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Theorem summolem2 13504
Description: Lemma for summo 13505. (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 5866 . . . . 5  |-  ( m  =  j  ->  ( ZZ>=
`  m )  =  ( ZZ>= `  j )
)
21sseq2d 3532 . . . 4  |-  ( m  =  j  ->  ( A  C_  ( ZZ>= `  m
)  <->  A  C_  ( ZZ>= `  j ) ) )
3 seqeq1 12079 . . . . 5  |-  ( m  =  j  ->  seq m (  +  ,  F )  =  seq j (  +  ,  F ) )
43breq1d 4457 . . . 4  |-  ( m  =  j  ->  (  seq m (  +  ,  F )  ~~>  x  <->  seq j
(  +  ,  F
)  ~~>  x ) )
52, 4anbi12d 710 . . 3  |-  ( m  =  j  ->  (
( A  C_  ( ZZ>=
`  m )  /\  seq m (  +  ,  F )  ~~>  x )  <-> 
( A  C_  ( ZZ>=
`  j )  /\  seq j (  +  ,  F )  ~~>  x ) ) )
65cbvrexv 3089 . 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 760 . . . . . . . . . . 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 759 . . . . . . . . . . . . . . 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 11102 . . . . . . . . . . . . . . . 16  |-  ( ZZ>= `  j )  C_  ZZ
10 zssre 10872 . . . . . . . . . . . . . . . 16  |-  ZZ  C_  RR
119, 10sstri 3513 . . . . . . . . . . . . . . 15  |-  ( ZZ>= `  j )  C_  RR
128, 11syl6ss 3516 . . . . . . . . . . . . . 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 9666 . . . . . . . . . . . . . 14  |-  <  Or  RR
14 soss 4818 . . . . . . . . . . . . . 14  |-  ( A 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  A ) )
1512, 13, 14mpisyl 18 . . . . . . . . . . . . 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 12051 . . . . . . . . . . . . . 14  |-  ( 1 ... m )  e. 
Fin
17 ovex 6310 . . . . . . . . . . . . . . . . 17  |-  ( 1 ... m )  e. 
_V
1817f1oen 7537 . . . . . . . . . . . . . . . 16  |-  ( f : ( 1 ... m ) -1-1-onto-> A  ->  ( 1 ... m )  ~~  A )
1918ad2antll 728 . . . . . . . . . . . . . . 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 7567 . . . . . . . . . . . . . 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 7738 . . . . . . . . . . . . . 14  |-  ( ( ( 1 ... m
)  e.  Fin  /\  A  ~~  ( 1 ... m ) )  ->  A  e.  Fin )
2216, 20, 21sylancr 663 . . . . . . . . . . . . 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 12478 . . . . . . . . . . . . 13  |-  ( (  <  Or  A  /\  A  e.  Fin )  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )
2415, 22, 23syl2anc 661 . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  |->  [_ (
g `  n )  /  k ]_ B
)  =  ( n  e.  NN  |->  [_ (
g `  n )  /  k ]_ B
)
31 simprll 761 . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . 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 13503 . . . . . . . . . . . . . 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 615 . . . . . . . . . . . . 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 1700 . . . . . . . . . . . 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 13341 . . . . . . . . . . 11  |-  ( (  seq j (  +  ,  F )  ~~>  x  /\  seq j (  +  ,  F )  ~~>  (  seq 1 (  +  ,  G ) `  m
) )  ->  x  =  (  seq 1
(  +  ,  G
) `  m )
)
417, 39, 40syl2anc 661 . . . . . . . . . 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 648 . . . . . . . . 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 2482 . . . . . . . . 9  |-  ( y  =  (  seq 1
(  +  ,  G
) `  m )  ->  ( x  =  y  <-> 
x  =  (  seq 1 (  +  ,  G ) `  m
) ) )
4442, 43syl5ibrcom 222 . . . . . . . 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 603 . . . . . . 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 1700 . . . . . 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 2955 . . . . 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 434 . . . 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 2955 . . 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 429 . 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 475 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   E.wrex 2815   [_csb 3435    C_ wss 3476   ifcif 3939   class class class wbr 4447    |-> cmpt 4505    Or wor 4799   -1-1-onto->wf1o 5587   ` cfv 5588    Isom wiso 5589  (class class class)co 6285    ~~ cen 7514   Fincfn 7517   CCcc 9491   RRcr 9492   0cc0 9493   1c1 9494    + caddc 9496    < clt 9629   NNcn 10537   ZZcz 10865   ZZ>=cuz 11083   ...cfz 11673    seqcseq 12076   #chash 12374    ~~> cli 13273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-oadd 7135  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-oi 7936  df-card 8321  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-fz 11674  df-fzo 11794  df-seq 12077  df-exp 12136  df-hash 12375  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-clim 13277
This theorem is referenced by:  summo  13505
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