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Theorem summolem2 12280
Description: Lemma for summo 12281. (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 5605 . . . . 5  |-  ( m  =  j  ->  ( ZZ>=
`  m )  =  ( ZZ>= `  j )
)
21sseq2d 3282 . . . 4  |-  ( m  =  j  ->  ( A  C_  ( ZZ>= `  m
)  <->  A  C_  ( ZZ>= `  j ) ) )
3 seqeq1 11138 . . . . 5  |-  ( m  =  j  ->  seq  m (  +  ,  F )  =  seq  j (  +  ,  F ) )
43breq1d 4112 . . . 4  |-  ( m  =  j  ->  (  seq  m (  +  ,  F )  ~~>  x  <->  seq  j (  +  ,  F )  ~~>  x ) )
52, 4anbi12d 691 . . 3  |-  ( m  =  j  ->  (
( A  C_  ( ZZ>=
`  m )  /\  seq  m (  +  ,  F )  ~~>  x )  <-> 
( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) ) )
65cbvrexv 2841 . 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 737 . . . . . . . . . . 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 736 . . . . . . . . . . . . . . 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 10336 . . . . . . . . . . . . . . . 16  |-  ( ZZ>= `  j )  C_  ZZ
10 zssre 10120 . . . . . . . . . . . . . . . 16  |-  ZZ  C_  RR
119, 10sstri 3264 . . . . . . . . . . . . . . 15  |-  ( ZZ>= `  j )  C_  RR
128, 11syl6ss 3267 . . . . . . . . . . . . . 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 8990 . . . . . . . . . . . . . 14  |-  <  Or  RR
14 soss 4411 . . . . . . . . . . . . . 14  |-  ( A 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  A ) )
1512, 13, 14ee10 1376 . . . . . . . . . . . . 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 11123 . . . . . . . . . . . . . 14  |-  ( 1 ... m )  e. 
Fin
17 ovex 5967 . . . . . . . . . . . . . . . . 17  |-  ( 1 ... m )  e. 
_V
1817f1oen 6967 . . . . . . . . . . . . . . . 16  |-  ( f : ( 1 ... m ) -1-1-onto-> A  ->  ( 1 ... m )  ~~  A )
1918ad2antll 709 . . . . . . . . . . . . . . 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 )
20 ensym 6995 . . . . . . . . . . . . . . 15  |-  ( ( 1 ... m ) 
~~  A  ->  A  ~~  ( 1 ... m
) )
2119, 20syl 15 . . . . . . . . . . . . . 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 ) )
22 enfii 7165 . . . . . . . . . . . . . 14  |-  ( ( ( 1 ... m
)  e.  Fin  /\  A  ~~  ( 1 ... m ) )  ->  A  e.  Fin )
2316, 21, 22sylancr 644 . . . . . . . . . . . . 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 )
24 fz1iso 11490 . . . . . . . . . . . . 13  |-  ( (  <  Or  A  /\  A  e.  Fin )  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )
2515, 23, 24syl2anc 642 . . . . . . . . . . . 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
) )
26 summo.1 . . . . . . . . . . . . . . 15  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
27 simplll 734 . . . . . . . . . . . . . . . 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 )
28 summo.2 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
2927, 28sylan 457 . . . . . . . . . . . . . . 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 )
30 summo.3 . . . . . . . . . . . . . . 15  |-  G  =  ( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
31 eqid 2358 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  |->  [_ (
g `  n )  /  k ]_ B
)  =  ( n  e.  NN  |->  [_ (
g `  n )  /  k ]_ B
)
32 simprll 738 . . . . . . . . . . . . . . 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 )
33 simpllr 735 . . . . . . . . . . . . . . 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 )
34 simplrl 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 ) ) )  ->  A  C_  ( ZZ>=
`  j ) )
35 simprlr 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 ) ) )  ->  f :
( 1 ... m
)
-1-1-onto-> A )
36 simprr 733 . . . . . . . . . . . . . . 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
) )
3726, 29, 30, 31, 32, 33, 34, 35, 36summolem2a 12279 . . . . . . . . . . . . . 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 ) )
3837expr 598 . . . . . . . . . . . . 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 ) ) )
3938exlimdv 1636 . . . . . . . . . . . 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
) ) )
4025, 39mpd 14 . . . . . . . . . . 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
) )
41 climuni 12116 . . . . . . . . . . 11  |-  ( (  seq  j (  +  ,  F )  ~~>  x  /\  seq  j (  +  ,  F )  ~~>  (  seq  1 (  +  ,  G ) `  m
) )  ->  x  =  (  seq  1
(  +  ,  G
) `  m )
)
427, 40, 41syl2anc 642 . . . . . . . . . 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
) )
4342anassrs 629 . . . . . . . . 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 )
)
44 eqeq2 2367 . . . . . . . . 9  |-  ( y  =  (  seq  1
(  +  ,  G
) `  m )  ->  ( x  =  y  <-> 
x  =  (  seq  1 (  +  ,  G ) `  m
) ) )
4543, 44syl5ibrcom 213 . . . . . . . 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 ) )
4645expimpd 586 . . . . . . 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 )
)
4746exlimdv 1636 . . . . . 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 )
)
4847rexlimdva 2743 . . . . 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 ) )
4948ex 423 . . . 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 ) ) )
5049rexlimdva 2743 . . 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 ) ) )
5150imp 418 . 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 )
)
526, 51sylan2b 461 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 358   E.wex 1541    = wceq 1642    e. wcel 1710   E.wrex 2620   [_csb 3157    C_ wss 3228   ifcif 3641   class class class wbr 4102    e. cmpt 4156    Or wor 4392   -1-1-onto->wf1o 5333   ` cfv 5334    Isom wiso 5335  (class class class)co 5942    ~~ cen 6945   Fincfn 6948   CCcc 8822   RRcr 8823   0cc0 8824   1c1 8825    + caddc 8827    < clt 8954   NNcn 9833   ZZcz 10113   ZZ>=cuz 10319   ...cfz 10871    seq cseq 11135   #chash 11427    ~~> cli 12048
This theorem is referenced by:  summo  12281
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-sup 7281  df-oi 7312  df-card 7659  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-rp 10444  df-fz 10872  df-fzo 10960  df-seq 11136  df-exp 11195  df-hash 11428  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-clim 12052
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