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Theorem prodmolem3 25251
Description: Lemma for prodmo 25254. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypotheses
Ref Expression
prodmo.1  |-  F  =  ( k  e.  ZZ  |->  if ( k  e.  A ,  B ,  1 ) )
prodmo.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
prodmo.3  |-  G  =  ( j  e.  NN  |->  [_ ( f `  j
)  /  k ]_ B )
prodmolem3.4  |-  H  =  ( j  e.  NN  |->  [_ ( K `  j
)  /  k ]_ B )
prodmolem3.5  |-  ( ph  ->  ( M  e.  NN  /\  N  e.  NN ) )
prodmolem3.6  |-  ( ph  ->  f : ( 1 ... M ) -1-1-onto-> A )
prodmolem3.7  |-  ( ph  ->  K : ( 1 ... N ) -1-1-onto-> A )
Assertion
Ref Expression
prodmolem3  |-  ( ph  ->  (  seq  1 (  x.  ,  G ) `
 M )  =  (  seq  1 (  x.  ,  H ) `
 N ) )
Distinct variable groups:    A, k    k, F    ph, k    B, j   
f, j, k    j, G    j, k, ph    j, K   
j, M
Allowed substitution hints:    ph( f)    A( f, j)    B( f, k)    F( f, j)    G( f, k)    H( f, j, k)    K( f, k)    M( f, k)    N( f, j, k)

Proof of Theorem prodmolem3
Dummy variables  i  m  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulcl 9066 . . . 4  |-  ( ( m  e.  CC  /\  j  e.  CC )  ->  ( m  x.  j
)  e.  CC )
21adantl 453 . . 3  |-  ( (
ph  /\  ( m  e.  CC  /\  j  e.  CC ) )  -> 
( m  x.  j
)  e.  CC )
3 mulcom 9068 . . . 4  |-  ( ( m  e.  CC  /\  j  e.  CC )  ->  ( m  x.  j
)  =  ( j  x.  m ) )
43adantl 453 . . 3  |-  ( (
ph  /\  ( m  e.  CC  /\  j  e.  CC ) )  -> 
( m  x.  j
)  =  ( j  x.  m ) )
5 mulass 9070 . . . 4  |-  ( ( m  e.  CC  /\  j  e.  CC  /\  z  e.  CC )  ->  (
( m  x.  j
)  x.  z )  =  ( m  x.  ( j  x.  z
) ) )
65adantl 453 . . 3  |-  ( (
ph  /\  ( m  e.  CC  /\  j  e.  CC  /\  z  e.  CC ) )  -> 
( ( m  x.  j )  x.  z
)  =  ( m  x.  ( j  x.  z ) ) )
7 prodmolem3.5 . . . . 5  |-  ( ph  ->  ( M  e.  NN  /\  N  e.  NN ) )
87simpld 446 . . . 4  |-  ( ph  ->  M  e.  NN )
9 nnuz 10513 . . . 4  |-  NN  =  ( ZZ>= `  1 )
108, 9syl6eleq 2525 . . 3  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
11 ssid 3359 . . . 4  |-  CC  C_  CC
1211a1i 11 . . 3  |-  ( ph  ->  CC  C_  CC )
13 prodmolem3.6 . . . . . 6  |-  ( ph  ->  f : ( 1 ... M ) -1-1-onto-> A )
14 f1ocnv 5679 . . . . . 6  |-  ( f : ( 1 ... M ) -1-1-onto-> A  ->  `' f : A -1-1-onto-> ( 1 ... M
) )
1513, 14syl 16 . . . . 5  |-  ( ph  ->  `' f : A -1-1-onto-> (
1 ... M ) )
16 prodmolem3.7 . . . . 5  |-  ( ph  ->  K : ( 1 ... N ) -1-1-onto-> A )
17 f1oco 5690 . . . . 5  |-  ( ( `' f : A -1-1-onto-> (
1 ... M )  /\  K : ( 1 ... N ) -1-1-onto-> A )  ->  ( `' f  o.  K
) : ( 1 ... N ) -1-1-onto-> ( 1 ... M ) )
1815, 16, 17syl2anc 643 . . . 4  |-  ( ph  ->  ( `' f  o.  K ) : ( 1 ... N ) -1-1-onto-> ( 1 ... M ) )
19 ovex 6098 . . . . . . . . . 10  |-  ( 1 ... N )  e. 
_V
2019f1oen 7120 . . . . . . . . 9  |-  ( ( `' f  o.  K
) : ( 1 ... N ) -1-1-onto-> ( 1 ... M )  -> 
( 1 ... N
)  ~~  ( 1 ... M ) )
2118, 20syl 16 . . . . . . . 8  |-  ( ph  ->  ( 1 ... N
)  ~~  ( 1 ... M ) )
22 fzfi 11303 . . . . . . . . 9  |-  ( 1 ... N )  e. 
Fin
23 fzfi 11303 . . . . . . . . 9  |-  ( 1 ... M )  e. 
Fin
24 hashen 11623 . . . . . . . . 9  |-  ( ( ( 1 ... N
)  e.  Fin  /\  ( 1 ... M
)  e.  Fin )  ->  ( ( # `  (
1 ... N ) )  =  ( # `  (
1 ... M ) )  <-> 
( 1 ... N
)  ~~  ( 1 ... M ) ) )
2522, 23, 24mp2an 654 . . . . . . . 8  |-  ( (
# `  ( 1 ... N ) )  =  ( # `  (
1 ... M ) )  <-> 
( 1 ... N
)  ~~  ( 1 ... M ) )
2621, 25sylibr 204 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  ( # `  (
1 ... M ) ) )
277simprd 450 . . . . . . . . 9  |-  ( ph  ->  N  e.  NN )
2827nnnn0d 10266 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
29 hashfz1 11622 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
3028, 29syl 16 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  N )
318nnnn0d 10266 . . . . . . . 8  |-  ( ph  ->  M  e.  NN0 )
32 hashfz1 11622 . . . . . . . 8  |-  ( M  e.  NN0  ->  ( # `  ( 1 ... M
) )  =  M )
3331, 32syl 16 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  M )
3426, 30, 333eqtr3rd 2476 . . . . . 6  |-  ( ph  ->  M  =  N )
3534oveq2d 6089 . . . . 5  |-  ( ph  ->  ( 1 ... M
)  =  ( 1 ... N ) )
36 f1oeq2 5658 . . . . 5  |-  ( ( 1 ... M )  =  ( 1 ... N )  ->  (
( `' f  o.  K ) : ( 1 ... M ) -1-1-onto-> ( 1 ... M )  <-> 
( `' f  o.  K ) : ( 1 ... N ) -1-1-onto-> ( 1 ... M ) ) )
3735, 36syl 16 . . . 4  |-  ( ph  ->  ( ( `' f  o.  K ) : ( 1 ... M
)
-1-1-onto-> ( 1 ... M
)  <->  ( `' f  o.  K ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... M
) ) )
3818, 37mpbird 224 . . 3  |-  ( ph  ->  ( `' f  o.  K ) : ( 1 ... M ) -1-1-onto-> ( 1 ... M ) )
39 elfznn 11072 . . . . . 6  |-  ( m  e.  ( 1 ... M )  ->  m  e.  NN )
4039adantl 453 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  m  e.  NN )
41 f1of 5666 . . . . . . . 8  |-  ( f : ( 1 ... M ) -1-1-onto-> A  ->  f :
( 1 ... M
) --> A )
4213, 41syl 16 . . . . . . 7  |-  ( ph  ->  f : ( 1 ... M ) --> A )
4342ffvelrnda 5862 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  (
f `  m )  e.  A )
44 prodmo.2 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
4544ralrimiva 2781 . . . . . . 7  |-  ( ph  ->  A. k  e.  A  B  e.  CC )
4645adantr 452 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  A. k  e.  A  B  e.  CC )
47 nfcsb1v 3275 . . . . . . . 8  |-  F/_ k [_ ( f `  m
)  /  k ]_ B
4847nfel1 2581 . . . . . . 7  |-  F/ k
[_ ( f `  m )  /  k ]_ B  e.  CC
49 csbeq1a 3251 . . . . . . . 8  |-  ( k  =  ( f `  m )  ->  B  =  [_ ( f `  m )  /  k ]_ B )
5049eleq1d 2501 . . . . . . 7  |-  ( k  =  ( f `  m )  ->  ( B  e.  CC  <->  [_ ( f `
 m )  / 
k ]_ B  e.  CC ) )
5148, 50rspc 3038 . . . . . 6  |-  ( ( f `  m )  e.  A  ->  ( A. k  e.  A  B  e.  CC  ->  [_ ( f `  m
)  /  k ]_ B  e.  CC )
)
5243, 46, 51sylc 58 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  [_ (
f `  m )  /  k ]_ B  e.  CC )
53 fveq2 5720 . . . . . . 7  |-  ( j  =  m  ->  (
f `  j )  =  ( f `  m ) )
5453csbeq1d 3249 . . . . . 6  |-  ( j  =  m  ->  [_ (
f `  j )  /  k ]_ B  =  [_ ( f `  m )  /  k ]_ B )
55 prodmo.3 . . . . . 6  |-  G  =  ( j  e.  NN  |->  [_ ( f `  j
)  /  k ]_ B )
5654, 55fvmptg 5796 . . . . 5  |-  ( ( m  e.  NN  /\  [_ ( f `  m
)  /  k ]_ B  e.  CC )  ->  ( G `  m
)  =  [_ (
f `  m )  /  k ]_ B
)
5740, 52, 56syl2anc 643 . . . 4  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  ( G `  m )  =  [_ ( f `  m )  /  k ]_ B )
5857, 52eqeltrd 2509 . . 3  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  ( G `  m )  e.  CC )
59 f1oeq2 5658 . . . . . . . . . . . 12  |-  ( ( 1 ... M )  =  ( 1 ... N )  ->  ( K : ( 1 ... M ) -1-1-onto-> A  <->  K : ( 1 ... N ) -1-1-onto-> A ) )
6035, 59syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( K : ( 1 ... M ) -1-1-onto-> A  <-> 
K : ( 1 ... N ) -1-1-onto-> A ) )
6116, 60mpbird 224 . . . . . . . . . 10  |-  ( ph  ->  K : ( 1 ... M ) -1-1-onto-> A )
62 f1of 5666 . . . . . . . . . 10  |-  ( K : ( 1 ... M ) -1-1-onto-> A  ->  K :
( 1 ... M
) --> A )
6361, 62syl 16 . . . . . . . . 9  |-  ( ph  ->  K : ( 1 ... M ) --> A )
64 fvco3 5792 . . . . . . . . 9  |-  ( ( K : ( 1 ... M ) --> A  /\  i  e.  ( 1 ... M ) )  ->  ( ( `' f  o.  K
) `  i )  =  ( `' f `
 ( K `  i ) ) )
6563, 64sylan 458 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( `' f  o.  K ) `  i
)  =  ( `' f `  ( K `
 i ) ) )
6665fveq2d 5724 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
f `  ( ( `' f  o.  K
) `  i )
)  =  ( f `
 ( `' f `
 ( K `  i ) ) ) )
6713adantr 452 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  f : ( 1 ... M ) -1-1-onto-> A )
6863ffvelrnda 5862 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( K `  i )  e.  A )
69 f1ocnvfv2 6007 . . . . . . . 8  |-  ( ( f : ( 1 ... M ) -1-1-onto-> A  /\  ( K `  i )  e.  A )  -> 
( f `  ( `' f `  ( K `  i )
) )  =  ( K `  i ) )
7067, 68, 69syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
f `  ( `' f `  ( K `  i ) ) )  =  ( K `  i ) )
7166, 70eqtrd 2467 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
f `  ( ( `' f  o.  K
) `  i )
)  =  ( K `
 i ) )
7271csbeq1d 3249 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  [_ (
f `  ( ( `' f  o.  K
) `  i )
)  /  k ]_ B  =  [_ ( K `
 i )  / 
k ]_ B )
7372fveq2d 5724 . . . 4  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (  _I  `  [_ ( f `
 ( ( `' f  o.  K ) `
 i ) )  /  k ]_ B
)  =  (  _I 
`  [_ ( K `  i )  /  k ]_ B ) )
74 f1of 5666 . . . . . . 7  |-  ( ( `' f  o.  K
) : ( 1 ... M ) -1-1-onto-> ( 1 ... M )  -> 
( `' f  o.  K ) : ( 1 ... M ) --> ( 1 ... M
) )
7538, 74syl 16 . . . . . 6  |-  ( ph  ->  ( `' f  o.  K ) : ( 1 ... M ) --> ( 1 ... M
) )
7675ffvelrnda 5862 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( `' f  o.  K ) `  i
)  e.  ( 1 ... M ) )
77 elfznn 11072 . . . . 5  |-  ( ( ( `' f  o.  K ) `  i
)  e.  ( 1 ... M )  -> 
( ( `' f  o.  K ) `  i )  e.  NN )
78 fveq2 5720 . . . . . . 7  |-  ( j  =  ( ( `' f  o.  K ) `
 i )  -> 
( f `  j
)  =  ( f `
 ( ( `' f  o.  K ) `
 i ) ) )
7978csbeq1d 3249 . . . . . 6  |-  ( j  =  ( ( `' f  o.  K ) `
 i )  ->  [_ ( f `  j
)  /  k ]_ B  =  [_ ( f `
 ( ( `' f  o.  K ) `
 i ) )  /  k ]_ B
)
8079, 55fvmpti 5797 . . . . 5  |-  ( ( ( `' f  o.  K ) `  i
)  e.  NN  ->  ( G `  ( ( `' f  o.  K
) `  i )
)  =  (  _I 
`  [_ ( f `  ( ( `' f  o.  K ) `  i ) )  / 
k ]_ B ) )
8176, 77, 803syl 19 . . . 4  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G `  ( ( `' f  o.  K
) `  i )
)  =  (  _I 
`  [_ ( f `  ( ( `' f  o.  K ) `  i ) )  / 
k ]_ B ) )
82 elfznn 11072 . . . . . 6  |-  ( i  e.  ( 1 ... M )  ->  i  e.  NN )
8382adantl 453 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  i  e.  NN )
84 fveq2 5720 . . . . . . 7  |-  ( j  =  i  ->  ( K `  j )  =  ( K `  i ) )
8584csbeq1d 3249 . . . . . 6  |-  ( j  =  i  ->  [_ ( K `  j )  /  k ]_ B  =  [_ ( K `  i )  /  k ]_ B )
86 prodmolem3.4 . . . . . 6  |-  H  =  ( j  e.  NN  |->  [_ ( K `  j
)  /  k ]_ B )
8785, 86fvmpti 5797 . . . . 5  |-  ( i  e.  NN  ->  ( H `  i )  =  (  _I  `  [_ ( K `  i )  /  k ]_ B
) )
8883, 87syl 16 . . . 4  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( H `  i )  =  (  _I  `  [_ ( K `  i )  /  k ]_ B
) )
8973, 81, 883eqtr4rd 2478 . . 3  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( H `  i )  =  ( G `  ( ( `' f  o.  K ) `  i ) ) )
902, 4, 6, 10, 12, 38, 58, 89seqf1o 11356 . 2  |-  ( ph  ->  (  seq  1 (  x.  ,  H ) `
 M )  =  (  seq  1 (  x.  ,  G ) `
 M ) )
9134fveq2d 5724 . 2  |-  ( ph  ->  (  seq  1 (  x.  ,  H ) `
 M )  =  (  seq  1 (  x.  ,  H ) `
 N ) )
9290, 91eqtr3d 2469 1  |-  ( ph  ->  (  seq  1 (  x.  ,  G ) `
 M )  =  (  seq  1 (  x.  ,  H ) `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   [_csb 3243    C_ wss 3312   ifcif 3731   class class class wbr 4204    e. cmpt 4258    _I cid 4485   `'ccnv 4869    o. ccom 4874   -->wf 5442   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073    ~~ cen 7098   Fincfn 7101   CCcc 8980   1c1 8983    x. cmul 8987   NNcn 9992   NN0cn0 10213   ZZcz 10274   ZZ>=cuz 10480   ...cfz 11035    seq cseq 11315   #chash 11610
This theorem is referenced by:  prodmolem2a  25252  prodmo  25254
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611
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