Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fprodsplit1f Structured version   Unicode version

Theorem fprodsplit1f 31832
Description: Separate out a term in a finite product. A version of fprodsplit1 31838 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
fprodsplit1f.kph  |-  F/ k
ph
fprodsplit1f.fk  |-  ( ph  -> 
F/_ k D )
fprodsplit1f.a  |-  ( ph  ->  A  e.  Fin )
fprodsplit1f.b  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
fprodsplit1f.c  |-  ( ph  ->  C  e.  A )
fprodsplit1f.d  |-  ( (
ph  /\  k  =  C )  ->  B  =  D )
Assertion
Ref Expression
fprodsplit1f  |-  ( ph  ->  prod_ k  e.  A  B  =  ( D  x.  prod_ k  e.  ( A  \  { C } ) B ) )
Distinct variable groups:    A, k    C, k
Allowed substitution hints:    ph( k)    B( k)    D( k)

Proof of Theorem fprodsplit1f
StepHypRef Expression
1 fprodsplit1f.kph . . 3  |-  F/ k
ph
2 disjdif 3888 . . . 4  |-  ( { C }  i^i  ( A  \  { C }
) )  =  (/)
32a1i 11 . . 3  |-  ( ph  ->  ( { C }  i^i  ( A  \  { C } ) )  =  (/) )
4 fprodsplit1f.c . . . . . 6  |-  ( ph  ->  C  e.  A )
5 snssi 4160 . . . . . 6  |-  ( C  e.  A  ->  { C }  C_  A )
64, 5syl 16 . . . . 5  |-  ( ph  ->  { C }  C_  A )
7 undif 3896 . . . . 5  |-  ( { C }  C_  A  <->  ( { C }  u.  ( A  \  { C } ) )  =  A )
86, 7sylib 196 . . . 4  |-  ( ph  ->  ( { C }  u.  ( A  \  { C } ) )  =  A )
98eqcomd 2462 . . 3  |-  ( ph  ->  A  =  ( { C }  u.  ( A  \  { C }
) ) )
10 fprodsplit1f.a . . 3  |-  ( ph  ->  A  e.  Fin )
11 fprodsplit1f.b . . 3  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  CC )
121, 3, 9, 10, 11fprodsplitf 31828 . 2  |-  ( ph  ->  prod_ k  e.  A  B  =  ( prod_ k  e.  { C } B  x.  prod_ k  e.  ( A  \  { C } ) B ) )
13 fprodsplit1f.fk . . . . . . 7  |-  ( ph  -> 
F/_ k D )
14 fprodsplit1f.d . . . . . . 7  |-  ( (
ph  /\  k  =  C )  ->  B  =  D )
151, 13, 4, 14csbiedf 3441 . . . . . 6  |-  ( ph  ->  [_ C  /  k ]_ B  =  D
)
1615eqcomd 2462 . . . . . . 7  |-  ( ph  ->  D  =  [_ C  /  k ]_ B
)
174ancli 549 . . . . . . . 8  |-  ( ph  ->  ( ph  /\  C  e.  A ) )
18 nfcv 2616 . . . . . . . . 9  |-  F/_ k C
19 nfv 1712 . . . . . . . . . . 11  |-  F/ k  C  e.  A
201, 19nfan 1933 . . . . . . . . . 10  |-  F/ k ( ph  /\  C  e.  A )
2118nfcsb1 3435 . . . . . . . . . . 11  |-  F/_ k [_ C  /  k ]_ B
22 nfcv 2616 . . . . . . . . . . 11  |-  F/_ k CC
2321, 22nfel 2629 . . . . . . . . . 10  |-  F/ k
[_ C  /  k ]_ B  e.  CC
2420, 23nfim 1925 . . . . . . . . 9  |-  F/ k ( ( ph  /\  C  e.  A )  ->  [_ C  /  k ]_ B  e.  CC )
25 eleq1 2526 . . . . . . . . . . 11  |-  ( k  =  C  ->  (
k  e.  A  <->  C  e.  A ) )
2625anbi2d 701 . . . . . . . . . 10  |-  ( k  =  C  ->  (
( ph  /\  k  e.  A )  <->  ( ph  /\  C  e.  A ) ) )
27 csbeq1a 3429 . . . . . . . . . . 11  |-  ( k  =  C  ->  B  =  [_ C  /  k ]_ B )
2827eleq1d 2523 . . . . . . . . . 10  |-  ( k  =  C  ->  ( B  e.  CC  <->  [_ C  / 
k ]_ B  e.  CC ) )
2926, 28imbi12d 318 . . . . . . . . 9  |-  ( k  =  C  ->  (
( ( ph  /\  k  e.  A )  ->  B  e.  CC )  <-> 
( ( ph  /\  C  e.  A )  ->  [_ C  /  k ]_ B  e.  CC ) ) )
3018, 24, 29, 11vtoclgf 3162 . . . . . . . 8  |-  ( C  e.  A  ->  (
( ph  /\  C  e.  A )  ->  [_ C  /  k ]_ B  e.  CC ) )
314, 17, 30sylc 60 . . . . . . 7  |-  ( ph  ->  [_ C  /  k ]_ B  e.  CC )
3216, 31eqeltrd 2542 . . . . . 6  |-  ( ph  ->  D  e.  CC )
3315, 32eqeltrd 2542 . . . . 5  |-  ( ph  ->  [_ C  /  k ]_ B  e.  CC )
34 prodsns 13858 . . . . 5  |-  ( ( C  e.  A  /\  [_ C  /  k ]_ B  e.  CC )  ->  prod_ k  e.  { C } B  =  [_ C  /  k ]_ B
)
354, 33, 34syl2anc 659 . . . 4  |-  ( ph  ->  prod_ k  e.  { C } B  =  [_ C  /  k ]_ B
)
3635, 15eqtrd 2495 . . 3  |-  ( ph  ->  prod_ k  e.  { C } B  =  D )
3736oveq1d 6285 . 2  |-  ( ph  ->  ( prod_ k  e.  { C } B  x.  prod_ k  e.  ( A  \  { C } ) B )  =  ( D  x.  prod_ k  e.  ( A  \  { C } ) B ) )
3812, 37eqtrd 2495 1  |-  ( ph  ->  prod_ k  e.  A  B  =  ( D  x.  prod_ k  e.  ( A  \  { C } ) B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398   F/wnf 1621    e. wcel 1823   F/_wnfc 2602   [_csb 3420    \ cdif 3458    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3783   {csn 4016  (class class class)co 6270   Fincfn 7509   CCcc 9479    x. cmul 9486   prod_cprod 13794
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-fal 1404  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 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-prod 13795
This theorem is referenced by:  fprodsplit1  31838  fprodeq0g  31840  fprod0  31842  dvmptfprodlem  31980
  Copyright terms: Public domain W3C validator