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Theorem usgra2wlkspthlem2 30444
Description: Lemma 2 for usgra2wlkspth 30445. (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 6207 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  2  ->  (
0 ... ( # `  F
) )  =  ( 0 ... 2 ) )
21feq2d 5654 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  2  ->  ( P : ( 0 ... ( # `  F
) ) --> V  <->  P :
( 0 ... 2
) --> V ) )
3 fz0tp 11629 . . . . . . . . . . . . . . . 16  |-  ( 0 ... 2 )  =  { 0 ,  1 ,  2 }
43feq2i 5659 . . . . . . . . . . . . . . 15  |-  ( P : ( 0 ... 2 ) --> V  <->  P : { 0 ,  1 ,  2 } --> V )
54biimpi 194 . . . . . . . . . . . . . 14  |-  ( P : ( 0 ... 2 ) --> V  ->  P : { 0 ,  1 ,  2 } --> V )
62, 5syl6bi 228 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  2  ->  ( P : ( 0 ... ( # `  F
) ) --> V  ->  P : { 0 ,  1 ,  2 } --> V ) )
76ad2antll 728 . . . . . . . . . . . 12  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  ( # `  F )  =  2 ) )  ->  ( P :
( 0 ... ( # `
 F ) ) --> V  ->  P : { 0 ,  1 ,  2 } --> V ) )
87adantr 465 . . . . . . . . . . 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 ,  1 ,  2 } --> V ) )
98impcom 430 . . . . . . . . . 10  |-  ( ( 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 2473 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B )  -> 
( ( P ` 
0 )  =  ( P `  2 )  <-> 
A  =  B ) )
1110biimpd 207 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B )  -> 
( ( P ` 
0 )  =  ( P `  2 )  ->  A  =  B ) )
1211necon3d 2675 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B )  -> 
( A  =/=  B  ->  ( P `  0
)  =/=  ( P `
 2 ) ) )
13123impia 1185 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B  /\  A  =/=  B )  ->  ( P `  0 )  =/=  ( P `  2
) )
1413adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( 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 . . . . . . . . . . . . . . . . 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 `  2
) )
1615adantr 465 . . . . . . . . . . . . . . . 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 ... 2
) --> V )  -> 
( P `  0
)  =/=  ( P `
 2 ) )
17 simplr 754 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } )  ->  V USGrph  E )
18 usgrafun 23428 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( V USGrph  E  ->  Fun  E )
19 wrdf 12357 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( F  e. Word  dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
20 oveq2 6207 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 2 ) )
2120feq2d 5654 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
# `  F )  =  2  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  <->  F :
( 0..^ 2 ) --> dom  E ) )
22 c0ex 9490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  0  e.  _V
2322prid1 4090 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  0  e.  { 0 ,  1 }
24 fzo0to2pr 11730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( 0..^ 2 )  =  {
0 ,  1 }
2523, 24eleqtrri 2541 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  0  e.  ( 0..^ 2 )
26 ffvelrn 5949 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( F : ( 0..^ 2 ) --> dom  E  /\  0  e.  (
0..^ 2 ) )  ->  ( F ` 
0 )  e.  dom  E )
2725, 26mpan2 671 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( F : ( 0..^ 2 ) --> dom  E  ->  ( F `  0 )  e.  dom  E )
2821, 27syl6bi 228 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( (
# `  F )  =  2  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( F `  0 )  e.  dom  E ) )
2919, 28mpan9 469 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  ->  ( F ` 
0 )  e.  dom  E )
30 fvelrn 5947 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( Fun  E  /\  ( F `  0 )  e.  dom  E )  -> 
( E `  ( F `  0 )
)  e.  ran  E
)
3118, 29, 30syl2anr 478 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  V USGrph  E )  ->  ( E `  ( F `  0 )
)  e.  ran  E
)
3231adantr 465 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( 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 2526 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( E `  ( F `
 0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  ->  (
( E `  ( F `  0 )
)  e.  ran  E  <->  { ( P `  0
) ,  ( P `
 1 ) }  e.  ran  E ) )
3433adantl 466 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( 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 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( 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 23451 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( V USGrph  E  /\  { ( P `  0 ) ,  ( P ` 
1 ) }  e.  ran  E )  ->  ( P `  0 )  =/=  ( P `  1
) )
3717, 35, 36syl2anc 661 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } )  ->  ( P `  0 )  =/=  ( P `  1
) )
3837ex 434 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  V USGrph  E )  ->  ( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  ->  ( P `  0 )  =/=  ( P ` 
1 ) ) )
39 simplr 754 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  V USGrph  E )
40 1ex 9491 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  1  e.  _V
4140prid2 4091 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  1  e.  { 0 ,  1 }
4241, 24eleqtrri 2541 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  1  e.  ( 0..^ 2 )
43 ffvelrn 5949 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( F : ( 0..^ 2 ) --> dom  E  /\  1  e.  (
0..^ 2 ) )  ->  ( F ` 
1 )  e.  dom  E )
4442, 43mpan2 671 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( F : ( 0..^ 2 ) --> dom  E  ->  ( F `  1 )  e.  dom  E )
4521, 44syl6bi 228 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( (
# `  F )  =  2  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  ->  ( F `  1 )  e.  dom  E ) )
4619, 45mpan9 469 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  ->  ( F ` 
1 )  e.  dom  E )
47 fvelrn 5947 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( Fun  E  /\  ( F `  1 )  e.  dom  E )  -> 
( E `  ( F `  1 )
)  e.  ran  E
)
4818, 46, 47syl2anr 478 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  V USGrph  E )  ->  ( E `  ( F `  1 )
)  e.  ran  E
)
4948adantr 465 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( 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 2526 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) }  ->  (
( E `  ( F `  1 )
)  e.  ran  E  <->  { ( P `  1
) ,  ( P `
 2 ) }  e.  ran  E ) )
5150adantl 466 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( 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 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( 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 23451 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( V USGrph  E  /\  { ( P `  1 ) ,  ( P ` 
2 ) }  e.  ran  E )  ->  ( P `  1 )  =/=  ( P `  2
) )
5439, 52, 53syl2anc 661 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  ( P `  1 )  =/=  ( P `  2
) )
5554ex 434 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  V USGrph  E )  ->  ( ( E `  ( F `  1 ) )  =  { ( P `  1 ) ,  ( P ` 
2 ) }  ->  ( P `  1 )  =/=  ( P ` 
2 ) ) )
5638, 55anim12d 563 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 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 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 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 . . . . . . . . . . . . . . . . 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
) ) )
5958adantr 465 . . . . . . . . . . . . . . . 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 ... 2
) --> V )  -> 
( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  1
)  =/=  ( P `
 2 ) ) )
60 3anan12 978 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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 . . . . . . . . . . . . . . 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 `  0
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) ) )
6261exp41 610 . . . . . . . . . . . . . 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
) ) ) ) ) )
6362impcom 430 . . . . . . . . . . . . 13  |-  ( ( 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 5798 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  F )  =  2  ->  ( P `  ( # `  F
) )  =  ( P `  2 ) )
6564eqeq1d 2456 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  2  ->  (
( P `  ( # `
 F ) )  =  B  <->  ( P `  2 )  =  B ) )
66653anbi2d 1295 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  2  ->  (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/=  B )  <->  ( ( P `  0 )  =  A  /\  ( P `  2 )  =  B  /\  A  =/= 
B ) ) )
6720, 24syl6eq 2511 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  {
0 ,  1 } )
6867raleqdv 3027 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  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 23614 . . . . . . . . . . . . . . . . 17  |-  ( 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 . . . . . . . . . . . . . . . 16  |-  ( (
# `  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 . . . . . . . . . . . . . . 15  |-  ( (
# `  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 . . . . . . . . . . . . . . 15  |-  ( (
# `  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 . . . . . . . . . . . . . 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 ... ( # `  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 . . . . . . . . . . . . 13  |-  ( ( 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 . . . . . . . . . . . 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 ) ) ) ) )
7675imp 429 . . . . . . . . . . 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
) ) ) )
7776impcom 430 . . . . . . . . . 10  |-  ( ( 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
) ) )
789, 77jca 532 . . . . . . . . 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  /\  ( ( P ` 
0 )  =/=  ( P `  1 )  /\  ( P `  0
)  =/=  ( P `
 2 )  /\  ( P `  1 )  =/=  ( P ` 
2 ) ) ) )
79 2z 10788 . . . . . . . . . . . 12  |-  2  e.  ZZ
8022, 40, 793pm3.2i 1166 . . . . . . . . . . 11  |-  ( 0  e.  _V  /\  1  e.  _V  /\  2  e.  ZZ )
81 0ne1 10499 . . . . . . . . . . . 12  |-  0  =/=  1
82 0ne2 10643 . . . . . . . . . . . 12  |-  0  =/=  2
83 1ne2 10644 . . . . . . . . . . . 12  |-  1  =/=  2
8481, 82, 833pm3.2i 1166 . . . . . . . . . . 11  |-  ( 0  =/=  1  /\  0  =/=  2  /\  1  =/=  2 )
8580, 84pm3.2i 455 . . . . . . . . . 10  |-  ( ( 0  e.  _V  /\  1  e.  _V  /\  2  e.  ZZ )  /\  (
0  =/=  1  /\  0  =/=  2  /\  1  =/=  2 ) )
86 eqid 2454 . . . . . . . . . . 11  |-  { 0 ,  1 ,  2 }  =  { 0 ,  1 ,  2 }
8786f13dfv 30294 . . . . . . . . . 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 ) ) ) ) )
8885, 87mp1i 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
) ) ) ) )
8978, 88mpbird 232 . . . . . . . 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 )
90 df-f1 5530 . . . . . . . 8  |-  ( P : { 0 ,  1 ,  2 }
-1-1-> V  <->  ( P : { 0 ,  1 ,  2 } --> V  /\  Fun  `' P ) )
9189, 90sylib 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 ) )
9291simprd 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
)
9392expcom 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
) )
9493ex 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 ) ) )
9594com23 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 ) ) )
9695impancom 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 ) ) )
9796impcom 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 965    = wceq 1370    e. wcel 1758    =/= wne 2647   A.wral 2798   _Vcvv 3076   {cpr 3986   {ctp 3988   class class class wbr 4399   `'ccnv 4946   dom cdm 4947   ran crn 4948   Fun wfun 5519   -->wf 5521   -1-1->wf1 5522   ` cfv 5525  (class class class)co 6199   0cc0 9392   1c1 9393    + caddc 9395   2c2 10481   ZZcz 10756   ...cfz 11553  ..^cfzo 11664   #chash 12219  Word cword 12338   USGrph cusg 23415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-n0 10690  df-z 10757  df-uz 10972  df-fz 11554  df-fzo 11665  df-hash 12220  df-word 12346  df-usgra 23417
This theorem is referenced by:  usgra2wlkspth  30445
  Copyright terms: Public domain W3C validator