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Theorem prodmolem3 27445
Description: Lemma for prodmo 27448. (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 9365 . . . 4  |-  ( ( m  e.  CC  /\  j  e.  CC )  ->  ( m  x.  j
)  e.  CC )
21adantl 466 . . 3  |-  ( (
ph  /\  ( m  e.  CC  /\  j  e.  CC ) )  -> 
( m  x.  j
)  e.  CC )
3 mulcom 9367 . . . 4  |-  ( ( m  e.  CC  /\  j  e.  CC )  ->  ( m  x.  j
)  =  ( j  x.  m ) )
43adantl 466 . . 3  |-  ( (
ph  /\  ( m  e.  CC  /\  j  e.  CC ) )  -> 
( m  x.  j
)  =  ( j  x.  m ) )
5 mulass 9369 . . . 4  |-  ( ( m  e.  CC  /\  j  e.  CC  /\  z  e.  CC )  ->  (
( m  x.  j
)  x.  z )  =  ( m  x.  ( j  x.  z
) ) )
65adantl 466 . . 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 459 . . . 4  |-  ( ph  ->  M  e.  NN )
9 nnuz 10895 . . . 4  |-  NN  =  ( ZZ>= `  1 )
108, 9syl6eleq 2532 . . 3  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
11 ssid 3374 . . . 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 5652 . . . . . 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 5662 . . . . 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 661 . . . 4  |-  ( ph  ->  ( `' f  o.  K ) : ( 1 ... N ) -1-1-onto-> ( 1 ... M ) )
19 ovex 6115 . . . . . . . . . 10  |-  ( 1 ... N )  e. 
_V
2019f1oen 7329 . . . . . . . . 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 11793 . . . . . . . . 9  |-  ( 1 ... N )  e. 
Fin
23 fzfi 11793 . . . . . . . . 9  |-  ( 1 ... M )  e. 
Fin
24 hashen 12117 . . . . . . . . 9  |-  ( ( ( 1 ... N
)  e.  Fin  /\  ( 1 ... M
)  e.  Fin )  ->  ( ( # `  (
1 ... N ) )  =  ( # `  (
1 ... M ) )  <-> 
( 1 ... N
)  ~~  ( 1 ... M ) ) )
2522, 23, 24mp2an 672 . . . . . . . 8  |-  ( (
# `  ( 1 ... N ) )  =  ( # `  (
1 ... M ) )  <-> 
( 1 ... N
)  ~~  ( 1 ... M ) )
2621, 25sylibr 212 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  ( # `  (
1 ... M ) ) )
277simprd 463 . . . . . . . . 9  |-  ( ph  ->  N  e.  NN )
2827nnnn0d 10635 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
29 hashfz1 12116 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
3028, 29syl 16 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  N )
318nnnn0d 10635 . . . . . . . 8  |-  ( ph  ->  M  e.  NN0 )
32 hashfz1 12116 . . . . . . . 8  |-  ( M  e.  NN0  ->  ( # `  ( 1 ... M
) )  =  M )
3331, 32syl 16 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  M )
3426, 30, 333eqtr3rd 2483 . . . . . 6  |-  ( ph  ->  M  =  N )
3534oveq2d 6106 . . . . 5  |-  ( ph  ->  ( 1 ... M
)  =  ( 1 ... N ) )
36 f1oeq2 5632 . . . . 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 232 . . 3  |-  ( ph  ->  ( `' f  o.  K ) : ( 1 ... M ) -1-1-onto-> ( 1 ... M ) )
39 elfznn 11477 . . . . . 6  |-  ( m  e.  ( 1 ... M )  ->  m  e.  NN )
4039adantl 466 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  m  e.  NN )
41 f1of 5640 . . . . . . . 8  |-  ( f : ( 1 ... M ) -1-1-onto-> A  ->  f :
( 1 ... M
) --> A )
4213, 41syl 16 . . . . . . 7  |-  ( ph  ->  f : ( 1 ... M ) --> A )
4342ffvelrnda 5842 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  (
f `  m )  e.  A )
44 prodmo.2 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
4544ralrimiva 2798 . . . . . . 7  |-  ( ph  ->  A. k  e.  A  B  e.  CC )
4645adantr 465 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  A. k  e.  A  B  e.  CC )
47 nfcsb1v 3303 . . . . . . . 8  |-  F/_ k [_ ( f `  m
)  /  k ]_ B
4847nfel1 2588 . . . . . . 7  |-  F/ k
[_ ( f `  m )  /  k ]_ B  e.  CC
49 csbeq1a 3296 . . . . . . . 8  |-  ( k  =  ( f `  m )  ->  B  =  [_ ( f `  m )  /  k ]_ B )
5049eleq1d 2508 . . . . . . 7  |-  ( k  =  ( f `  m )  ->  ( B  e.  CC  <->  [_ ( f `
 m )  / 
k ]_ B  e.  CC ) )
5148, 50rspc 3066 . . . . . 6  |-  ( ( f `  m )  e.  A  ->  ( A. k  e.  A  B  e.  CC  ->  [_ ( f `  m
)  /  k ]_ B  e.  CC )
)
5243, 46, 51sylc 60 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  [_ (
f `  m )  /  k ]_ B  e.  CC )
53 fveq2 5690 . . . . . . 7  |-  ( j  =  m  ->  (
f `  j )  =  ( f `  m ) )
5453csbeq1d 3294 . . . . . 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 5771 . . . . 5  |-  ( ( m  e.  NN  /\  [_ ( f `  m
)  /  k ]_ B  e.  CC )  ->  ( G `  m
)  =  [_ (
f `  m )  /  k ]_ B
)
5740, 52, 56syl2anc 661 . . . 4  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  ( G `  m )  =  [_ ( f `  m )  /  k ]_ B )
5857, 52eqeltrd 2516 . . 3  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  ( G `  m )  e.  CC )
59 f1oeq2 5632 . . . . . . . . . . . 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 232 . . . . . . . . . 10  |-  ( ph  ->  K : ( 1 ... M ) -1-1-onto-> A )
62 f1of 5640 . . . . . . . . . 10  |-  ( K : ( 1 ... M ) -1-1-onto-> A  ->  K :
( 1 ... M
) --> A )
6361, 62syl 16 . . . . . . . . 9  |-  ( ph  ->  K : ( 1 ... M ) --> A )
64 fvco3 5767 . . . . . . . . 9  |-  ( ( K : ( 1 ... M ) --> A  /\  i  e.  ( 1 ... M ) )  ->  ( ( `' f  o.  K
) `  i )  =  ( `' f `
 ( K `  i ) ) )
6563, 64sylan 471 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( `' f  o.  K ) `  i
)  =  ( `' f `  ( K `
 i ) ) )
6665fveq2d 5694 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
f `  ( ( `' f  o.  K
) `  i )
)  =  ( f `
 ( `' f `
 ( K `  i ) ) ) )
6713adantr 465 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  f : ( 1 ... M ) -1-1-onto-> A )
6863ffvelrnda 5842 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( K `  i )  e.  A )
69 f1ocnvfv2 5983 . . . . . . . 8  |-  ( ( f : ( 1 ... M ) -1-1-onto-> A  /\  ( K `  i )  e.  A )  -> 
( f `  ( `' f `  ( K `  i )
) )  =  ( K `  i ) )
7067, 68, 69syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
f `  ( `' f `  ( K `  i ) ) )  =  ( K `  i ) )
7166, 70eqtrd 2474 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
f `  ( ( `' f  o.  K
) `  i )
)  =  ( K `
 i ) )
7271csbeq1d 3294 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  [_ (
f `  ( ( `' f  o.  K
) `  i )
)  /  k ]_ B  =  [_ ( K `
 i )  / 
k ]_ B )
7372fveq2d 5694 . . . 4  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (  _I  `  [_ ( f `
 ( ( `' f  o.  K ) `
 i ) )  /  k ]_ B
)  =  (  _I 
`  [_ ( K `  i )  /  k ]_ B ) )
74 f1of 5640 . . . . . . 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 5842 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( `' f  o.  K ) `  i
)  e.  ( 1 ... M ) )
77 elfznn 11477 . . . . 5  |-  ( ( ( `' f  o.  K ) `  i
)  e.  ( 1 ... M )  -> 
( ( `' f  o.  K ) `  i )  e.  NN )
78 fveq2 5690 . . . . . . 7  |-  ( j  =  ( ( `' f  o.  K ) `
 i )  -> 
( f `  j
)  =  ( f `
 ( ( `' f  o.  K ) `
 i ) ) )
7978csbeq1d 3294 . . . . . 6  |-  ( j  =  ( ( `' f  o.  K ) `
 i )  ->  [_ ( f `  j
)  /  k ]_ B  =  [_ ( f `
 ( ( `' f  o.  K ) `
 i ) )  /  k ]_ B
)
8079, 55fvmpti 5772 . . . . 5  |-  ( ( ( `' f  o.  K ) `  i
)  e.  NN  ->  ( G `  ( ( `' f  o.  K
) `  i )
)  =  (  _I 
`  [_ ( f `  ( ( `' f  o.  K ) `  i ) )  / 
k ]_ B ) )
8176, 77, 803syl 20 . . . 4  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G `  ( ( `' f  o.  K
) `  i )
)  =  (  _I 
`  [_ ( f `  ( ( `' f  o.  K ) `  i ) )  / 
k ]_ B ) )
82 elfznn 11477 . . . . . 6  |-  ( i  e.  ( 1 ... M )  ->  i  e.  NN )
8382adantl 466 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  i  e.  NN )
84 fveq2 5690 . . . . . . 7  |-  ( j  =  i  ->  ( K `  j )  =  ( K `  i ) )
8584csbeq1d 3294 . . . . . 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 5772 . . . . 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 2485 . . 3  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( H `  i )  =  ( G `  ( ( `' f  o.  K ) `  i ) ) )
902, 4, 6, 10, 12, 38, 58, 89seqf1o 11846 . 2  |-  ( ph  ->  (  seq 1 (  x.  ,  H ) `
 M )  =  (  seq 1 (  x.  ,  G ) `
 M ) )
9134fveq2d 5694 . 2  |-  ( ph  ->  (  seq 1 (  x.  ,  H ) `
 M )  =  (  seq 1 (  x.  ,  H ) `
 N ) )
9290, 91eqtr3d 2476 1  |-  ( ph  ->  (  seq 1 (  x.  ,  G ) `
 M )  =  (  seq 1 (  x.  ,  H ) `
 N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2714   [_csb 3287    C_ wss 3327   ifcif 3790   class class class wbr 4291    e. cmpt 4349    _I cid 4630   `'ccnv 4838    o. ccom 4843   -->wf 5413   -1-1-onto->wf1o 5416   ` cfv 5417  (class class class)co 6090    ~~ cen 7306   Fincfn 7309   CCcc 9279   1c1 9282    x. cmul 9286   NNcn 10321   NN0cn0 10578   ZZcz 10645   ZZ>=cuz 10860   ...cfz 11436    seqcseq 11805   #chash 12102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-fzo 11548  df-seq 11806  df-hash 12103
This theorem is referenced by:  prodmolem2a  27446  prodmo  27448
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