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Theorem 2wlklem1 25376
Description: Lemma 1 for constr2wlk 25377. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
Hypotheses
Ref Expression
2trlY.i  |-  ( I  e.  U  /\  J  e.  W )
2trlY.f  |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }
2trlY.p  |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }
Assertion
Ref Expression
2wlklem1  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  A. k  e.  { 0 ,  1 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )
Distinct variable groups:    k, E    k, F    P, k
Allowed substitution hints:    A( k)    B( k)    C( k)    U( k)    I( k)    J( k)    V( k)    W( k)    X( k)    Y( k)

Proof of Theorem 2wlklem1
StepHypRef Expression
1 simprl 769 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  ( E `  I )  =  { A ,  B } )
2 2trlY.f . . . . 5  |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }
3 fveq1 5887 . . . . . 6  |-  ( F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }  ->  ( F `  0 )  =  ( { <. 0 ,  I >. , 
<. 1 ,  J >. } `  0 ) )
4 0ne1 10705 . . . . . . 7  |-  0  =/=  1
5 c0ex 9663 . . . . . . . 8  |-  0  e.  _V
6 2trlY.i . . . . . . . . 9  |-  ( I  e.  U  /\  J  e.  W )
7 elex 3066 . . . . . . . . . 10  |-  ( I  e.  U  ->  I  e.  _V )
87adantr 471 . . . . . . . . 9  |-  ( ( I  e.  U  /\  J  e.  W )  ->  I  e.  _V )
96, 8ax-mp 5 . . . . . . . 8  |-  I  e. 
_V
105, 9fvpr1 6131 . . . . . . 7  |-  ( 0  =/=  1  ->  ( { <. 0 ,  I >. ,  <. 1 ,  J >. } `  0 )  =  I )
114, 10ax-mp 5 . . . . . 6  |-  ( {
<. 0 ,  I >. ,  <. 1 ,  J >. } `  0 )  =  I
123, 11syl6eq 2512 . . . . 5  |-  ( F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }  ->  ( F `  0 )  =  I )
132, 12mp1i 13 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  ( F `  0 )  =  I )
1413fveq2d 5892 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  ( E `  ( F `  0 ) )  =  ( E `  I ) )
15 2trlY.p . . . . . . 7  |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }
16152wlklemA 25333 . . . . . 6  |-  ( A  e.  V  ->  ( P `  0 )  =  A )
17163ad2ant1 1035 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( P `  0
)  =  A )
1817ad2antlr 738 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  ( P `  0 )  =  A )
19152wlklemB 25334 . . . . . 6  |-  ( B  e.  V  ->  ( P `  1 )  =  B )
20193ad2ant2 1036 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( P `  1
)  =  B )
2120ad2antlr 738 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  ( P `  1 )  =  B )
2218, 21preq12d 4072 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  { ( P `  0 ) ,  ( P ` 
1 ) }  =  { A ,  B }
)
231, 14, 223eqtr4d 2506 . 2  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  ( E `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) } )
24 simprr 771 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  ( E `  J )  =  { B ,  C } )
25 fveq1 5887 . . . . . 6  |-  ( F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }  ->  ( F `  1 )  =  ( { <. 0 ,  I >. , 
<. 1 ,  J >. } `  1 ) )
26 1ex 9664 . . . . . . . 8  |-  1  e.  _V
27 elex 3066 . . . . . . . . . 10  |-  ( J  e.  W  ->  J  e.  _V )
2827adantl 472 . . . . . . . . 9  |-  ( ( I  e.  U  /\  J  e.  W )  ->  J  e.  _V )
296, 28ax-mp 5 . . . . . . . 8  |-  J  e. 
_V
3026, 29fvpr2 6132 . . . . . . 7  |-  ( 0  =/=  1  ->  ( { <. 0 ,  I >. ,  <. 1 ,  J >. } `  1 )  =  J )
314, 30ax-mp 5 . . . . . 6  |-  ( {
<. 0 ,  I >. ,  <. 1 ,  J >. } `  1 )  =  J
3225, 31syl6eq 2512 . . . . 5  |-  ( F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }  ->  ( F `  1 )  =  J )
332, 32mp1i 13 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  ( F `  1 )  =  J )
3433fveq2d 5892 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  ( E `  ( F `  1 ) )  =  ( E `  J ) )
35152wlklemC 25335 . . . . . 6  |-  ( C  e.  V  ->  ( P `  2 )  =  C )
36353ad2ant3 1037 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( P `  2
)  =  C )
3736ad2antlr 738 . . . 4  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  ( P `  2 )  =  C )
3821, 37preq12d 4072 . . 3  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  { ( P `  1 ) ,  ( P ` 
2 ) }  =  { B ,  C }
)
3924, 34, 383eqtr4d 2506 . 2  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } )
40 2wlklem 25343 . 2  |-  ( 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 ) } ) )
4123, 39, 40sylanbrc 675 1  |-  ( ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
)  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
) )  ->  A. k  e.  { 0 ,  1 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   A.wral 2749   _Vcvv 3057   {cpr 3982   {ctp 3984   <.cop 3986   ` cfv 5601  (class class class)co 6315   0cc0 9565   1c1 9566    + caddc 9568   2c2 10687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-nn 10638  df-2 10696  df-z 10967
This theorem is referenced by:  constr2wlk  25377  constr2trl  25378
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