MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  summolem2 Structured version   Unicode version

Theorem summolem2 12515
Description: Lemma for summo 12516. (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 5731 . . . . 5  |-  ( m  =  j  ->  ( ZZ>=
`  m )  =  ( ZZ>= `  j )
)
21sseq2d 3378 . . . 4  |-  ( m  =  j  ->  ( A  C_  ( ZZ>= `  m
)  <->  A  C_  ( ZZ>= `  j ) ) )
3 seqeq1 11331 . . . . 5  |-  ( m  =  j  ->  seq  m (  +  ,  F )  =  seq  j (  +  ,  F ) )
43breq1d 4225 . . . 4  |-  ( m  =  j  ->  (  seq  m (  +  ,  F )  ~~>  x  <->  seq  j (  +  ,  F )  ~~>  x ) )
52, 4anbi12d 693 . . 3  |-  ( m  =  j  ->  (
( A  C_  ( ZZ>=
`  m )  /\  seq  m (  +  ,  F )  ~~>  x )  <-> 
( A  C_  ( ZZ>=
`  j )  /\  seq  j (  +  ,  F )  ~~>  x ) ) )
65cbvrexv 2935 . 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 739 . . . . . . . . . . 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 738 . . . . . . . . . . . . . . 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 10510 . . . . . . . . . . . . . . . 16  |-  ( ZZ>= `  j )  C_  ZZ
10 zssre 10294 . . . . . . . . . . . . . . . 16  |-  ZZ  C_  RR
119, 10sstri 3359 . . . . . . . . . . . . . . 15  |-  ( ZZ>= `  j )  C_  RR
128, 11syl6ss 3362 . . . . . . . . . . . . . 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 9161 . . . . . . . . . . . . . 14  |-  <  Or  RR
14 soss 4524 . . . . . . . . . . . . . 14  |-  ( A 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  A ) )
1512, 13, 14ee10 1386 . . . . . . . . . . . . 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 11316 . . . . . . . . . . . . . 14  |-  ( 1 ... m )  e. 
Fin
17 ovex 6109 . . . . . . . . . . . . . . . . 17  |-  ( 1 ... m )  e. 
_V
1817f1oen 7131 . . . . . . . . . . . . . . . 16  |-  ( f : ( 1 ... m ) -1-1-onto-> A  ->  ( 1 ... m )  ~~  A )
1918ad2antll 711 . . . . . . . . . . . . . . 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 7161 . . . . . . . . . . . . . 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 7329 . . . . . . . . . . . . . 14  |-  ( ( ( 1 ... m
)  e.  Fin  /\  A  ~~  ( 1 ... m ) )  ->  A  e.  Fin )
2216, 20, 21sylancr 646 . . . . . . . . . . . . 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 11716 . . . . . . . . . . . . 13  |-  ( (  <  Or  A  /\  A  e.  Fin )  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )
2415, 22, 23syl2anc 644 . . . . . . . . . . . 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 736 . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  |->  [_ (
g `  n )  /  k ]_ B
)  =  ( n  e.  NN  |->  [_ (
g `  n )  /  k ]_ B
)
31 simprll 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 ) ) )  ->  m  e.  NN )
32 simpllr 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 ) ) )  ->  j  e.  ZZ )
33 simplrl 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 ) ) )  ->  A  C_  ( ZZ>=
`  j ) )
34 simprlr 741 . . . . . . . . . . . . . . 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 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 ) ) )  ->  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )
3625, 28, 29, 30, 31, 32, 33, 34, 35summolem2a 12514 . . . . . . . . . . . . . 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 600 . . . . . . . . . . . . 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 1647 . . . . . . . . . . . 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 12351 . . . . . . . . . . 11  |-  ( (  seq  j (  +  ,  F )  ~~>  x  /\  seq  j (  +  ,  F )  ~~>  (  seq  1 (  +  ,  G ) `  m
) )  ->  x  =  (  seq  1
(  +  ,  G
) `  m )
)
417, 39, 40syl2anc 644 . . . . . . . . . 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 631 . . . . . . . . 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 2447 . . . . . . . . 9  |-  ( y  =  (  seq  1
(  +  ,  G
) `  m )  ->  ( x  =  y  <-> 
x  =  (  seq  1 (  +  ,  G ) `  m
) ) )
4442, 43syl5ibrcom 215 . . . . . . . 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 588 . . . . . . 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 1647 . . . . . 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 2832 . . . . 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 425 . . . 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 2832 . . 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 420 . 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 463 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 360   E.wex 1551    = wceq 1653    e. wcel 1726   E.wrex 2708   [_csb 3253    C_ wss 3322   ifcif 3741   class class class wbr 4215    e. cmpt 4269    Or wor 4505   -1-1-onto->wf1o 5456   ` cfv 5457    Isom wiso 5458  (class class class)co 6084    ~~ cen 7109   Fincfn 7112   CCcc 8993   RRcr 8994   0cc0 8995   1c1 8996    + caddc 8998    < clt 9125   NNcn 10005   ZZcz 10287   ZZ>=cuz 10493   ...cfz 11048    seq cseq 11328   #chash 11623    ~~> cli 12283
This theorem is referenced by:  summo  12516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287
  Copyright terms: Public domain W3C validator