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Theorem summolem2 12473
Description: Lemma for summo 12474. (Contributed by Mario Carneiro, 3-Apr-2014.)
Hypotheses
Ref Expression
summo.1  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
summo.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
summo.3  |-  G  =  ( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
Assertion
Ref Expression
summolem2  |-  ( (
ph  /\  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  F )  ~~>  x ) )  ->  ( E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq  1 (  +  ,  G ) `  m ) )  ->  x  =  y )
)
Distinct variable groups:    f, k, m, n, x, y, A   
f, F, k, m, n, x, y    k, G, m, n, x, y    ph, k, m, n, y    B, f, m, n, x, y    ph, x, f
Allowed substitution hints:    B( k)    G( f)

Proof of Theorem summolem2
Dummy variables  g 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5695 . . . . 5  |-  ( m  =  j  ->  ( ZZ>=
`  m )  =  ( ZZ>= `  j )
)
21sseq2d 3344 . . . 4  |-  ( m  =  j  ->  ( A  C_  ( ZZ>= `  m
)  <->  A  C_  ( ZZ>= `  j ) ) )
3 seqeq1 11289 . . . . 5  |-  ( m  =  j  ->  seq  m (  +  ,  F )  =  seq  j (  +  ,  F ) )
43breq1d 4190 . . . 4  |-  ( m  =  j  ->  (  seq  m (  +  ,  F )  ~~>  x  <->  seq  j (  +  ,  F )  ~~>  x ) )
52, 4anbi12d 692 . . 3  |-  ( m  =  j  ->  (
( A  C_  ( ZZ>=
`  m )  /\  seq  m (  +  ,  F )  ~~>  x )  <-> 
( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) ) )
65cbvrexv 2901 . 2  |-  ( E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m
)  /\  seq  m (  +  ,  F )  ~~>  x )  <->  E. j  e.  ZZ  ( A  C_  ( ZZ>= `  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )
7 simplrr 738 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  seq  j (  +  ,  F )  ~~>  x )
8 simplrl 737 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  A  C_  ( ZZ>= `  j
) )
9 uzssz 10469 . . . . . . . . . . . . . . . 16  |-  ( ZZ>= `  j )  C_  ZZ
10 zssre 10253 . . . . . . . . . . . . . . . 16  |-  ZZ  C_  RR
119, 10sstri 3325 . . . . . . . . . . . . . . 15  |-  ( ZZ>= `  j )  C_  RR
128, 11syl6ss 3328 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  A  C_  RR )
13 ltso 9120 . . . . . . . . . . . . . 14  |-  <  Or  RR
14 soss 4489 . . . . . . . . . . . . . 14  |-  ( A 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  A ) )
1512, 13, 14ee10 1382 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  <  Or  A )
16 fzfi 11274 . . . . . . . . . . . . . 14  |-  ( 1 ... m )  e. 
Fin
17 ovex 6073 . . . . . . . . . . . . . . . . 17  |-  ( 1 ... m )  e. 
_V
1817f1oen 7095 . . . . . . . . . . . . . . . 16  |-  ( f : ( 1 ... m ) -1-1-onto-> A  ->  ( 1 ... m )  ~~  A )
1918ad2antll 710 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  -> 
( 1 ... m
)  ~~  A )
2019ensymd 7125 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  A  ~~  ( 1 ... m ) )
21 enfii 7293 . . . . . . . . . . . . . 14  |-  ( ( ( 1 ... m
)  e.  Fin  /\  A  ~~  ( 1 ... m ) )  ->  A  e.  Fin )
2216, 20, 21sylancr 645 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  A  e.  Fin )
23 fz1iso 11674 . . . . . . . . . . . . 13  |-  ( (  <  Or  A  /\  A  e.  Fin )  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )
2415, 22, 23syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )
25 summo.1 . . . . . . . . . . . . . . 15  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
26 simplll 735 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A ) ) )  ->  ph )
27 summo.2 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
2826, 27sylan 458 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>= `  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A ) ) )  /\  k  e.  A )  ->  B  e.  CC )
29 summo.3 . . . . . . . . . . . . . . 15  |-  G  =  ( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
30 eqid 2412 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  |->  [_ (
g `  n )  /  k ]_ B
)  =  ( n  e.  NN  |->  [_ (
g `  n )  /  k ]_ B
)
31 simprll 739 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A ) ) )  ->  m  e.  NN )
32 simpllr 736 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A ) ) )  ->  j  e.  ZZ )
33 simplrl 737 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A ) ) )  ->  A  C_  ( ZZ>=
`  j ) )
34 simprlr 740 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A ) ) )  ->  f :
( 1 ... m
)
-1-1-onto-> A )
35 simprr 734 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A ) ) )  ->  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )
3625, 28, 29, 30, 31, 32, 33, 34, 35summolem2a 12472 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A )  /\  g  Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A ) ) )  ->  seq  j (  +  ,  F )  ~~>  (  seq  1 (  +  ,  G ) `
 m ) )
3736expr 599 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  -> 
( g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
)  ->  seq  j (  +  ,  F )  ~~>  (  seq  1 (  +  ,  G ) `
 m ) ) )
3837exlimdv 1643 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  -> 
( E. g  g 
Isom  <  ,  <  (
( 1 ... ( # `
 A ) ) ,  A )  ->  seq  j (  +  ,  F )  ~~>  (  seq  1 (  +  ,  G ) `  m
) ) )
3924, 38mpd 15 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  seq  j (  +  ,  F )  ~~>  (  seq  1 (  +  ,  G ) `  m
) )
40 climuni 12309 . . . . . . . . . . 11  |-  ( (  seq  j (  +  ,  F )  ~~>  x  /\  seq  j (  +  ,  F )  ~~>  (  seq  1 (  +  ,  G ) `  m
) )  ->  x  =  (  seq  1
(  +  ,  G
) `  m )
)
417, 39, 40syl2anc 643 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  ( m  e.  NN  /\  f : ( 1 ... m ) -1-1-onto-> A ) )  ->  x  =  (  seq  1 (  +  ,  G ) `  m
) )
4241anassrs 630 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>= `  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  m  e.  NN )  /\  f : ( 1 ... m ) -1-1-onto-> A )  ->  x  =  (  seq  1
(  +  ,  G
) `  m )
)
43 eqeq2 2421 . . . . . . . . 9  |-  ( y  =  (  seq  1
(  +  ,  G
) `  m )  ->  ( x  =  y  <-> 
x  =  (  seq  1 (  +  ,  G ) `  m
) ) )
4442, 43syl5ibrcom 214 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>= `  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  m  e.  NN )  /\  f : ( 1 ... m ) -1-1-onto-> A )  ->  (
y  =  (  seq  1 (  +  ,  G ) `  m
)  ->  x  =  y ) )
4544expimpd 587 . . . . . . 7  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  m  e.  NN )  ->  (
( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq  1 (  +  ,  G ) `  m ) )  ->  x  =  y )
)
4645exlimdv 1643 . . . . . 6  |-  ( ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  /\  m  e.  NN )  ->  ( E. f ( f : ( 1 ... m
)
-1-1-onto-> A  /\  y  =  (  seq  1 (  +  ,  G ) `  m ) )  ->  x  =  y )
)
4746rexlimdva 2798 . . . . 5  |-  ( ( ( ph  /\  j  e.  ZZ )  /\  ( A  C_  ( ZZ>= `  j
)  /\  seq  j (  +  ,  F )  ~~>  x ) )  -> 
( E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq  1 (  +  ,  G ) `  m
) )  ->  x  =  y ) )
4847ex 424 . . . 4  |-  ( (
ph  /\  j  e.  ZZ )  ->  ( ( A  C_  ( ZZ>= `  j )  /\  seq  j (  +  ,  F )  ~~>  x )  ->  ( E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq  1 (  +  ,  G ) `  m
) )  ->  x  =  y ) ) )
4948rexlimdva 2798 . . 3  |-  ( ph  ->  ( E. j  e.  ZZ  ( A  C_  ( ZZ>= `  j )  /\  seq  j (  +  ,  F )  ~~>  x )  ->  ( E. m  e.  NN  E. f ( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq  1 (  +  ,  G ) `  m
) )  ->  x  =  y ) ) )
5049imp 419 . 2  |-  ( (
ph  /\  E. j  e.  ZZ  ( A  C_  ( ZZ>= `  j )  /\  seq  j (  +  ,  F )  ~~>  x ) )  ->  ( E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq  1 (  +  ,  G ) `  m ) )  ->  x  =  y )
)
516, 50sylan2b 462 1  |-  ( (
ph  /\  E. m  e.  ZZ  ( A  C_  ( ZZ>= `  m )  /\  seq  m (  +  ,  F )  ~~>  x ) )  ->  ( E. m  e.  NN  E. f
( f : ( 1 ... m ) -1-1-onto-> A  /\  y  =  (  seq  1 (  +  ,  G ) `  m ) )  ->  x  =  y )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   E.wrex 2675   [_csb 3219    C_ wss 3288   ifcif 3707   class class class wbr 4180    e. cmpt 4234    Or wor 4470   -1-1-onto->wf1o 5420   ` cfv 5421    Isom wiso 5422  (class class class)co 6048    ~~ cen 7073   Fincfn 7076   CCcc 8952   RRcr 8953   0cc0 8954   1c1 8955    + caddc 8957    < clt 9084   NNcn 9964   ZZcz 10246   ZZ>=cuz 10452   ...cfz 11007    seq cseq 11286   #chash 11581    ~~> cli 12241
This theorem is referenced by:  summo  12474
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-fz 11008  df-fzo 11099  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245
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