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Theorem spthonepeq 21540
Description: The endpoints of a simple path between two vertices are equal if and only if the path is of length 0 (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
Assertion
Ref Expression
spthonepeq  |-  ( F ( A ( V SPathOn  E ) B ) P  ->  ( A  =  B  <->  ( # `  F
)  =  0 ) )

Proof of Theorem spthonepeq
StepHypRef Expression
1 spthonprp 21538 . 2  |-  ( F ( A ( V SPathOn  E ) B ) P  ->  ( (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  /\  ( F ( A ( V WalkOn  E ) B ) P  /\  F
( V SPaths  E ) P ) ) )
2 iswlkon 21484 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( F ( A ( V WalkOn  E ) B ) P  <->  ( F
( V Walks  E ) P  /\  ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B ) ) )
3 isspth 21522 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( F
( V SPaths  E ) P 
<->  ( F ( V Trails  E ) P  /\  Fun  `' P ) ) )
433adant3 977 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  ( F ( V SPaths  E
) P  <->  ( F
( V Trails  E ) P  /\  Fun  `' P
) ) )
52, 4anbi12d 692 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  (
( F ( A ( V WalkOn  E ) B ) P  /\  F ( V SPaths  E
) P )  <->  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  /\  ( F ( V Trails  E ) P  /\  Fun  `' P ) ) ) )
6 2mwlk 21481 . . . . . . . 8  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
7 lencl 11690 . . . . . . . . 9  |-  ( F  e. Word  dom  E  ->  (
# `  F )  e.  NN0 )
87anim1i 552 . . . . . . . 8  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( ( # `  F )  e.  NN0  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
9 df-f1 5418 . . . . . . . . . . . 12  |-  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  <->  ( P : ( 0 ... ( # `  F
) ) --> V  /\  Fun  `' P ) )
10 eqeq2 2413 . . . . . . . . . . . . . 14  |-  ( A  =  B  ->  (
( P `  0
)  =  A  <->  ( P `  0 )  =  B ) )
11 eqtr3 2423 . . . . . . . . . . . . . . . 16  |-  ( ( ( P `  ( # `
 F ) )  =  B  /\  ( P `  0 )  =  B )  ->  ( P `  ( # `  F
) )  =  ( P `  0 ) )
12 elnn0uz 10479 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  F )  e.  NN0  <->  ( # `  F
)  e.  ( ZZ>= ` 
0 ) )
13 eluzfz2 11021 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  F )  e.  ( ZZ>= `  0 )  ->  ( # `  F
)  e.  ( 0 ... ( # `  F
) ) )
1412, 13sylbi 188 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  e.  ( 0 ... ( # `  F
) ) )
15 eluzfz1 11020 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  F )  e.  ( ZZ>= `  0 )  ->  0  e.  ( 0 ... ( # `  F
) ) )
1612, 15sylbi 188 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  e.  NN0  ->  0  e.  ( 0 ... ( # `
 F ) ) )
1714, 16jca 519 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  F )  e.  NN0  ->  ( ( # `
 F )  e.  ( 0 ... ( # `
 F ) )  /\  0  e.  ( 0 ... ( # `  F ) ) ) )
18 f1veqaeq 5964 . . . . . . . . . . . . . . . . . . 19  |-  ( ( P : ( 0 ... ( # `  F
) ) -1-1-> V  /\  ( ( # `  F
)  e.  ( 0 ... ( # `  F
) )  /\  0  e.  ( 0 ... ( # `
 F ) ) ) )  ->  (
( P `  ( # `
 F ) )  =  ( P ` 
0 )  ->  ( # `
 F )  =  0 ) )
1917, 18sylan2 461 . . . . . . . . . . . . . . . . . 18  |-  ( ( P : ( 0 ... ( # `  F
) ) -1-1-> V  /\  ( # `  F )  e.  NN0 )  -> 
( ( P `  ( # `  F ) )  =  ( P `
 0 )  -> 
( # `  F )  =  0 ) )
2019ex 424 . . . . . . . . . . . . . . . . 17  |-  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  -> 
( ( # `  F
)  e.  NN0  ->  ( ( P `  ( # `
 F ) )  =  ( P ` 
0 )  ->  ( # `
 F )  =  0 ) ) )
2120com13 76 . . . . . . . . . . . . . . . 16  |-  ( ( P `  ( # `  F ) )  =  ( P `  0
)  ->  ( ( # `
 F )  e. 
NN0  ->  ( P :
( 0 ... ( # `
 F ) )
-1-1-> V  ->  ( # `  F
)  =  0 ) ) )
2211, 21syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( P `  ( # `
 F ) )  =  B  /\  ( P `  0 )  =  B )  ->  (
( # `  F )  e.  NN0  ->  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  -> 
( # `  F )  =  0 ) ) )
2322expcom 425 . . . . . . . . . . . . . 14  |-  ( ( P `  0 )  =  B  ->  (
( P `  ( # `
 F ) )  =  B  ->  (
( # `  F )  e.  NN0  ->  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  -> 
( # `  F )  =  0 ) ) ) )
2410, 23syl6bi 220 . . . . . . . . . . . . 13  |-  ( A  =  B  ->  (
( P `  0
)  =  A  -> 
( ( P `  ( # `  F ) )  =  B  -> 
( ( # `  F
)  e.  NN0  ->  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  -> 
( # `  F )  =  0 ) ) ) ) )
2524com15 89 . . . . . . . . . . . 12  |-  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  -> 
( ( P ` 
0 )  =  A  ->  ( ( P `
 ( # `  F
) )  =  B  ->  ( ( # `  F )  e.  NN0  ->  ( A  =  B  ->  ( # `  F
)  =  0 ) ) ) ) )
269, 25sylbir 205 . . . . . . . . . . 11  |-  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  Fun  `' P )  ->  (
( P `  0
)  =  A  -> 
( ( P `  ( # `  F ) )  =  B  -> 
( ( # `  F
)  e.  NN0  ->  ( A  =  B  -> 
( # `  F )  =  0 ) ) ) ) )
2726expcom 425 . . . . . . . . . 10  |-  ( Fun  `' P  ->  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( P ` 
0 )  =  A  ->  ( ( P `
 ( # `  F
) )  =  B  ->  ( ( # `  F )  e.  NN0  ->  ( A  =  B  ->  ( # `  F
)  =  0 ) ) ) ) ) )
2827com15 89 . . . . . . . . 9  |-  ( (
# `  F )  e.  NN0  ->  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( P ` 
0 )  =  A  ->  ( ( P `
 ( # `  F
) )  =  B  ->  ( Fun  `' P  ->  ( A  =  B  ->  ( # `  F
)  =  0 ) ) ) ) ) )
2928imp 419 . . . . . . . 8  |-  ( ( ( # `  F
)  e.  NN0  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( ( P `
 0 )  =  A  ->  ( ( P `  ( # `  F
) )  =  B  ->  ( Fun  `' P  ->  ( A  =  B  ->  ( # `  F
)  =  0 ) ) ) ) )
306, 8, 293syl 19 . . . . . . 7  |-  ( F ( V Walks  E ) P  ->  ( ( P `  0 )  =  A  ->  ( ( P `  ( # `  F ) )  =  B  ->  ( Fun  `' P  ->  ( A  =  B  ->  ( # `  F )  =  0 ) ) ) ) )
31303imp1 1166 . . . . . 6  |-  ( ( ( F ( V Walks 
E ) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  /\  Fun  `' P )  ->  ( A  =  B  ->  (
# `  F )  =  0 ) )
32 fveq2 5687 . . . . . . . . . . . 12  |-  ( (
# `  F )  =  0  ->  ( P `  ( # `  F
) )  =  ( P `  0 ) )
3332eqeq1d 2412 . . . . . . . . . . 11  |-  ( (
# `  F )  =  0  ->  (
( P `  ( # `
 F ) )  =  B  <->  ( P `  0 )  =  B ) )
3433anbi2d 685 . . . . . . . . . 10  |-  ( (
# `  F )  =  0  ->  (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B )  <-> 
( ( P ` 
0 )  =  A  /\  ( P ` 
0 )  =  B ) ) )
35 eqtr2 2422 . . . . . . . . . 10  |-  ( ( ( P `  0
)  =  A  /\  ( P `  0 )  =  B )  ->  A  =  B )
3634, 35syl6bi 220 . . . . . . . . 9  |-  ( (
# `  F )  =  0  ->  (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B )  ->  A  =  B ) )
3736com12 29 . . . . . . . 8  |-  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B )  ->  (
( # `  F )  =  0  ->  A  =  B ) )
38373adant1 975 . . . . . . 7  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  ( ( # `
 F )  =  0  ->  A  =  B ) )
3938adantr 452 . . . . . 6  |-  ( ( ( F ( V Walks 
E ) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  /\  Fun  `' P )  ->  (
( # `  F )  =  0  ->  A  =  B ) )
4031, 39impbid 184 . . . . 5  |-  ( ( ( F ( V Walks 
E ) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  /\  Fun  `' P )  ->  ( A  =  B  <->  ( # `  F
)  =  0 ) )
4140adantrl 697 . . . 4  |-  ( ( ( F ( V Walks 
E ) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  /\  ( F ( V Trails  E ) P  /\  Fun  `' P ) )  -> 
( A  =  B  <-> 
( # `  F )  =  0 ) )
425, 41syl6bi 220 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V
) )  ->  (
( F ( A ( V WalkOn  E ) B ) P  /\  F ( V SPaths  E
) P )  -> 
( A  =  B  <-> 
( # `  F )  =  0 ) ) )
4342imp 419 . 2  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V )
)  /\  ( F
( A ( V WalkOn  E ) B ) P  /\  F ( V SPaths  E ) P ) )  ->  ( A  =  B  <->  ( # `  F
)  =  0 ) )
441, 43syl 16 1  |-  ( F ( A ( V SPathOn  E ) B ) P  ->  ( A  =  B  <->  ( # `  F
)  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916   class class class wbr 4172   `'ccnv 4836   dom cdm 4837   Fun wfun 5407   -->wf 5409   -1-1->wf1 5410   ` cfv 5413  (class class class)co 6040   0cc0 8946   NN0cn0 10177   ZZ>=cuz 10444   ...cfz 10999   #chash 11573  Word cword 11672   Walks cwalk 21459   Trails ctrail 21460   SPaths cspath 21462   WalkOn cwlkon 21463   SPathOn cspthon 21466
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-hash 11574  df-word 11678  df-wlk 21469  df-trail 21470  df-pth 21471  df-spth 21472  df-wlkon 21475  df-spthon 21478
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