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Theorem prodmolem3 13752
Description: Lemma for prodmo 13755. (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 9593 . . . 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 9595 . . . 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 9597 . . . 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 11141 . . . 4  |-  NN  =  ( ZZ>= `  1 )
108, 9syl6eleq 2555 . . 3  |-  ( ph  ->  M  e.  ( ZZ>= ` 
1 ) )
11 ssid 3518 . . . 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 5834 . . . . . 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 5844 . . . . 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 6324 . . . . . . . . . 10  |-  ( 1 ... N )  e. 
_V
2019f1oen 7555 . . . . . . . . 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 12085 . . . . . . . . 9  |-  ( 1 ... N )  e. 
Fin
23 fzfi 12085 . . . . . . . . 9  |-  ( 1 ... M )  e. 
Fin
24 hashen 12423 . . . . . . . . 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 10873 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
29 hashfz1 12422 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( # `  ( 1 ... N
) )  =  N )
3028, 29syl 16 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... N ) )  =  N )
318nnnn0d 10873 . . . . . . . 8  |-  ( ph  ->  M  e.  NN0 )
32 hashfz1 12422 . . . . . . . 8  |-  ( M  e.  NN0  ->  ( # `  ( 1 ... M
) )  =  M )
3331, 32syl 16 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... M ) )  =  M )
3426, 30, 333eqtr3rd 2507 . . . . . 6  |-  ( ph  ->  M  =  N )
3534oveq2d 6312 . . . . 5  |-  ( ph  ->  ( 1 ... M
)  =  ( 1 ... N ) )
36 f1oeq2 5814 . . . . 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 11739 . . . . . 6  |-  ( m  e.  ( 1 ... M )  ->  m  e.  NN )
4039adantl 466 . . . . 5  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  m  e.  NN )
41 f1of 5822 . . . . . . . 8  |-  ( f : ( 1 ... M ) -1-1-onto-> A  ->  f :
( 1 ... M
) --> A )
4213, 41syl 16 . . . . . . 7  |-  ( ph  ->  f : ( 1 ... M ) --> A )
4342ffvelrnda 6032 . . . . . 6  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  (
f `  m )  e.  A )
44 prodmo.2 . . . . . . . 8  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
4544ralrimiva 2871 . . . . . . 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 3446 . . . . . . . 8  |-  F/_ k [_ ( f `  m
)  /  k ]_ B
4847nfel1 2635 . . . . . . 7  |-  F/ k
[_ ( f `  m )  /  k ]_ B  e.  CC
49 csbeq1a 3439 . . . . . . . 8  |-  ( k  =  ( f `  m )  ->  B  =  [_ ( f `  m )  /  k ]_ B )
5049eleq1d 2526 . . . . . . 7  |-  ( k  =  ( f `  m )  ->  ( B  e.  CC  <->  [_ ( f `
 m )  / 
k ]_ B  e.  CC ) )
5148, 50rspc 3204 . . . . . 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 5872 . . . . . . 7  |-  ( j  =  m  ->  (
f `  j )  =  ( f `  m ) )
5453csbeq1d 3437 . . . . . 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 5954 . . . . 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 2545 . . 3  |-  ( (
ph  /\  m  e.  ( 1 ... M
) )  ->  ( G `  m )  e.  CC )
59 f1oeq2 5814 . . . . . . . . . . . 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 5822 . . . . . . . . . 10  |-  ( K : ( 1 ... M ) -1-1-onto-> A  ->  K :
( 1 ... M
) --> A )
6361, 62syl 16 . . . . . . . . 9  |-  ( ph  ->  K : ( 1 ... M ) --> A )
64 fvco3 5950 . . . . . . . . 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 5876 . . . . . . 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 6032 . . . . . . . 8  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( K `  i )  e.  A )
69 f1ocnvfv2 6184 . . . . . . . 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 2498 . . . . . 6  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
f `  ( ( `' f  o.  K
) `  i )
)  =  ( K `
 i ) )
7271csbeq1d 3437 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  [_ (
f `  ( ( `' f  o.  K
) `  i )
)  /  k ]_ B  =  [_ ( K `
 i )  / 
k ]_ B )
7372fveq2d 5876 . . . 4  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (  _I  `  [_ ( f `
 ( ( `' f  o.  K ) `
 i ) )  /  k ]_ B
)  =  (  _I 
`  [_ ( K `  i )  /  k ]_ B ) )
74 f1of 5822 . . . . . . 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 6032 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( `' f  o.  K ) `  i
)  e.  ( 1 ... M ) )
77 elfznn 11739 . . . . 5  |-  ( ( ( `' f  o.  K ) `  i
)  e.  ( 1 ... M )  -> 
( ( `' f  o.  K ) `  i )  e.  NN )
78 fveq2 5872 . . . . . . 7  |-  ( j  =  ( ( `' f  o.  K ) `
 i )  -> 
( f `  j
)  =  ( f `
 ( ( `' f  o.  K ) `
 i ) ) )
7978csbeq1d 3437 . . . . . 6  |-  ( j  =  ( ( `' f  o.  K ) `
 i )  ->  [_ ( f `  j
)  /  k ]_ B  =  [_ ( f `
 ( ( `' f  o.  K ) `
 i ) )  /  k ]_ B
)
8079, 55fvmpti 5955 . . . . 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 11739 . . . . . 6  |-  ( i  e.  ( 1 ... M )  ->  i  e.  NN )
8382adantl 466 . . . . 5  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  i  e.  NN )
84 fveq2 5872 . . . . . . 7  |-  ( j  =  i  ->  ( K `  j )  =  ( K `  i ) )
8584csbeq1d 3437 . . . . . 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 5955 . . . . 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 2509 . . 3  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( H `  i )  =  ( G `  ( ( `' f  o.  K ) `  i ) ) )
902, 4, 6, 10, 12, 38, 58, 89seqf1o 12151 . 2  |-  ( ph  ->  (  seq 1 (  x.  ,  H ) `
 M )  =  (  seq 1 (  x.  ,  G ) `
 M ) )
9134fveq2d 5876 . 2  |-  ( ph  ->  (  seq 1 (  x.  ,  H ) `
 M )  =  (  seq 1 (  x.  ,  H ) `
 N ) )
9290, 91eqtr3d 2500 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 973    = wceq 1395    e. wcel 1819   A.wral 2807   [_csb 3430    C_ wss 3471   ifcif 3944   class class class wbr 4456    |-> cmpt 4515    _I cid 4799   `'ccnv 5007    o. ccom 5012   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296    ~~ cen 7532   Fincfn 7535   CCcc 9507   1c1 9510    x. cmul 9514   NNcn 10556   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   ...cfz 11697    seqcseq 12110   #chash 12408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409
This theorem is referenced by:  prodmolem2a  13753  prodmo  13755
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