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Theorem usgra2wlkspthlem2 24324
Description: Lemma 2 for usgra2wlkspth 24325. (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 6292 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  2  ->  (
0 ... ( # `  F
) )  =  ( 0 ... 2 ) )
21feq2d 5718 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  2  ->  ( P : ( 0 ... ( # `  F
) ) --> V  <->  P :
( 0 ... 2
) --> V ) )
3 fz0tp 11775 . . . . . . . . . . . . . . . 16  |-  ( 0 ... 2 )  =  { 0 ,  1 ,  2 }
43feq2i 5724 . . . . . . . . . . . . . . 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 2486 . . . . . . . . . . . . . . . . . . . . . 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 2691 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B )  -> 
( A  =/=  B  ->  ( P `  0
)  =/=  ( P `
 2 ) ) )
13123impia 1193 . . . . . . . . . . . . . . . . . . 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 24053 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( V USGrph  E  ->  Fun  E )
19 wrdf 12519 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( F  e. Word  dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
20 oveq2 6292 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 2 ) )
2120feq2d 5718 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
# `  F )  =  2  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  <->  F :
( 0..^ 2 ) --> dom  E ) )
22 c0ex 9590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  0  e.  _V
2322prid1 4135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  0  e.  { 0 ,  1 }
24 fzo0to2pr 11867 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( 0..^ 2 )  =  {
0 ,  1 }
2523, 24eleqtrri 2554 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  0  e.  ( 0..^ 2 )
26 ffvelrn 6019 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6017 . . . . . . . . . . . . . . . . . . . . . . . . 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 2539 . . . . . . . . . . . . . . . . . . . . . . . 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 24085 . . . . . . . . . . . . . . . . . . . . . 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 9591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  1  e.  _V
4140prid2 4136 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  1  e.  { 0 ,  1 }
4241, 24eleqtrri 2554 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  1  e.  ( 0..^ 2 )
43 ffvelrn 6019 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6017 . . . . . . . . . . . . . . . . . . . . . . . . 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 2539 . . . . . . . . . . . . . . . . . . . . . . . 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 24085 . . . . . . . . . . . . . . . . . . . . . 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 986 . . . . . . . . . . . . . . . 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 5866 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  F )  =  2  ->  ( P `  ( # `  F
) )  =  ( P `  2 ) )
6564eqeq1d 2469 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  2  ->  (
( P `  ( # `
 F ) )  =  B  <->  ( P `  2 )  =  B ) )
66653anbi2d 1304 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  2  ->  (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/=  B )  <->  ( ( P `  0 )  =  A  /\  ( P `  2 )  =  B  /\  A  =/= 
B ) ) )
6720, 24syl6eq 2524 . . . . . . . . . . . . . . . . . 18  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  {
0 ,  1 } )
6867raleqdv 3064 . . . . . . . . . . . . . . . . 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 24270 . . . . . . . . . . . . . . . . 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 10896 . . . . . . . . . . . 12  |-  2  e.  ZZ
8022, 40, 793pm3.2i 1174 . . . . . . . . . . 11  |-  ( 0  e.  _V  /\  1  e.  _V  /\  2  e.  ZZ )
81 0ne1 10603 . . . . . . . . . . . 12  |-  0  =/=  1
82 0ne2 10747 . . . . . . . . . . . 12  |-  0  =/=  2
83 1ne2 10748 . . . . . . . . . . . 12  |-  1  =/=  2
8481, 82, 833pm3.2i 1174 . . . . . . . . . . 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 2467 . . . . . . . . . . 11  |-  { 0 ,  1 ,  2 }  =  { 0 ,  1 ,  2 }
8786f13dfv 6168 . . . . . . . . . 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 5593 . . . . . . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113   {cpr 4029   {ctp 4031   class class class wbr 4447   `'ccnv 4998   dom cdm 4999   ran crn 5000   Fun wfun 5582   -->wf 5584   -1-1->wf1 5585   ` cfv 5588  (class class class)co 6284   0cc0 9492   1c1 9493    + caddc 9495   2c2 10585   ZZcz 10864   ...cfz 11672  ..^cfzo 11792   #chash 12373  Word cword 12500   USGrph cusg 24034
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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-usgra 24037
This theorem is referenced by:  usgra2wlkspth  24325
  Copyright terms: Public domain W3C validator