MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2wlklem Structured version   Visualization version   Unicode version

Theorem 2wlklem 25373
Description: Lemma for is2wlk 25374 and 2wlklemA 25363. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
Assertion
Ref Expression
2wlklem  |-  ( A. k  e.  { 0 ,  1 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } ) )
Distinct variable groups:    k, E    k, F    P, k

Proof of Theorem 2wlklem
StepHypRef Expression
1 c0ex 9655 . 2  |-  0  e.  _V
2 1ex 9656 . 2  |-  1  e.  _V
3 fveq2 5879 . . . 4  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
43fveq2d 5883 . . 3  |-  ( k  =  0  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  0 )
) )
5 fveq2 5879 . . . 4  |-  ( k  =  0  ->  ( P `  k )  =  ( P ` 
0 ) )
6 oveq1 6315 . . . . . 6  |-  ( k  =  0  ->  (
k  +  1 )  =  ( 0  +  1 ) )
7 0p1e1 10743 . . . . . 6  |-  ( 0  +  1 )  =  1
86, 7syl6eq 2521 . . . . 5  |-  ( k  =  0  ->  (
k  +  1 )  =  1 )
98fveq2d 5883 . . . 4  |-  ( k  =  0  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
1 ) )
105, 9preq12d 4050 . . 3  |-  ( k  =  0  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
0 ) ,  ( P `  1 ) } )
114, 10eqeq12d 2486 . 2  |-  ( k  =  0  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } ) )
12 fveq2 5879 . . . 4  |-  ( k  =  1  ->  ( F `  k )  =  ( F ` 
1 ) )
1312fveq2d 5883 . . 3  |-  ( k  =  1  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  1 )
) )
14 fveq2 5879 . . . 4  |-  ( k  =  1  ->  ( P `  k )  =  ( P ` 
1 ) )
15 oveq1 6315 . . . . . 6  |-  ( k  =  1  ->  (
k  +  1 )  =  ( 1  +  1 ) )
16 1p1e2 10745 . . . . . 6  |-  ( 1  +  1 )  =  2
1715, 16syl6eq 2521 . . . . 5  |-  ( k  =  1  ->  (
k  +  1 )  =  2 )
1817fveq2d 5883 . . . 4  |-  ( k  =  1  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
2 ) )
1914, 18preq12d 4050 . . 3  |-  ( k  =  1  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
1 ) ,  ( P `  2 ) } )
2013, 19eqeq12d 2486 . 2  |-  ( k  =  1  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } ) )
211, 2, 11, 20ralpr 4016 1  |-  ( A. k  e.  { 0 ,  1 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452   A.wral 2756   {cpr 3961   ` cfv 5589  (class class class)co 6308   0cc0 9557   1c1 9558    + caddc 9560   2c2 10681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-ltxr 9698  df-2 10690
This theorem is referenced by:  is2wlk  25374  2wlklem1  25406  usgra2wlkspthlem1  25426  usgra2wlkspthlem2  25427  usgrcyclnl2  25448  4cycl4v4e  25473  4cycl4dv4e  25475  uspgrn2crct  39986  1wlk2v2elem2  40044  usgra2pthspth  40173  usgra2pthlem1  40175  usgra2pth  40176
  Copyright terms: Public domain W3C validator