MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  s7eqd Unicode version

Theorem s7eqd 11758
Description: Equality theorem for a length 7 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
s2eqd.1  |-  ( ph  ->  A  =  N )
s2eqd.2  |-  ( ph  ->  B  =  O )
s3eqd.3  |-  ( ph  ->  C  =  P )
s4eqd.4  |-  ( ph  ->  D  =  Q )
s5eqd.5  |-  ( ph  ->  E  =  R )
s6eqd.6  |-  ( ph  ->  F  =  S )
s7eqd.6  |-  ( ph  ->  G  =  T )
Assertion
Ref Expression
s7eqd  |-  ( ph  ->  <" A B C D E F G ">  =  <" N O P Q R S T "> )

Proof of Theorem s7eqd
StepHypRef Expression
1 s2eqd.1 . . . 4  |-  ( ph  ->  A  =  N )
2 s2eqd.2 . . . 4  |-  ( ph  ->  B  =  O )
3 s3eqd.3 . . . 4  |-  ( ph  ->  C  =  P )
4 s4eqd.4 . . . 4  |-  ( ph  ->  D  =  Q )
5 s5eqd.5 . . . 4  |-  ( ph  ->  E  =  R )
6 s6eqd.6 . . . 4  |-  ( ph  ->  F  =  S )
71, 2, 3, 4, 5, 6s6eqd 11757 . . 3  |-  ( ph  ->  <" A B C D E F ">  =  <" N O P Q R S "> )
8 s7eqd.6 . . . 4  |-  ( ph  ->  G  =  T )
98s1eqd 11681 . . 3  |-  ( ph  ->  <" G ">  =  <" T "> )
107, 9oveq12d 6038 . 2  |-  ( ph  ->  ( <" A B C D E F "> concat  <" G "> )  =  (
<" N O P Q R S "> concat 
<" T "> ) )
11 df-s7 11744 . 2  |-  <" A B C D E F G ">  =  ( <" A B C D E F "> concat  <" G "> )
12 df-s7 11744 . 2  |-  <" N O P Q R S T ">  =  ( <" N O P Q R S "> concat  <" T "> )
1310, 11, 123eqtr4g 2444 1  |-  ( ph  ->  <" A B C D E F G ">  =  <" N O P Q R S T "> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649  (class class class)co 6020   concat cconcat 11645   <"cs1 11646   <"cs6 11736   <"cs7 11737
This theorem is referenced by:  s8eqd  11759
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-ov 6023  df-s1 11652  df-s2 11739  df-s3 11740  df-s4 11741  df-s5 11742  df-s6 11743  df-s7 11744
  Copyright terms: Public domain W3C validator