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Theorem summolem2 13623
Description: Lemma for summo 13624. (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 5848 . . . . 5  |-  ( m  =  j  ->  ( ZZ>=
`  m )  =  ( ZZ>= `  j )
)
21sseq2d 3517 . . . 4  |-  ( m  =  j  ->  ( A  C_  ( ZZ>= `  m
)  <->  A  C_  ( ZZ>= `  j ) ) )
3 seqeq1 12095 . . . . 5  |-  ( m  =  j  ->  seq m (  +  ,  F )  =  seq j (  +  ,  F ) )
43breq1d 4449 . . . 4  |-  ( m  =  j  ->  (  seq m (  +  ,  F )  ~~>  x  <->  seq j
(  +  ,  F
)  ~~>  x ) )
52, 4anbi12d 708 . . 3  |-  ( m  =  j  ->  (
( A  C_  ( ZZ>=
`  m )  /\  seq m (  +  ,  F )  ~~>  x )  <-> 
( A  C_  ( ZZ>=
`  j )  /\  seq j (  +  ,  F )  ~~>  x ) ) )
65cbvrexv 3082 . 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 . . . . . . . . 9  |-  ( ( ( ( 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 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 11101 . . . . . . . . . . . . . 14  |-  ( ZZ>= `  j )  C_  ZZ
10 zssre 10867 . . . . . . . . . . . . . 14  |-  ZZ  C_  RR
119, 10sstri 3498 . . . . . . . . . . . . 13  |-  ( ZZ>= `  j )  C_  RR
128, 11syl6ss 3501 . . . . . . . . . . . 12  |-  ( ( ( ( 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 9654 . . . . . . . . . . . 12  |-  <  Or  RR
14 soss 4807 . . . . . . . . . . . 12  |-  ( A 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  A ) )
1512, 13, 14mpisyl 18 . . . . . . . . . . 11  |-  ( ( ( ( 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 12067 . . . . . . . . . . . 12  |-  ( 1 ... m )  e. 
Fin
17 ovex 6298 . . . . . . . . . . . . . . 15  |-  ( 1 ... m )  e. 
_V
1817f1oen 7529 . . . . . . . . . . . . . 14  |-  ( f : ( 1 ... m ) -1-1-onto-> A  ->  ( 1 ... m )  ~~  A )
1918ad2antll 726 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 7559 . . . . . . . . . . . 12  |-  ( ( ( ( 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 7730 . . . . . . . . . . . 12  |-  ( ( ( 1 ... m
)  e.  Fin  /\  A  ~~  ( 1 ... m ) )  ->  A  e.  Fin )
2216, 20, 21sylancr 661 . . . . . . . . . . 11  |-  ( ( ( ( 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 12498 . . . . . . . . . . 11  |-  ( (  <  Or  A  /\  A  e.  Fin )  ->  E. g  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )
2415, 22, 23syl2anc 659 . . . . . . . . . 10  |-  ( ( ( ( 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 . . . . . . . . . . . . 13  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  0 ) )
26 simplll 757 . . . . . . . . . . . . . 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 ) ) )  ->  ph )
27 summo.2 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
2826, 27sylan 469 . . . . . . . . . . . . 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 ) ) )  /\  k  e.  A )  ->  B  e.  CC )
29 summo.3 . . . . . . . . . . . . 13  |-  G  =  ( n  e.  NN  |->  [_ ( f `  n
)  /  k ]_ B )
30 eqid 2454 . . . . . . . . . . . . 13  |-  ( n  e.  NN  |->  [_ (
g `  n )  /  k ]_ B
)  =  ( n  e.  NN  |->  [_ (
g `  n )  /  k ]_ B
)
31 simprll 761 . . . . . . . . . . . . 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 ) ) )  ->  m  e.  NN )
32 simpllr 758 . . . . . . . . . . . . 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 ) ) )  ->  j  e.  ZZ )
33 simplrl 759 . . . . . . . . . . . . 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 ) ) )  ->  A  C_  ( ZZ>=
`  j ) )
34 simprlr 762 . . . . . . . . . . . . 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 ) ) )  ->  f :
( 1 ... m
)
-1-1-onto-> A )
35 simprr 755 . . . . . . . . . . . . 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 ) ) )  ->  g  Isom  <  ,  <  ( ( 1 ... ( # `  A
) ) ,  A
) )
3625, 28, 29, 30, 31, 32, 33, 34, 35summolem2a 13622 . . . . . . . . . . . 12  |-  ( ( ( ( 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 613 . . . . . . . . . . 11  |-  ( ( ( ( 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 1729 . . . . . . . . . 10  |-  ( ( ( ( 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 . . . . . . . . 9  |-  ( ( ( ( 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 13460 . . . . . . . . 9  |-  ( (  seq j (  +  ,  F )  ~~>  x  /\  seq j (  +  ,  F )  ~~>  (  seq 1 (  +  ,  G ) `  m
) )  ->  x  =  (  seq 1
(  +  ,  G
) `  m )
)
417, 39, 40syl2anc 659 . . . . . . . 8  |-  ( ( ( ( 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 646 . . . . . . 7  |-  ( ( ( ( ( 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 2469 . . . . . . 7  |-  ( y  =  (  seq 1
(  +  ,  G
) `  m )  ->  ( x  =  y  <-> 
x  =  (  seq 1 (  +  ,  G ) `  m
) ) )
4442, 43syl5ibrcom 222 . . . . . 6  |-  ( ( ( ( ( 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 601 . . . . 5  |-  ( ( ( ( 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 1729 . . . 4  |-  ( ( ( ( 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 2946 . . 3  |-  ( ( ( 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 ) )
4847r19.29an 2995 . 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 )
)
496, 48sylan2b 473 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 367    = wceq 1398   E.wex 1617    e. wcel 1823   E.wrex 2805   [_csb 3420    C_ wss 3461   ifcif 3929   class class class wbr 4439    |-> cmpt 4497    Or wor 4788   -1-1-onto->wf1o 5569   ` cfv 5570    Isom wiso 5571  (class class class)co 6270    ~~ cen 7506   Fincfn 7509   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    < clt 9617   NNcn 10531   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675    seqcseq 12092   #chash 12390    ~~> cli 13392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-seq 12093  df-exp 12152  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-clim 13396
This theorem is referenced by:  summo  13624
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