MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  usgra2wlkspthlem2 Structured version   Unicode version

Theorem usgra2wlkspthlem2 24598
Description: Lemma 2 for usgra2wlkspth 24599. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
Assertion
Ref Expression
usgra2wlkspthlem2  |-  ( ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  ( V USGrph  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )  ->  ( (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/=  B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  ->  Fun  `' P ) )
Distinct variable groups:    i, E    i, F    P, i
Allowed substitution hints:    A( i)    B( i)    V( i)

Proof of Theorem usgra2wlkspthlem2
StepHypRef Expression
1 oveq2 6289 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  2  ->  (
0 ... ( # `  F
) )  =  ( 0 ... 2 ) )
21feq2d 5708 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  2  ->  ( P : ( 0 ... ( # `  F
) ) --> V  <->  P :
( 0 ... 2
) --> V ) )
3 fz0tp 11788 . . . . . . . . . . . . . . 15  |-  ( 0 ... 2 )  =  { 0 ,  1 ,  2 }
43feq2i 5714 . . . . . . . . . . . . . 14  |-  ( P : ( 0 ... 2 ) --> V  <->  P : { 0 ,  1 ,  2 } --> V )
54biimpi 194 . . . . . . . . . . . . 13  |-  ( P : ( 0 ... 2 ) --> V  ->  P : { 0 ,  1 ,  2 } --> V )
62, 5syl6bi 228 . . . . . . . . . . . 12  |-  ( (
# `  F )  =  2  ->  ( P : ( 0 ... ( # `  F
) ) --> V  ->  P : { 0 ,  1 ,  2 } --> V ) )
76ad2antll 728 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  ( # `  F )  =  2 ) )  ->  ( P :
( 0 ... ( # `
 F ) ) --> V  ->  P : { 0 ,  1 ,  2 } --> V ) )
87adantr 465 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 ) )  /\  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/= 
B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  -> 
( P : ( 0 ... ( # `  F ) ) --> V  ->  P : {
0 ,  1 ,  2 } --> V ) )
98impcom 430 . . . . . . . . 9  |-  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 ) )  /\  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/= 
B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) ) )  ->  P : {
0 ,  1 ,  2 } --> V )
10 eqeq12 2462 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B )  -> 
( ( P ` 
0 )  =  ( P `  2 )  <-> 
A  =  B ) )
1110biimpd 207 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B )  -> 
( ( P ` 
0 )  =  ( P `  2 )  ->  A  =  B ) )
1211necon3d 2667 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B )  -> 
( A  =/=  B  ->  ( P `  0
)  =/=  ( P `
 2 ) ) )
13123impia 1194 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B  /\  A  =/=  B )  ->  ( P `  0 )  =/=  ( P `  2
) )
1413adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( P ` 
0 )  =  A  /\  ( P ` 
2 )  =  B  /\  A  =/=  B
)  /\  ( ( E `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) )  ->  ( P ` 
0 )  =/=  ( P `  2 )
)
1514adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( (
( P `  0
)  =  A  /\  ( P `  2 )  =  B  /\  A  =/=  B )  /\  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) ) )  ->  ( P `  0 )  =/=  ( P `  2
) )
1615adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E )  /\  (
( ( P ` 
0 )  =  A  /\  ( P ` 
2 )  =  B  /\  A  =/=  B
)  /\  ( ( E `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) ) )  /\  P :
( 0 ... 2
) --> V )  -> 
( P `  0
)  =/=  ( P `
 2 ) )
17 simplr 755 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } )  ->  V USGrph  E )
18 usgrafun 24327 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( V USGrph  E  ->  Fun  E )
19 wrdf 12535 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( F  e. Word  dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
20 oveq2 6289 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 2 ) )
2120feq2d 5708 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( (
# `  F )  =  2  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  <->  F :
( 0..^ 2 ) --> dom  E ) )
22 c0ex 9593 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  0  e.  _V
2322prid1 4123 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  0  e.  { 0 ,  1 }
24 fzo0to2pr 11881 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( 0..^ 2 )  =  {
0 ,  1 }
2523, 24eleqtrri 2530 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  0  e.  ( 0..^ 2 )
26 ffvelrn 6014 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( F : ( 0..^ 2 ) --> dom  E  /\  0  e.  (
0..^ 2 ) )  ->  ( F ` 
0 )  e.  dom  E )
2725, 26mpan2 671 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( F : ( 0..^ 2 ) --> dom  E  ->  ( F `  0 )  e.  dom  E )
2821, 27syl6bi 228 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
# `  F )  =  2  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( F `  0 )  e.  dom  E ) )
2919, 28mpan9 469 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  ->  ( F ` 
0 )  e.  dom  E )
30 fvelrn 6009 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( Fun  E  /\  ( F `  0 )  e.  dom  E )  -> 
( E `  ( F `  0 )
)  e.  ran  E
)
3118, 29, 30syl2anr 478 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  V USGrph  E )  ->  ( E `  ( F `  0 )
)  e.  ran  E
)
3231adantr 465 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } )  ->  ( E `  ( F `  0
) )  e.  ran  E )
33 eleq1 2515 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( E `  ( F `
 0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  ->  (
( E `  ( F `  0 )
)  e.  ran  E  <->  { ( P `  0
) ,  ( P `
 1 ) }  e.  ran  E ) )
3433adantl 466 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } )  ->  ( ( E `  ( F `  0 ) )  e.  ran  E  <->  { ( P `  0 ) ,  ( P ` 
1 ) }  e.  ran  E ) )
3532, 34mpbid 210 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } )  ->  { ( P `  0 ) ,  ( P ` 
1 ) }  e.  ran  E )
36 usgraedgrn 24359 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( V USGrph  E  /\  { ( P `  0 ) ,  ( P ` 
1 ) }  e.  ran  E )  ->  ( P `  0 )  =/=  ( P `  1
) )
3717, 35, 36syl2anc 661 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } )  ->  ( P `  0 )  =/=  ( P `  1
) )
3837ex 434 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  V USGrph  E )  ->  ( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  ->  ( P `  0 )  =/=  ( P ` 
1 ) ) )
39 simplr 755 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  V USGrph  E )
40 1ex 9594 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  1  e.  _V
4140prid2 4124 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  1  e.  { 0 ,  1 }
4241, 24eleqtrri 2530 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  1  e.  ( 0..^ 2 )
43 ffvelrn 6014 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( F : ( 0..^ 2 ) --> dom  E  /\  1  e.  (
0..^ 2 ) )  ->  ( F ` 
1 )  e.  dom  E )
4442, 43mpan2 671 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( F : ( 0..^ 2 ) --> dom  E  ->  ( F `  1 )  e.  dom  E )
4521, 44syl6bi 228 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
# `  F )  =  2  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( F `  1 )  e.  dom  E ) )
4619, 45mpan9 469 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  ->  ( F ` 
1 )  e.  dom  E )
47 fvelrn 6009 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( Fun  E  /\  ( F `  1 )  e.  dom  E )  -> 
( E `  ( F `  1 )
)  e.  ran  E
)
4818, 46, 47syl2anr 478 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  V USGrph  E )  ->  ( E `  ( F `  1 )
)  e.  ran  E
)
4948adantr 465 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  ( E `  ( F `  1
) )  e.  ran  E )
50 eleq1 2515 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) }  ->  (
( E `  ( F `  1 )
)  e.  ran  E  <->  { ( P `  1
) ,  ( P `
 2 ) }  e.  ran  E ) )
5150adantl 466 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  ( ( E `  ( F `  1 ) )  e.  ran  E  <->  { ( P `  1 ) ,  ( P ` 
2 ) }  e.  ran  E ) )
5249, 51mpbid 210 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  { ( P `  1 ) ,  ( P ` 
2 ) }  e.  ran  E )
53 usgraedgrn 24359 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( V USGrph  E  /\  { ( P `  1 ) ,  ( P ` 
2 ) }  e.  ran  E )  ->  ( P `  1 )  =/=  ( P `  2
) )
5439, 52, 53syl2anc 661 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  ( P `  1 )  =/=  ( P `  2
) )
5554ex 434 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  V USGrph  E )  ->  ( ( E `  ( F `  1 ) )  =  { ( P `  1 ) ,  ( P ` 
2 ) }  ->  ( P `  1 )  =/=  ( P ` 
2 ) ) )
5638, 55anim12d 563 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  V USGrph  E )  ->  ( ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } )  ->  (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) ) ) )
5756adantld 467 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  V USGrph  E )  ->  ( ( ( ( P `  0 )  =  A  /\  ( P `  2 )  =  B  /\  A  =/= 
B )  /\  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) )  ->  ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  1 )  =/=  ( P `  2
) ) ) )
5857imp 429 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( (
( P `  0
)  =  A  /\  ( P `  2 )  =  B  /\  A  =/=  B )  /\  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) ) )  ->  ( ( P `  0 )  =/=  ( P `  1
)  /\  ( P `  1 )  =/=  ( P `  2
) ) )
5958adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E )  /\  (
( ( P ` 
0 )  =  A  /\  ( P ` 
2 )  =  B  /\  A  =/=  B
)  /\  ( ( E `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) ) )  /\  P :
( 0 ... 2
) --> V )  -> 
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  1
)  =/=  ( P `
 2 ) ) )
60 3anan12 987 . . . . . . . . . . . . . . 15  |-  ( ( ( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  1 )  =/=  ( P `  2
) )  <->  ( ( P `  0 )  =/=  ( P `  2
)  /\  ( ( P `  0 )  =/=  ( P `  1
)  /\  ( P `  1 )  =/=  ( P `  2
) ) ) )
6116, 59, 60sylanbrc 664 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E )  /\  (
( ( P ` 
0 )  =  A  /\  ( P ` 
2 )  =  B  /\  A  =/=  B
)  /\  ( ( E `  ( F `  0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) ) )  /\  P :
( 0 ... 2
) --> V )  -> 
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) ) )
6261exp41 610 . . . . . . . . . . . . 13  |-  ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  ->  ( V USGrph  E  ->  ( ( ( ( P `  0 )  =  A  /\  ( P `  2 )  =  B  /\  A  =/= 
B )  /\  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) )  ->  ( P :
( 0 ... 2
) --> V  ->  (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  1 )  =/=  ( P `  2
) ) ) ) ) )
6362impcom 430 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  ( # `  F )  =  2 ) )  ->  ( ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B  /\  A  =/=  B )  /\  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) )  ->  ( P :
( 0 ... 2
) --> V  ->  (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  1 )  =/=  ( P `  2
) ) ) ) )
64 fveq2 5856 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  2  ->  ( P `  ( # `  F
) )  =  ( P `  2 ) )
6564eqeq1d 2445 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  2  ->  (
( P `  ( # `
 F ) )  =  B  <->  ( P `  2 )  =  B ) )
66653anbi2d 1305 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  2  ->  (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/=  B )  <->  ( ( P `  0 )  =  A  /\  ( P `  2 )  =  B  /\  A  =/= 
B ) ) )
6720, 24syl6eq 2500 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  {
0 ,  1 } )
6867raleqdv 3046 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  2  ->  ( A. i  e.  (
0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  <->  A. i  e.  {
0 ,  1 }  ( E `  ( F `  i )
)  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )
69 2wlklem 24544 . . . . . . . . . . . . . . . 16  |-  ( A. i  e.  { 0 ,  1 }  ( E `  ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  <->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } ) )
7068, 69syl6bb 261 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  2  ->  ( A. i  e.  (
0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }  <->  ( ( E `
 ( F ` 
0 ) )  =  { ( P ` 
0 ) ,  ( P `  1 ) }  /\  ( E `
 ( F ` 
1 ) )  =  { ( P ` 
1 ) ,  ( P `  2 ) } ) ) )
7166, 70anbi12d 710 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  2  ->  (
( ( ( P `
 0 )  =  A  /\  ( P `
 ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )  <->  ( (
( P `  0
)  =  A  /\  ( P `  2 )  =  B  /\  A  =/=  B )  /\  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) ) ) )
722imbi1d 317 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  2  ->  (
( P : ( 0 ... ( # `  F ) ) --> V  ->  ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) ) )  <->  ( P : ( 0 ... 2 ) --> V  -> 
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) ) ) ) )
7371, 72imbi12d 320 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  2  ->  (
( ( ( ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B  /\  A  =/=  B
)  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
 i ) )  =  { ( P `
 i ) ,  ( P `  (
i  +  1 ) ) } )  -> 
( P : ( 0 ... ( # `  F ) ) --> V  ->  ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) ) ) )  <-> 
( ( ( ( P `  0 )  =  A  /\  ( P `  2 )  =  B  /\  A  =/= 
B )  /\  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) )  ->  ( P :
( 0 ... 2
) --> V  ->  (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  1 )  =/=  ( P `  2
) ) ) ) ) )
7473ad2antll 728 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  ( # `  F )  =  2 ) )  ->  ( ( ( ( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/=  B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  ->  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) ) ) )  <->  ( ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B  /\  A  =/=  B )  /\  (
( E `  ( F `  0 )
)  =  { ( P `  0 ) ,  ( P ` 
1 ) }  /\  ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } ) )  ->  ( P :
( 0 ... 2
) --> V  ->  (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  1 )  =/=  ( P `  2
) ) ) ) ) )
7563, 74mpbird 232 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  ( # `  F )  =  2 ) )  ->  ( ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/= 
B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  ->  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) ) ) ) )
7675imp 429 . . . . . . . . . 10  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 ) )  /\  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/= 
B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  -> 
( P : ( 0 ... ( # `  F ) ) --> V  ->  ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) ) ) )
7776impcom 430 . . . . . . . . 9  |-  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 ) )  /\  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/= 
B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) ) )  ->  ( ( P `
 0 )  =/=  ( P `  1
)  /\  ( P `  0 )  =/=  ( P `  2
)  /\  ( P `  1 )  =/=  ( P `  2
) ) )
78 2z 10903 . . . . . . . . . . . 12  |-  2  e.  ZZ
7922, 40, 783pm3.2i 1175 . . . . . . . . . . 11  |-  ( 0  e.  _V  /\  1  e.  _V  /\  2  e.  ZZ )
80 0ne1 10610 . . . . . . . . . . . 12  |-  0  =/=  1
81 0ne2 10754 . . . . . . . . . . . 12  |-  0  =/=  2
82 1ne2 10755 . . . . . . . . . . . 12  |-  1  =/=  2
8380, 81, 823pm3.2i 1175 . . . . . . . . . . 11  |-  ( 0  =/=  1  /\  0  =/=  2  /\  1  =/=  2 )
8479, 83pm3.2i 455 . . . . . . . . . 10  |-  ( ( 0  e.  _V  /\  1  e.  _V  /\  2  e.  ZZ )  /\  (
0  =/=  1  /\  0  =/=  2  /\  1  =/=  2 ) )
85 eqid 2443 . . . . . . . . . . 11  |-  { 0 ,  1 ,  2 }  =  { 0 ,  1 ,  2 }
8685f13dfv 6165 . . . . . . . . . 10  |-  ( ( ( 0  e.  _V  /\  1  e.  _V  /\  2  e.  ZZ )  /\  ( 0  =/=  1  /\  0  =/=  2  /\  1  =/=  2
) )  ->  ( P : { 0 ,  1 ,  2 }
-1-1-> V  <->  ( P : { 0 ,  1 ,  2 } --> V  /\  ( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) ) ) ) )
8784, 86mp1i 12 . . . . . . . . 9  |-  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 ) )  /\  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/= 
B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) ) )  ->  ( P : { 0 ,  1 ,  2 } -1-1-> V  <->  ( P : { 0 ,  1 ,  2 } --> V  /\  (
( P `  0
)  =/=  ( P `
 1 )  /\  ( P `  0 )  =/=  ( P ` 
2 )  /\  ( P `  1 )  =/=  ( P `  2
) ) ) ) )
889, 77, 87mpbir2and 922 . . . . . . . 8  |-  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 ) )  /\  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/= 
B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) ) )  ->  P : {
0 ,  1 ,  2 } -1-1-> V )
89 df-f1 5583 . . . . . . . 8  |-  ( P : { 0 ,  1 ,  2 }
-1-1-> V  <->  ( P : { 0 ,  1 ,  2 } --> V  /\  Fun  `' P ) )
9088, 89sylib 196 . . . . . . 7  |-  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 ) )  /\  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/= 
B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) ) )  ->  ( P : { 0 ,  1 ,  2 } --> V  /\  Fun  `' P ) )
9190simprd 463 . . . . . 6  |-  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 ) )  /\  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/= 
B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) ) )  ->  Fun  `' P
)
9291expcom 435 . . . . 5  |-  ( ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 ) )  /\  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/= 
B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } ) )  -> 
( P : ( 0 ... ( # `  F ) ) --> V  ->  Fun  `' P
) )
9392ex 434 . . . 4  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  ( # `  F )  =  2 ) )  ->  ( ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/= 
B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  ->  ( P : ( 0 ... ( # `  F
) ) --> V  ->  Fun  `' P ) ) )
9493com23 78 . . 3  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  ( # `  F )  =  2 ) )  ->  ( P :
( 0 ... ( # `
 F ) ) --> V  ->  ( (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/=  B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  ->  Fun  `' P ) ) )
9594impancom 440 . 2  |-  ( ( V USGrph  E  /\  P :
( 0 ... ( # `
 F ) ) --> V )  ->  (
( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  ->  ( ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/= 
B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  ->  Fun  `' P ) ) )
9695impcom 430 1  |-  ( ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  ( V USGrph  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )  ->  ( (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/=  B )  /\  A. i  e.  ( 0..^ ( # `  F
) ) ( E `
 ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) } )  ->  Fun  `' P ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   _Vcvv 3095   {cpr 4016   {ctp 4018   class class class wbr 4437   `'ccnv 4988   dom cdm 4989   ran crn 4990   Fun wfun 5572   -->wf 5574   -1-1->wf1 5575   ` cfv 5578  (class class class)co 6281   0cc0 9495   1c1 9496    + caddc 9498   2c2 10592   ZZcz 10871   ...cfz 11683  ..^cfzo 11806   #chash 12387  Word cword 12516   USGrph cusg 24308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-fzo 11807  df-hash 12388  df-word 12524  df-usgra 24311
This theorem is referenced by:  usgra2wlkspth  24599
  Copyright terms: Public domain W3C validator