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

Theorem usgra2wlkspthlem2 24741
Description: Lemma 2 for usgra2wlkspth 24742. (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 6204 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  2  ->  (
0 ... ( # `  F
) )  =  ( 0 ... 2 ) )
21feq2d 5626 . . . . . . . . . . . . 13  |-  ( (
# `  F )  =  2  ->  ( P : ( 0 ... ( # `  F
) ) --> V  <->  P :
( 0 ... 2
) --> V ) )
3 fz0tp 11699 . . . . . . . . . . . . . . 15  |-  ( 0 ... 2 )  =  { 0 ,  1 ,  2 }
43feq2i 5632 . . . . . . . . . . . . . 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 726 . . . . . . . . . . 11  |-  ( ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  ( # `  F )  =  2 ) )  ->  ( P :
( 0 ... ( # `
 F ) ) --> V  ->  P : { 0 ,  1 ,  2 } --> V ) )
87adantr 463 . . . . . . . . . 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 428 . . . . . . . . 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 2401 . . . . . . . . . . . . . . . . . . . . 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 2606 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B )  -> 
( A  =/=  B  ->  ( P `  0
)  =/=  ( P `
 2 ) ) )
13123impia 1191 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( P `  0
)  =  A  /\  ( P `  2 )  =  B  /\  A  =/=  B )  ->  ( P `  0 )  =/=  ( P `  2
) )
1413adantr 463 . . . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . 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 753 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } )  ->  V USGrph  E )
18 usgrafun 24470 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( V USGrph  E  ->  Fun  E )
19 wrdf 12458 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( F  e. Word  dom  E  ->  F : ( 0..^ (
# `  F )
) --> dom  E )
20 oveq2 6204 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  ( 0..^ 2 ) )
2120feq2d 5626 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( (
# `  F )  =  2  ->  ( F : ( 0..^ (
# `  F )
) --> dom  E  <->  F :
( 0..^ 2 ) --> dom  E ) )
22 c0ex 9501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  0  e.  _V
2322prid1 4052 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  0  e.  { 0 ,  1 }
24 fzo0to2pr 11798 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( 0..^ 2 )  =  {
0 ,  1 }
2523, 24eleqtrri 2469 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  0  e.  ( 0..^ 2 )
26 ffvelrn 5931 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( F : ( 0..^ 2 ) --> dom  E  /\  0  e.  (
0..^ 2 ) )  ->  ( F ` 
0 )  e.  dom  E )
2725, 26mpan2 669 . . . . . . . . . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  ->  ( F ` 
0 )  e.  dom  E )
30 fvelrn 5926 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( Fun  E  /\  ( F `  0 )  e.  dom  E )  -> 
( E `  ( F `  0 )
)  e.  ran  E
)
3118, 29, 30syl2anr 476 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  V USGrph  E )  ->  ( E `  ( F `  0 )
)  e.  ran  E
)
3231adantr 463 . . . . . . . . . . . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( E `  ( F `
 0 ) )  =  { ( P `
 0 ) ,  ( P `  1
) }  ->  (
( E `  ( F `  0 )
)  e.  ran  E  <->  { ( P `  0
) ,  ( P `
 1 ) }  e.  ran  E ) )
3433adantl 464 . . . . . . . . . . . . . . . . . . . . . 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 24502 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( V USGrph  E  /\  { ( P `  0 ) ,  ( P ` 
1 ) }  e.  ran  E )  ->  ( P `  0 )  =/=  ( P `  1
) )
3717, 35, 36syl2anc 659 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  0
) )  =  {
( P `  0
) ,  ( P `
 1 ) } )  ->  ( P `  0 )  =/=  ( P `  1
) )
3837ex 432 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  V USGrph  E )  ->  ( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P ` 
1 ) }  ->  ( P `  0 )  =/=  ( P ` 
1 ) ) )
39 simplr 753 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  V USGrph  E )
40 1ex 9502 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  1  e.  _V
4140prid2 4053 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  1  e.  { 0 ,  1 }
4241, 24eleqtrri 2469 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  1  e.  ( 0..^ 2 )
43 ffvelrn 5931 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( F : ( 0..^ 2 ) --> dom  E  /\  1  e.  (
0..^ 2 ) )  ->  ( F ` 
1 )  e.  dom  E )
4442, 43mpan2 669 . . . . . . . . . . . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  ->  ( F ` 
1 )  e.  dom  E )
47 fvelrn 5926 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( Fun  E  /\  ( F `  1 )  e.  dom  E )  -> 
( E `  ( F `  1 )
)  e.  ran  E
)
4818, 46, 47syl2anr 476 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  V USGrph  E )  ->  ( E `  ( F `  1 )
)  e.  ran  E
)
4948adantr 463 . . . . . . . . . . . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( E `  ( F `
 1 ) )  =  { ( P `
 1 ) ,  ( P `  2
) }  ->  (
( E `  ( F `  1 )
)  e.  ran  E  <->  { ( P `  1
) ,  ( P `
 2 ) }  e.  ran  E ) )
5150adantl 464 . . . . . . . . . . . . . . . . . . . . . 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 24502 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( V USGrph  E  /\  { ( P `  1 ) ,  ( P ` 
2 ) }  e.  ran  E )  ->  ( P `  1 )  =/=  ( P `  2
) )
5439, 52, 53syl2anc 659 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( F  e. Word  dom  E  /\  ( # `  F )  =  2 )  /\  V USGrph  E
)  /\  ( E `  ( F `  1
) )  =  {
( P `  1
) ,  ( P `
 2 ) } )  ->  ( P `  1 )  =/=  ( P `  2
) )
5554ex 432 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( F  e. Word  dom  E  /\  ( # `  F
)  =  2 )  /\  V USGrph  E )  ->  ( ( E `  ( F `  1 ) )  =  { ( P `  1 ) ,  ( P ` 
2 ) }  ->  ( P `  1 )  =/=  ( P ` 
2 ) ) )
5638, 55anim12d 561 . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . 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 427 . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . . 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 984 . . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . . 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 608 . . . . . . . . . . . . 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 428 . . . . . . . . . . . 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 5774 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  2  ->  ( P `  ( # `  F
) )  =  ( P `  2 ) )
6564eqeq1d 2384 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  2  ->  (
( P `  ( # `
 F ) )  =  B  <->  ( P `  2 )  =  B ) )
66653anbi2d 1302 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  2  ->  (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B  /\  A  =/=  B )  <->  ( ( P `  0 )  =  A  /\  ( P `  2 )  =  B  /\  A  =/= 
B ) ) )
6720, 24syl6eq 2439 . . . . . . . . . . . . . . . . 17  |-  ( (
# `  F )  =  2  ->  (
0..^ ( # `  F
) )  =  {
0 ,  1 } )
6867raleqdv 2985 . . . . . . . . . . . . . . . 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 24687 . . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . 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 315 . . . . . . . . . . . . . 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 318 . . . . . . . . . . . . 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 726 . . . . . . . . . . . 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 427 . . . . . . . . . 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 428 . . . . . . . . 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 10813 . . . . . . . . . . . 12  |-  2  e.  ZZ
7922, 40, 783pm3.2i 1172 . . . . . . . . . . 11  |-  ( 0  e.  _V  /\  1  e.  _V  /\  2  e.  ZZ )
80 0ne1 10520 . . . . . . . . . . . 12  |-  0  =/=  1
81 0ne2 10664 . . . . . . . . . . . 12  |-  0  =/=  2
82 1ne2 10665 . . . . . . . . . . . 12  |-  1  =/=  2
8380, 81, 823pm3.2i 1172 . . . . . . . . . . 11  |-  ( 0  =/=  1  /\  0  =/=  2  /\  1  =/=  2 )
8479, 83pm3.2i 453 . . . . . . . . . 10  |-  ( ( 0  e.  _V  /\  1  e.  _V  /\  2  e.  ZZ )  /\  (
0  =/=  1  /\  0  =/=  2  /\  1  =/=  2 ) )
85 eqid 2382 . . . . . . . . . . 11  |-  { 0 ,  1 ,  2 }  =  { 0 ,  1 ,  2 }
8685f13dfv 6081 . . . . . . . . . 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 920 . . . . . . . 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 5501 . . . . . . . 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 461 . . . . . 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 433 . . . . 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 432 . . . 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 438 . 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 428 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   _Vcvv 3034   {cpr 3946   {ctp 3948   class class class wbr 4367   `'ccnv 4912   dom cdm 4913   ran crn 4914   Fun wfun 5490   -->wf 5492   -1-1->wf1 5493   ` cfv 5496  (class class class)co 6196   0cc0 9403   1c1 9404    + caddc 9406   2c2 10502   ZZcz 10781   ...cfz 11593  ..^cfzo 11717   #chash 12307  Word cword 12438   USGrph cusg 24451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-hash 12308  df-word 12446  df-usgra 24454
This theorem is referenced by:  usgra2wlkspth  24742
  Copyright terms: Public domain W3C validator