Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prodeq1d Structured version   Unicode version

Theorem prodeq1d 25278
Description: Equality deduction for sum. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypothesis
Ref Expression
prodeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
prodeq1d  |-  ( ph  ->  prod_ k  e.  A C  =  prod_ k  e.  B C )
Distinct variable groups:    A, k    B, k
Allowed substitution hints:    ph( k)    C( k)

Proof of Theorem prodeq1d
StepHypRef Expression
1 prodeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 prodeq1 25266 . 2  |-  ( A  =  B  ->  prod_ k  e.  A C  = 
prod_ k  e.  B C )
31, 2syl 16 1  |-  ( ph  ->  prod_ k  e.  A C  =  prod_ k  e.  B C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   prod_cprod 25262
This theorem is referenced by:  prodeq12dv  25283  prodeq12rdv  25284  fprodf1o  25303  prodss  25304  fprod1  25318  fprodp1  25323  fprodfac  25327  fprodabs  25328  fprodefsum  25329  fprod2d  25336  fprodcom2  25339  risefacval  25355  fallfacval  25356  risefacval2  25357  fallfacval2  25358  risefacp1  25376  fallfacp1  25377  fallfacval4  25390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-cnv 4915  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-recs 6662  df-rdg 6697  df-seq 11355  df-prod 25263
  Copyright terms: Public domain W3C validator