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Theorem 11wlkdlem4 40028
Description: Lemma 4 for 11wlkd 40029. (Contributed by AV, 22-Jan-2021.)
Hypotheses
Ref Expression
11wlkd.p  |-  P  = 
<" X Y ">
11wlkd.f  |-  F  = 
<" J ">
11wlkd.x  |-  ( ph  ->  X  e.  V )
11wlkd.y  |-  ( ph  ->  Y  e.  V )
11wlkd.l  |-  ( (
ph  /\  X  =  Y )  ->  (
I `  J )  =  { X } )
11wlkd.j  |-  ( (
ph  /\  X  =/=  Y )  ->  { X ,  Y }  C_  (
I `  J )
)
Assertion
Ref Expression
11wlkdlem4  |-  ( ph  ->  A. k  e.  ( 0..^ ( # `  F
) )if- ( ( P `  k )  =  ( P `  ( k  +  1 ) ) ,  ( I `  ( F `
 k ) )  =  { ( P `
 k ) } ,  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  C_  (
I `  ( F `  k ) ) ) )
Distinct variable groups:    k, F    k, I    P, k
Allowed substitution hints:    ph( k)    J( k)    V( k)    X( k)    Y( k)

Proof of Theorem 11wlkdlem4
StepHypRef Expression
1 11wlkd.f . . . . . . . . . 10  |-  F  = 
<" J ">
21fveq1i 5880 . . . . . . . . 9  |-  ( F `
 0 )  =  ( <" J "> `  0 )
3 11wlkd.p . . . . . . . . . . . 12  |-  P  = 
<" X Y ">
4 11wlkd.x . . . . . . . . . . . 12  |-  ( ph  ->  X  e.  V )
5 11wlkd.y . . . . . . . . . . . 12  |-  ( ph  ->  Y  e.  V )
6 11wlkd.l . . . . . . . . . . . 12  |-  ( (
ph  /\  X  =  Y )  ->  (
I `  J )  =  { X } )
7 11wlkd.j . . . . . . . . . . . 12  |-  ( (
ph  /\  X  =/=  Y )  ->  { X ,  Y }  C_  (
I `  J )
)
83, 1, 4, 5, 6, 711wlkdlem2 40026 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  ( I `
 J ) )
98elfvexd 5907 . . . . . . . . . 10  |-  ( ph  ->  J  e.  _V )
10 s1fv 12801 . . . . . . . . . 10  |-  ( J  e.  _V  ->  ( <" J "> `  0 )  =  J )
119, 10syl 17 . . . . . . . . 9  |-  ( ph  ->  ( <" J "> `  0 )  =  J )
122, 11syl5eq 2517 . . . . . . . 8  |-  ( ph  ->  ( F `  0
)  =  J )
1312fveq2d 5883 . . . . . . 7  |-  ( ph  ->  ( I `  ( F `  0 )
)  =  ( I `
 J ) )
1413adantr 472 . . . . . 6  |-  ( (
ph  /\  X  =  Y )  ->  (
I `  ( F `  0 ) )  =  ( I `  J ) )
1514, 6eqtrd 2505 . . . . 5  |-  ( (
ph  /\  X  =  Y )  ->  (
I `  ( F `  0 ) )  =  { X }
)
16 df-ne 2643 . . . . . . 7  |-  ( X  =/=  Y  <->  -.  X  =  Y )
1716, 7sylan2br 484 . . . . . 6  |-  ( (
ph  /\  -.  X  =  Y )  ->  { X ,  Y }  C_  (
I `  J )
)
1813adantr 472 . . . . . 6  |-  ( (
ph  /\  -.  X  =  Y )  ->  (
I `  ( F `  0 ) )  =  ( I `  J ) )
1917, 18sseqtr4d 3455 . . . . 5  |-  ( (
ph  /\  -.  X  =  Y )  ->  { X ,  Y }  C_  (
I `  ( F `  0 ) ) )
2015, 19ifpimpda 999 . . . 4  |-  ( ph  -> if- ( X  =  Y ,  ( I `  ( F `  0 ) )  =  { X } ,  { X ,  Y }  C_  (
I `  ( F `  0 ) ) ) )
213fveq1i 5880 . . . . . 6  |-  ( P `
 0 )  =  ( <" X Y "> `  0
)
22 s2fv0 13041 . . . . . . 7  |-  ( X  e.  V  ->  ( <" X Y "> `  0 )  =  X )
234, 22syl 17 . . . . . 6  |-  ( ph  ->  ( <" X Y "> `  0
)  =  X )
2421, 23syl5eq 2517 . . . . 5  |-  ( ph  ->  ( P `  0
)  =  X )
253fveq1i 5880 . . . . . 6  |-  ( P `
 1 )  =  ( <" X Y "> `  1
)
26 s2fv1 13042 . . . . . . 7  |-  ( Y  e.  V  ->  ( <" X Y "> `  1 )  =  Y )
275, 26syl 17 . . . . . 6  |-  ( ph  ->  ( <" X Y "> `  1
)  =  Y )
2825, 27syl5eq 2517 . . . . 5  |-  ( ph  ->  ( P `  1
)  =  Y )
29 eqeq12 2484 . . . . . 6  |-  ( ( ( P `  0
)  =  X  /\  ( P `  1 )  =  Y )  -> 
( ( P ` 
0 )  =  ( P `  1 )  <-> 
X  =  Y ) )
30 sneq 3969 . . . . . . . 8  |-  ( ( P `  0 )  =  X  ->  { ( P `  0 ) }  =  { X } )
3130adantr 472 . . . . . . 7  |-  ( ( ( P `  0
)  =  X  /\  ( P `  1 )  =  Y )  ->  { ( P ` 
0 ) }  =  { X } )
3231eqeq2d 2481 . . . . . 6  |-  ( ( ( P `  0
)  =  X  /\  ( P `  1 )  =  Y )  -> 
( ( I `  ( F `  0 ) )  =  { ( P `  0 ) }  <->  ( I `  ( F `  0 ) )  =  { X } ) )
33 preq12 4044 . . . . . . 7  |-  ( ( ( P `  0
)  =  X  /\  ( P `  1 )  =  Y )  ->  { ( P ` 
0 ) ,  ( P `  1 ) }  =  { X ,  Y } )
3433sseq1d 3445 . . . . . 6  |-  ( ( ( P `  0
)  =  X  /\  ( P `  1 )  =  Y )  -> 
( { ( P `
 0 ) ,  ( P `  1
) }  C_  (
I `  ( F `  0 ) )  <->  { X ,  Y }  C_  ( I `  ( F `  0 )
) ) )
3529, 32, 34ifpbi123d 998 . . . . 5  |-  ( ( ( P `  0
)  =  X  /\  ( P `  1 )  =  Y )  -> 
(if- ( ( P `
 0 )  =  ( P `  1
) ,  ( I `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) } ,  { ( P ` 
0 ) ,  ( P `  1 ) }  C_  ( I `  ( F `  0
) ) )  <-> if- ( X  =  Y ,  ( I `
 ( F ` 
0 ) )  =  { X } ,  { X ,  Y }  C_  ( I `  ( F `  0 )
) ) ) )
3624, 28, 35syl2anc 673 . . . 4  |-  ( ph  ->  (if- ( ( P `
 0 )  =  ( P `  1
) ,  ( I `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) } ,  { ( P ` 
0 ) ,  ( P `  1 ) }  C_  ( I `  ( F `  0
) ) )  <-> if- ( X  =  Y ,  ( I `
 ( F ` 
0 ) )  =  { X } ,  { X ,  Y }  C_  ( I `  ( F `  0 )
) ) ) )
3720, 36mpbird 240 . . 3  |-  ( ph  -> if- ( ( P ` 
0 )  =  ( P `  1 ) ,  ( I `  ( F `  0 ) )  =  { ( P `  0 ) } ,  { ( P `  0 ) ,  ( P ` 
1 ) }  C_  ( I `  ( F `  0 )
) ) )
38 c0ex 9655 . . . 4  |-  0  e.  _V
39 fveq2 5879 . . . . . 6  |-  ( k  =  0  ->  ( P `  k )  =  ( P ` 
0 ) )
40 oveq1 6315 . . . . . . . 8  |-  ( k  =  0  ->  (
k  +  1 )  =  ( 0  +  1 ) )
41 0p1e1 10743 . . . . . . . 8  |-  ( 0  +  1 )  =  1
4240, 41syl6eq 2521 . . . . . . 7  |-  ( k  =  0  ->  (
k  +  1 )  =  1 )
4342fveq2d 5883 . . . . . 6  |-  ( k  =  0  ->  ( P `  ( k  +  1 ) )  =  ( P ` 
1 ) )
4439, 43eqeq12d 2486 . . . . 5  |-  ( k  =  0  ->  (
( P `  k
)  =  ( P `
 ( k  +  1 ) )  <->  ( P `  0 )  =  ( P `  1
) ) )
45 fveq2 5879 . . . . . . 7  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
4645fveq2d 5883 . . . . . 6  |-  ( k  =  0  ->  (
I `  ( F `  k ) )  =  ( I `  ( F `  0 )
) )
4739sneqd 3971 . . . . . 6  |-  ( k  =  0  ->  { ( P `  k ) }  =  { ( P `  0 ) } )
4846, 47eqeq12d 2486 . . . . 5  |-  ( k  =  0  ->  (
( I `  ( F `  k )
)  =  { ( P `  k ) }  <->  ( I `  ( F `  0 ) )  =  { ( P `  0 ) } ) )
4939, 43preq12d 4050 . . . . . 6  |-  ( k  =  0  ->  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  =  { ( P ` 
0 ) ,  ( P `  1 ) } )
5049, 46sseq12d 3447 . . . . 5  |-  ( k  =  0  ->  ( { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  C_  ( I `  ( F `  k
) )  <->  { ( P `  0 ) ,  ( P ` 
1 ) }  C_  ( I `  ( F `  0 )
) ) )
5144, 48, 50ifpbi123d 998 . . . 4  |-  ( k  =  0  ->  (if- ( ( P `  k )  =  ( P `  ( k  +  1 ) ) ,  ( I `  ( F `  k ) )  =  { ( P `  k ) } ,  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  C_  ( I `  ( F `  k )
) )  <-> if- ( ( P `  0 )  =  ( P ` 
1 ) ,  ( I `  ( F `
 0 ) )  =  { ( P `
 0 ) } ,  { ( P `
 0 ) ,  ( P `  1
) }  C_  (
I `  ( F `  0 ) ) ) ) )
5238, 51ralsn 4001 . . 3  |-  ( A. k  e.  { 0 }if- ( ( P `  k )  =  ( P `  ( k  +  1 ) ) ,  ( I `  ( F `  k ) )  =  { ( P `  k ) } ,  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  C_  ( I `  ( F `  k )
) )  <-> if- ( ( P `  0 )  =  ( P ` 
1 ) ,  ( I `  ( F `
 0 ) )  =  { ( P `
 0 ) } ,  { ( P `
 0 ) ,  ( P `  1
) }  C_  (
I `  ( F `  0 ) ) ) )
5337, 52sylibr 217 . 2  |-  ( ph  ->  A. k  e.  {
0 }if- ( ( P `  k )  =  ( P `  ( k  +  1 ) ) ,  ( I `  ( F `
 k ) )  =  { ( P `
 k ) } ,  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  C_  (
I `  ( F `  k ) ) ) )
541fveq2i 5882 . . . . . . 7  |-  ( # `  F )  =  (
# `  <" J "> )
55 s1len 12797 . . . . . . 7  |-  ( # `  <" J "> )  =  1
5654, 55eqtri 2493 . . . . . 6  |-  ( # `  F )  =  1
5756oveq2i 6319 . . . . 5  |-  ( 0..^ ( # `  F
) )  =  ( 0..^ 1 )
58 fzo01 12024 . . . . 5  |-  ( 0..^ 1 )  =  {
0 }
5957, 58eqtri 2493 . . . 4  |-  ( 0..^ ( # `  F
) )  =  {
0 }
6059a1i 11 . . 3  |-  ( ph  ->  ( 0..^ ( # `  F ) )  =  { 0 } )
6160raleqdv 2979 . 2  |-  ( ph  ->  ( A. k  e.  ( 0..^ ( # `  F ) )if- ( ( P `  k
)  =  ( P `
 ( k  +  1 ) ) ,  ( I `  ( F `  k )
)  =  { ( P `  k ) } ,  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  C_  ( I `  ( F `  k )
) )  <->  A. k  e.  { 0 }if- (
( P `  k
)  =  ( P `
 ( k  +  1 ) ) ,  ( I `  ( F `  k )
)  =  { ( P `  k ) } ,  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  C_  ( I `  ( F `  k )
) ) ) )
6253, 61mpbird 240 1  |-  ( ph  ->  A. k  e.  ( 0..^ ( # `  F
) )if- ( ( P `  k )  =  ( P `  ( k  +  1 ) ) ,  ( I `  ( F `
 k ) )  =  { ( P `
 k ) } ,  { ( P `
 k ) ,  ( P `  (
k  +  1 ) ) }  C_  (
I `  ( F `  k ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376  if-wif 983    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   _Vcvv 3031    C_ wss 3390   {csn 3959   {cpr 3961   ` cfv 5589  (class class class)co 6308   0cc0 9557   1c1 9558    + caddc 9560  ..^cfzo 11942   #chash 12553   <"cs1 12706   <"cs2 12996
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-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  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  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ifp 984  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-reu 2763  df-rmo 2764  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-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  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-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  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-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-s2 13003
This theorem is referenced by:  11wlkd  40029
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