MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  s8eqd Structured version   Unicode version

Theorem s8eqd 12495
Description: Equality theorem for a length 8 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 )
s8eqd.6  |-  ( ph  ->  H  =  U )
Assertion
Ref Expression
s8eqd  |-  ( ph  ->  <" A B C D E F G H ">  =  <" N O P Q R S T U "> )

Proof of Theorem s8eqd
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 )
7 s7eqd.6 . . . 4  |-  ( ph  ->  G  =  T )
81, 2, 3, 4, 5, 6, 7s7eqd 12494 . . 3  |-  ( ph  ->  <" A B C D E F G ">  =  <" N O P Q R S T "> )
9 s8eqd.6 . . . 4  |-  ( ph  ->  H  =  U )
109s1eqd 12292 . . 3  |-  ( ph  ->  <" H ">  =  <" U "> )
118, 10oveq12d 6109 . 2  |-  ( ph  ->  ( <" A B C D E F G "> concat  <" H "> )  =  (
<" N O P Q R S T "> concat  <" U "> ) )
12 df-s8 12481 . 2  |-  <" A B C D E F G H ">  =  ( <" A B C D E F G "> concat  <" H "> )
13 df-s8 12481 . 2  |-  <" N O P Q R S T U ">  =  ( <" N O P Q R S T "> concat  <" U "> )
1411, 12, 133eqtr4g 2500 1  |-  ( ph  ->  <" A B C D E F G H ">  =  <" N O P Q R S T U "> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369  (class class class)co 6091   concat cconcat 12223   <"cs1 12224   <"cs7 12473   <"cs8 12474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-iota 5381  df-fv 5426  df-ov 6094  df-s1 12232  df-s2 12475  df-s3 12476  df-s4 12477  df-s5 12478  df-s6 12479  df-s7 12480  df-s8 12481
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator