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Theorem 2wlklem1 24431
Description: Lemma 1 for constr2wlk 24432. (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 755 . . 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 5871 . . . . . 6  |-  ( F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }  ->  ( F `  0 )  =  ( { <. 0 ,  I >. , 
<. 1 ,  J >. } `  0 ) )
4 0ne1 10615 . . . . . . 7  |-  0  =/=  1
5 c0ex 9602 . . . . . . . 8  |-  0  e.  _V
6 2trlY.i . . . . . . . . 9  |-  ( I  e.  U  /\  J  e.  W )
7 elex 3127 . . . . . . . . . 10  |-  ( I  e.  U  ->  I  e.  _V )
87adantr 465 . . . . . . . . 9  |-  ( ( I  e.  U  /\  J  e.  W )  ->  I  e.  _V )
96, 8ax-mp 5 . . . . . . . 8  |-  I  e. 
_V
105, 9fvpr1 6115 . . . . . . 7  |-  ( 0  =/=  1  ->  ( { <. 0 ,  I >. ,  <. 1 ,  J >. } `  0 )  =  I )
114, 10ax-mp 5 . . . . . 6  |-  ( {
<. 0 ,  I >. ,  <. 1 ,  J >. } `  0 )  =  I
123, 11syl6eq 2524 . . . . 5  |-  ( F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }  ->  ( F `  0 )  =  I )
132, 12mp1i 12 . . . 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 5876 . . 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 24388 . . . . . 6  |-  ( A  e.  V  ->  ( P `  0 )  =  A )
17163ad2ant1 1017 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( P `  0
)  =  A )
1817ad2antlr 726 . . . 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 24389 . . . . . 6  |-  ( B  e.  V  ->  ( P `  1 )  =  B )
20193ad2ant2 1018 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( P `  1
)  =  B )
2120ad2antlr 726 . . . 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 4120 . . 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 2518 . 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 756 . . 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 5871 . . . . . 6  |-  ( F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }  ->  ( F `  1 )  =  ( { <. 0 ,  I >. , 
<. 1 ,  J >. } `  1 ) )
26 1ex 9603 . . . . . . . 8  |-  1  e.  _V
27 elex 3127 . . . . . . . . . 10  |-  ( J  e.  W  ->  J  e.  _V )
2827adantl 466 . . . . . . . . 9  |-  ( ( I  e.  U  /\  J  e.  W )  ->  J  e.  _V )
296, 28ax-mp 5 . . . . . . . 8  |-  J  e. 
_V
3026, 29fvpr2 6116 . . . . . . 7  |-  ( 0  =/=  1  ->  ( { <. 0 ,  I >. ,  <. 1 ,  J >. } `  1 )  =  J )
314, 30ax-mp 5 . . . . . 6  |-  ( {
<. 0 ,  I >. ,  <. 1 ,  J >. } `  1 )  =  J
3225, 31syl6eq 2524 . . . . 5  |-  ( F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }  ->  ( F `  1 )  =  J )
332, 32mp1i 12 . . . 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 5876 . . 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 24390 . . . . . 6  |-  ( C  e.  V  ->  ( P `  2 )  =  C )
36353ad2ant3 1019 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( P `  2
)  =  C )
3736ad2antlr 726 . . . 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 4120 . . 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 2518 . 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 24398 . 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 664 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   _Vcvv 3118   {cpr 4035   {ctp 4037   <.cop 4039   ` cfv 5594  (class class class)co 6295   0cc0 9504   1c1 9505    + caddc 9507   2c2 10597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-z 10877
This theorem is referenced by:  constr2wlk  24432  constr2trl  24433
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