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Theorem fsumcl2lem 12220
Description: - Lemma for finite sum closures. (The "-" before "Lemma" forces the math content to be displayed in the Statement List - NM 11-Feb-2008.) (Contributed by Mario Carneiro, 3-Jun-2014.)
Hypotheses
Ref Expression
fsumcllem.1  |-  ( ph  ->  S  C_  CC )
fsumcllem.2  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
fsumcllem.3  |-  ( ph  ->  A  e.  Fin )
fsumcllem.4  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  S )
fsumcl2lem.5  |-  ( ph  ->  A  =/=  (/) )
Assertion
Ref Expression
fsumcl2lem  |-  ( ph  -> 
sum_ k  e.  A  B  e.  S )
Distinct variable groups:    x, k,
y, A    x, B, y    ph, k, x, y    S, k, x, y
Allowed substitution hint:    B( k)

Proof of Theorem fsumcl2lem
Dummy variables  f  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsumcl2lem.5 . . . 4  |-  ( ph  ->  A  =/=  (/) )
21a1d 22 . . 3  |-  ( ph  ->  ( -.  sum_ k  e.  A  B  e.  S  ->  A  =/=  (/) ) )
32necon4bd 2521 . 2  |-  ( ph  ->  ( A  =  (/)  -> 
sum_ k  e.  A  B  e.  S )
)
4 sumfc 12198 . . . . . . 7  |-  sum_ m  e.  A  ( (
k  e.  A  |->  B ) `  m )  =  sum_ k  e.  A  B
5 fveq2 5541 . . . . . . . 8  |-  ( m  =  ( f `  x )  ->  (
( k  e.  A  |->  B ) `  m
)  =  ( ( k  e.  A  |->  B ) `  ( f `
 x ) ) )
6 simprl 732 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  ( # `
 A )  e.  NN )
7 simprr 733 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A )
8 fsumcllem.1 . . . . . . . . . 10  |-  ( ph  ->  S  C_  CC )
98ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  m  e.  A )  ->  S  C_  CC )
10 fsumcllem.4 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  S )
11 eqid 2296 . . . . . . . . . . . 12  |-  ( k  e.  A  |->  B )  =  ( k  e.  A  |->  B )
1210, 11fmptd 5700 . . . . . . . . . . 11  |-  ( ph  ->  ( k  e.  A  |->  B ) : A --> S )
1312adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (
k  e.  A  |->  B ) : A --> S )
14 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( ( k  e.  A  |->  B ) : A --> S  /\  m  e.  A
)  ->  ( (
k  e.  A  |->  B ) `  m )  e.  S )
1513, 14sylan 457 . . . . . . . . 9  |-  ( ( ( ph  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  m  e.  A )  ->  ( ( k  e.  A  |->  B ) `  m )  e.  S
)
169, 15sseldd 3194 . . . . . . . 8  |-  ( ( ( ph  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  m  e.  A )  ->  ( ( k  e.  A  |->  B ) `  m )  e.  CC )
17 f1of 5488 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  f :
( 1 ... ( # `
 A ) ) --> A )
187, 17syl 15 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  f : ( 1 ... ( # `  A
) ) --> A )
19 fvco3 5612 . . . . . . . . 9  |-  ( ( f : ( 1 ... ( # `  A
) ) --> A  /\  x  e.  ( 1 ... ( # `  A
) ) )  -> 
( ( ( k  e.  A  |->  B )  o.  f ) `  x )  =  ( ( k  e.  A  |->  B ) `  (
f `  x )
) )
2018, 19sylan 457 . . . . . . . 8  |-  ( ( ( ph  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  x  e.  ( 1 ... ( # `  A
) ) )  -> 
( ( ( k  e.  A  |->  B )  o.  f ) `  x )  =  ( ( k  e.  A  |->  B ) `  (
f `  x )
) )
215, 6, 7, 16, 20fsum 12209 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  sum_ m  e.  A  ( (
k  e.  A  |->  B ) `  m )  =  (  seq  1
(  +  ,  ( ( k  e.  A  |->  B )  o.  f
) ) `  ( # `
 A ) ) )
224, 21syl5eqr 2342 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  sum_ k  e.  A  B  =  (  seq  1 (  +  ,  ( ( k  e.  A  |->  B )  o.  f ) ) `
 ( # `  A
) ) )
23 nnuz 10279 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
246, 23syl6eleq 2386 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  ( # `
 A )  e.  ( ZZ>= `  1 )
)
25 fco 5414 . . . . . . . . 9  |-  ( ( ( k  e.  A  |->  B ) : A --> S  /\  f : ( 1 ... ( # `  A ) ) --> A )  ->  ( (
k  e.  A  |->  B )  o.  f ) : ( 1 ... ( # `  A
) ) --> S )
2613, 18, 25syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (
( k  e.  A  |->  B )  o.  f
) : ( 1 ... ( # `  A
) ) --> S )
27 ffvelrn 5679 . . . . . . . 8  |-  ( ( ( ( k  e.  A  |->  B )  o.  f ) : ( 1 ... ( # `  A ) ) --> S  /\  x  e.  ( 1 ... ( # `  A ) ) )  ->  ( ( ( k  e.  A  |->  B )  o.  f ) `
 x )  e.  S )
2826, 27sylan 457 . . . . . . 7  |-  ( ( ( ph  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  x  e.  ( 1 ... ( # `  A
) ) )  -> 
( ( ( k  e.  A  |->  B )  o.  f ) `  x )  e.  S
)
29 fsumcllem.2 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
3029adantlr 695 . . . . . . 7  |-  ( ( ( ph  /\  (
( # `  A )  e.  NN  /\  f : ( 1 ... ( # `  A
) ) -1-1-onto-> A ) )  /\  ( x  e.  S  /\  y  e.  S
) )  ->  (
x  +  y )  e.  S )
3124, 28, 30seqcl 11082 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  (  seq  1 (  +  , 
( ( k  e.  A  |->  B )  o.  f ) ) `  ( # `  A ) )  e.  S )
3222, 31eqeltrd 2370 . . . . 5  |-  ( (
ph  /\  ( ( # `
 A )  e.  NN  /\  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A ) )  ->  sum_ k  e.  A  B  e.  S )
3332expr 598 . . . 4  |-  ( (
ph  /\  ( # `  A
)  e.  NN )  ->  ( f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A  ->  sum_ k  e.  A  B  e.  S )
)
3433exlimdv 1626 . . 3  |-  ( (
ph  /\  ( # `  A
)  e.  NN )  ->  ( E. f 
f : ( 1 ... ( # `  A
) ) -1-1-onto-> A  ->  sum_ k  e.  A  B  e.  S
) )
3534expimpd 586 . 2  |-  ( ph  ->  ( ( ( # `  A )  e.  NN  /\ 
E. f  f : ( 1 ... ( # `
 A ) ) -1-1-onto-> A )  ->  sum_ k  e.  A  B  e.  S
) )
36 fsumcllem.3 . . 3  |-  ( ph  ->  A  e.  Fin )
37 fz1f1o 12199 . . 3  |-  ( A  e.  Fin  ->  ( A  =  (/)  \/  (
( # `  A )  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
3836, 37syl 15 . 2  |-  ( ph  ->  ( A  =  (/)  \/  ( ( # `  A
)  e.  NN  /\  E. f  f : ( 1 ... ( # `  A ) ) -1-1-onto-> A ) ) )
393, 35, 38mpjaod 370 1  |-  ( ph  -> 
sum_ k  e.  A  B  e.  S )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   (/)c0 3468    e. cmpt 4093    o. ccom 4709   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   Fincfn 6879   CCcc 8751   1c1 8754    + caddc 8756   NNcn 9762   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062   #chash 11353   sum_csu 12174
This theorem is referenced by:  fsumcllem  12221  fsumrpcl  12226
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175
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