MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seqeq1d Structured version   Unicode version

Theorem seqeq1d 11329
Description: Equality deduction for the sequence builder operation. (Contributed by Mario Carneiro, 7-Sep-2013.)
Hypothesis
Ref Expression
seqeqd.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
seqeq1d  |-  ( ph  ->  seq  A (  .+  ,  F )  =  seq  B (  .+  ,  F
) )

Proof of Theorem seqeq1d
StepHypRef Expression
1 seqeqd.1 . 2  |-  ( ph  ->  A  =  B )
2 seqeq1 11326 . 2  |-  ( A  =  B  ->  seq  A (  .+  ,  F
)  =  seq  B
(  .+  ,  F
) )
31, 2syl 16 1  |-  ( ph  ->  seq  A (  .+  ,  F )  =  seq  B (  .+  ,  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    seq cseq 11323
This theorem is referenced by:  seqeq123d  11332  seqf1olem2  11363  bcval5  11609  bcn2  11610  seqshft  11900  iserex  12450  isershft  12457  isercoll2  12462  isumsplit  12620  cvgrat  12660  eftlub  12710  gsumval2a  14782  gsumccat  14787  mulgnndir  14912  geolim3  20256  ntrivcvg  25225  ntrivcvgtail  25228  fprodser  25275  fmul01lt1lem2  27691  stirlinglem7  27805  stirlinglem12  27810
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fv 5462  df-recs 6633  df-rdg 6668  df-seq 11324
  Copyright terms: Public domain W3C validator