 10  |-  1  e.  { 1 ,  2 }
19 eqid 2438 . . . . . . . . . . . 12  |-  1  =  1
2019iftruei 3793 . . . . . . . . . . 11  |-  if ( 1  =  1 ,  F ,  G )  =  F
21 refsum2cnlem1.8 . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( J  Cn  K ) )
2220, 21syl5eqel 2522 . . . . . . . . . 10  |-  ( ph  ->  if ( 1  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
23 eqeq1 2444 . . . . . . . . . . . 12  |-  ( k  =  1  ->  (
k  =  1  <->  1  =  1 ) )
2423ifbid 3806 . . . . . . . . . . 11  |-  ( k  =  1  ->  if ( k  =  1 ,  F ,  G
)  =  if ( 1  =  1 ,  F ,  G ) )
2524, 2fvmptg 5767 . . . . . . . . . 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 2486 . . . . . . . 8  |-  ( ph  ->  ( A `  1
)  =  F )
2827adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( A `  1 )  =  F )
2928fveq1d 5688 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  1
) `  x )  =  ( F `  x ) )
30 eqid 2438 . . . . . . . . . . 11  |-  U. J  =  U. J
31 eqid 2438 . . . . . . . . . . 11  |-  U. K  =  U. K
3230, 31cnf 18830 . . . . . . . . . 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 18512 . . . . . . . . . . . 12  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
3634, 35syl 16 . . . . . . . . . . 11  |-  ( ph  ->  X  =  U. J
)
3736eqcomd 2443 . . . . . . . . . 10  |-  ( ph  ->  U. J  =  X )
38 refsum2cnlem1.6 . . . . . . . . . . . . 13  |-  K  =  ( topGen `  ran  (,) )
3938unieqi 4095 . . . . . . . . . . . 12  |-  U. K  =  U. ( topGen `  ran  (,) )
40 uniretop 20321 . . . . . . . . . . . 12  |-  RR  =  U. ( topGen `  ran  (,) )
4139, 40eqtr4i 2461 . . . . . . . . . . 11  |-  U. K  =  RR
4241a1i 11 . . . . . . . . . 10  |-  ( ph  ->  U. K  =  RR )
4337, 42feq23d 5549 . . . . . . . . 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 5836 . . . . . . 7  |-  ( ( F : X --> RR  /\  x  e.  X )  ->  ( F `  x
)  e.  RR )
47 recn 9364 . . . . . . 7  |-  ( ( F `  x )  e.  RR  ->  ( F `  x )  e.  CC )
4845, 46, 473syl 20 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  CC )
4929, 48eqeltrd 2512 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  1
) `  x )  e.  CC )
50 2ex 10385 . . . . . . . . . . 11  |-  2  e.  _V
5150prid2 3979 . . . . . . . . . 10  |-  2  e.  { 1 ,  2 }
52 1ne2 10526 . . . . . . . . . . . . . 14  |-  1  =/=  2
5352necomi 2689 . . . . . . . . . . . . 13  |-  2  =/=  1
54 df-ne 2603 . . . . . . . . . . . . 13  |-  ( 2  =/=  1  <->  -.  2  =  1 )
5553, 54mpbi 208 . . . . . . . . . . . 12  |-  -.  2  =  1
5655iffalsei 3795 . . . . . . . . . . 11  |-  if ( 2  =  1 ,  F ,  G )  =  G
57 refsum2cnlem1.9 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  ( J  Cn  K ) )
5856, 57syl5eqel 2522 . . . . . . . . . 10  |-  ( ph  ->  if ( 2  =  1 ,  F ,  G )  e.  ( J  Cn  K ) )
59 eqeq1 2444 . . . . . . . . . . . 12  |-  ( k  =  2  ->  (
k  =  1  <->  2  =  1 ) )
6059ifbid 3806 . . . . . . . . . . 11  |-  ( k  =  2  ->  if ( k  =  1 ,  F ,  G
)  =  if ( 2  =  1 ,  F ,  G ) )
6160, 2fvmptg 5767 . . . . . . . . . 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 2486 . . . . . . . 8  |-  ( ph  ->  ( A `  2
)  =  G )
6463adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  ( A `  2 )  =  G )
6564fveq1d 5688 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  2
) `  x )  =  ( G `  x ) )
6630, 31cnf 18830 . . . . . . . . . 10  |-  ( G  e.  ( J  Cn  K )  ->  G : U. J --> U. K
)
6757, 66syl 16 . . . . . . . . 9  |-  ( ph  ->  G : U. J --> U. K )
6837, 42feq23d 5549 . . . . . . . . 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 5836 . . . . . . 7  |-  ( ( G : X --> RR  /\  x  e.  X )  ->  ( G `  x
)  e.  RR )
72 recn 9364 . . . . . . 7  |-  ( ( G `  x )  e.  RR  ->  ( G `  x )  e.  CC )
7370, 71, 723syl 20 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  CC )
7465, 73eqeltrd 2512 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  (
( A `  2
) `  x )  e.  CC )
7552a1i 11 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  1  =/=  2 )
76 fveq2 5686 . . . . . . 7  |-  ( k  =  1  ->  ( A `  k )  =  ( A ` 
1 ) )
7776fveq1d 5688 . . . . . 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 5686 . . . . . . 7  |-  ( k  =  2  ->  ( A `  k )  =  ( A ` 
2 ) )
8079fveq1d 5688 . . . . . 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 29728 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  sum_ k  e.  { 1 ,  2 }  ( ( A `
 k ) `  x )  =  ( ( ( A ` 
1 ) `  x
)  +  ( ( A `  2 ) `
 x ) ) )
8329, 65oveq12d 6104 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( A ` 
1 ) `  x
)  +  ( ( A `  2 ) `
 x ) )  =  ( ( F `
 x )  +  ( G `  x
) ) )
8482, 83eqtrd 2470 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  sum_ k  e.  { 1 ,  2 }  ( ( A `
 k ) `  x )  =  ( ( F `  x
)  +  ( G `
 x ) ) )
851, 84mpteq2da 4372 . 2  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  {
1 ,  2 }  ( ( A `  k ) `  x
) )  =  ( x  e.  X  |->  ( ( F `  x
)  +  ( G `
 x ) ) ) )
86 prfi 7578 . . . 4  |-  { 1 ,  2 }  e.  Fin
8786a1i 11 . . 3  |-  ( ph  ->  { 1 ,  2 }  e.  Fin )
88 eqid 2438 . . . . . . . . . 10  |-  X  =  X
8988ax-gen 1591 . . . . . . . . 9  |-  A. x  X  =  X
90 refsum2cnlem1.1 . . . . . . . . . . . 12  |-  F/_ x A
91 nfcv 2574 . . . . . . . . . . . 12  |-  F/_ x
k
9290, 91nffv 5693 . . . . . . . . . . 11  |-  F/_ x
( A `  k
)
93 refsum2cnlem1.2 . . . . . . . . . . 11  |-  F/_ x F
9492, 93nfeq 2581 . . . . . . . . . 10  |-  F/ x
( A `  k
)  =  F
95 fveq1 5685 . . . . . . . . . . 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 2792 . . . . . . . . 9  |-  ( ( A `  k )  =  F  ->  A. x  e.  X  ( ( A `  k ) `  x )  =  ( F `  x ) )
98 mpteq12f 4363 . . . . . . . . 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 20322 . . . . . . . . . . . . 13  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
10238, 101eqeltri 2508 . . . . . . . . . . . 12  |-  K  e.  (TopOn `  RR )
103102a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  K  e.  (TopOn `  RR ) )
104 cnf2 18833 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )  /\  F  e.  ( J  Cn  K ) )  ->  F : X --> RR )
10534, 103, 21, 104syl3anc 1218 . . . . . . . . . 10  |-  ( ph  ->  F : X --> RR )
106 ffn 5554 . . . . . . . . . 10  |-  ( F : X --> RR  ->  F  Fn  X )
107105, 106syl 16 . . . . . . . . 9  |-  ( ph  ->  F  Fn  X )
10893dffn5f 5741 . . . . . . . . 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 2473 . . . . . 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 2512 . . . . 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 2581 . . . . . . . . . 10  |-  F/ x
( A `  k
)  =  G
117 fveq1 5685 . . . . . . . . . . 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 2792 . . . . . . . . 9  |-  ( ( A `  k )  =  G  ->  A. x  e.  X  ( ( A `  k ) `  x )  =  ( G `  x ) )
120 mpteq12f 4363 . . . . . . . . 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 18833 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  RR )  /\  G  e.  ( J  Cn  K ) )  ->  G : X --> RR )
12434, 103, 57, 123syl3anc 1218 . . . . . . . . . 10  |-  ( ph  ->  G : X --> RR )
125 ffn 5554 . . . . . . . . . 10  |-  ( G : X --> RR  ->  G  Fn  X )
126124, 125syl 16 . . . . . . . . 9  |-  ( ph  ->  G  Fn  X )
127115dffn5f 5741 . . . . . . . . 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 2473 . . . . . 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 2512 . . . . 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 3826 . . . . . . . . . 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 5776 . . . . . . . 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 3792 . . . . . . 7  |-  ( k  =  1  ->  if ( k  =  1 ,  F ,  G
)  =  F )
141139, 140sylan9eq 2490 . . . . . 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 2612 . . . . . . . . . . . 12  |-  ( k  =  2  ->  (
1  =/=  k  <->  1  =/=  2 ) )
14552, 144mpbiri 233 . . . . . . . . . . 11  |-  ( k  =  2  ->  1  =/=  k )
146145necomd 2690 . . . . . . . . . 10  |-  ( k  =  2  ->  k  =/=  1 )
147146neneqd 2619 . . . . . . . . 9  |-  ( k  =  2  ->  -.  k  =  1 )
148147adantl 466 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  { 1 ,  2 } )  /\  k  =  2 )  ->  -.  k  =  1
)
149 iffalse 3794 . . . . . . . 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 2470 . . . . . 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 3892 . . . . . 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 29723 . 2  |-  ( ph  ->  ( x  e.  X  |-> 
sum_ k  e.  {
1 ,  2 }  ( ( A `  k ) `  x
) )  e.  ( J  Cn  K ) )
15885, 157eqeltrrd 2513 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 1367    = wceq 1369   F/wnf 1589    e. wcel 1756   F/_wnfc 2561    =/= wne 2601   A.wral 2710   ifcif 3786   {cpr 3874   U.cuni 4086    e. cmpt 4345   ran crn 4836    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6086   Fincfn 7302   CCcc 9272   RRcr 9273   1c1 9275    + caddc 9277   2c2 10363   (,)cioo 11292   sum_csu 13155   topGenctg 14368  TopOnctopon 18479    Cn ccn 18808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-icc 11299  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cn 18811  df-cnp 18812  df-tx 19115  df-hmeo 19308  df-xms 19875  df-ms 19876  df-tms 19877
This theorem is referenced by:  refsum2cn  29731
  Copyright terms: Public domain W3C validator