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Theorem 2wlklem 24228
Description: Lemma for is2wlk 24229 and 2wlklemA 24218. (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 9579 . 2  |-  0  e.  _V
2 1ex 9580 . 2  |-  1  e.  _V
3 fveq2 5857 . . . 4  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
43fveq2d 5861 . . 3  |-  ( k  =  0  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  0 )
) )
5 fveq2 5857 . . . 4  |-  ( k  =  0  ->  ( P `  k )  =  ( P ` 
0 ) )
6 oveq1 6282 . . . . . 6  |-  ( k  =  0  ->  (
k  +  1 )  =  ( 0  +  1 ) )
7 0p1e1 10636 . . . . . 6  |-  ( 0  +  1 )  =  1
86, 7syl6eq 2517 . . . . 5  |-  ( k  =  0  ->  (
k  +  1 )  =  1 )
98fveq2d 5861 . . . 4  |-  ( k  =  0  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
1 ) )
105, 9preq12d 4107 . . 3  |-  ( k  =  0  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
0 ) ,  ( P `  1 ) } )
114, 10eqeq12d 2482 . 2  |-  ( k  =  0  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } ) )
12 fveq2 5857 . . . 4  |-  ( k  =  1  ->  ( F `  k )  =  ( F ` 
1 ) )
1312fveq2d 5861 . . 3  |-  ( k  =  1  ->  ( E `  ( F `  k ) )  =  ( E `  ( F `  1 )
) )
14 fveq2 5857 . . . 4  |-  ( k  =  1  ->  ( P `  k )  =  ( P ` 
1 ) )
15 oveq1 6282 . . . . . 6  |-  ( k  =  1  ->  (
k  +  1 )  =  ( 1  +  1 ) )
16 1p1e2 10638 . . . . . 6  |-  ( 1  +  1 )  =  2
1715, 16syl6eq 2517 . . . . 5  |-  ( k  =  1  ->  (
k  +  1 )  =  2 )
1817fveq2d 5861 . . . 4  |-  ( k  =  1  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
2 ) )
1914, 18preq12d 4107 . . 3  |-  ( k  =  1  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
1 ) ,  ( P `  2 ) } )
2013, 19eqeq12d 2482 . 2  |-  ( k  =  1  ->  (
( E `  ( F `  k )
)  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  <->  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } ) )
211, 2, 11, 20ralpr 4073 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 184    /\ wa 369    = wceq 1374   A.wral 2807   {cpr 4022   ` cfv 5579  (class class class)co 6275   0cc0 9481   1c1 9482    + caddc 9484   2c2 10574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-2 10583
This theorem is referenced by:  is2wlk  24229  2wlklem1  24261  usgra2wlkspthlem2  24282
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