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Theorem spthonepeq 25303
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 25301 . 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 25248 . . . . 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 25285 . . . . . 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 1025 . . . . 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 715 . . . 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 25235 . . . . . . . 8  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
7 lencl 12680 . . . . . . . . 9  |-  ( F  e. Word  dom  E  ->  (
# `  F )  e.  NN0 )
87anim1i 570 . . . . . . . 8  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( ( # `  F )  e.  NN0  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
9 df-f1 5603 . . . . . . . . . . . 12  |-  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  <->  ( P : ( 0 ... ( # `  F
) ) --> V  /\  Fun  `' P ) )
10 eqeq2 2437 . . . . . . . . . . . . . 14  |-  ( A  =  B  ->  (
( P `  0
)  =  A  <->  ( P `  0 )  =  B ) )
11 eqtr3 2450 . . . . . . . . . . . . . . . 16  |-  ( ( ( P `  ( # `
 F ) )  =  B  /\  ( P `  0 )  =  B )  ->  ( P `  ( # `  F
) )  =  ( P `  0 ) )
12 elnn0uz 11197 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  F )  e.  NN0  <->  ( # `  F
)  e.  ( ZZ>= ` 
0 ) )
13 eluzfz2 11808 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  F )  e.  ( ZZ>= `  0 )  ->  ( # `  F
)  e.  ( 0 ... ( # `  F
) ) )
1412, 13sylbi 198 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  e.  ( 0 ... ( # `  F
) ) )
15 0elfz 11890 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  e.  NN0  ->  0  e.  ( 0 ... ( # `
 F ) ) )
1614, 15jca 534 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  F )  e.  NN0  ->  ( ( # `
 F )  e.  ( 0 ... ( # `
 F ) )  /\  0  e.  ( 0 ... ( # `  F ) ) ) )
17 f1veqaeq 6173 . . . . . . . . . . . . . . . . . . 19  |-  ( ( P : ( 0 ... ( # `  F
) ) -1-1-> V  /\  ( ( # `  F
)  e.  ( 0 ... ( # `  F
) )  /\  0  e.  ( 0 ... ( # `
 F ) ) ) )  ->  (
( P `  ( # `
 F ) )  =  ( P ` 
0 )  ->  ( # `
 F )  =  0 ) )
1816, 17sylan2 476 . . . . . . . . . . . . . . . . . 18  |-  ( ( P : ( 0 ... ( # `  F
) ) -1-1-> V  /\  ( # `  F )  e.  NN0 )  -> 
( ( P `  ( # `  F ) )  =  ( P `
 0 )  -> 
( # `  F )  =  0 ) )
1918ex 435 . . . . . . . . . . . . . . . . 17  |-  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  -> 
( ( # `  F
)  e.  NN0  ->  ( ( P `  ( # `
 F ) )  =  ( P ` 
0 )  ->  ( # `
 F )  =  0 ) ) )
2019com13 83 . . . . . . . . . . . . . . . 16  |-  ( ( P `  ( # `  F ) )  =  ( P `  0
)  ->  ( ( # `
 F )  e. 
NN0  ->  ( P :
( 0 ... ( # `
 F ) )
-1-1-> V  ->  ( # `  F
)  =  0 ) ) )
2111, 20syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( P `  ( # `
 F ) )  =  B  /\  ( P `  0 )  =  B )  ->  (
( # `  F )  e.  NN0  ->  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  -> 
( # `  F )  =  0 ) ) )
2221expcom 436 . . . . . . . . . . . . . 14  |-  ( ( P `  0 )  =  B  ->  (
( P `  ( # `
 F ) )  =  B  ->  (
( # `  F )  e.  NN0  ->  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  -> 
( # `  F )  =  0 ) ) ) )
2310, 22syl6bi 231 . . . . . . . . . . . . 13  |-  ( A  =  B  ->  (
( P `  0
)  =  A  -> 
( ( P `  ( # `  F ) )  =  B  -> 
( ( # `  F
)  e.  NN0  ->  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  -> 
( # `  F )  =  0 ) ) ) ) )
2423com15 96 . . . . . . . . . . . 12  |-  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  -> 
( ( P ` 
0 )  =  A  ->  ( ( P `
 ( # `  F
) )  =  B  ->  ( ( # `  F )  e.  NN0  ->  ( A  =  B  ->  ( # `  F
)  =  0 ) ) ) ) )
259, 24sylbir 216 . . . . . . . . . . 11  |-  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  Fun  `' P )  ->  (
( P `  0
)  =  A  -> 
( ( P `  ( # `  F ) )  =  B  -> 
( ( # `  F
)  e.  NN0  ->  ( A  =  B  -> 
( # `  F )  =  0 ) ) ) ) )
2625expcom 436 . . . . . . . . . 10  |-  ( Fun  `' P  ->  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( P ` 
0 )  =  A  ->  ( ( P `
 ( # `  F
) )  =  B  ->  ( ( # `  F )  e.  NN0  ->  ( A  =  B  ->  ( # `  F
)  =  0 ) ) ) ) ) )
2726com15 96 . . . . . . . . 9  |-  ( (
# `  F )  e.  NN0  ->  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( P ` 
0 )  =  A  ->  ( ( P `
 ( # `  F
) )  =  B  ->  ( Fun  `' P  ->  ( A  =  B  ->  ( # `  F
)  =  0 ) ) ) ) ) )
2827imp 430 . . . . . . . 8  |-  ( ( ( # `  F
)  e.  NN0  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( ( P `
 0 )  =  A  ->  ( ( P `  ( # `  F
) )  =  B  ->  ( Fun  `' P  ->  ( A  =  B  ->  ( # `  F
)  =  0 ) ) ) ) )
296, 8, 283syl 18 . . . . . . 7  |-  ( F ( V Walks  E ) P  ->  ( ( P `  0 )  =  A  ->  ( ( P `  ( # `  F ) )  =  B  ->  ( Fun  `' P  ->  ( A  =  B  ->  ( # `  F )  =  0 ) ) ) ) )
30293imp1 1218 . . . . . 6  |-  ( ( ( F ( V Walks 
E ) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  /\  Fun  `' P )  ->  ( A  =  B  ->  (
# `  F )  =  0 ) )
31 fveq2 5878 . . . . . . . . . . . 12  |-  ( (
# `  F )  =  0  ->  ( P `  ( # `  F
) )  =  ( P `  0 ) )
3231eqeq1d 2424 . . . . . . . . . . 11  |-  ( (
# `  F )  =  0  ->  (
( P `  ( # `
 F ) )  =  B  <->  ( P `  0 )  =  B ) )
3332anbi2d 708 . . . . . . . . . 10  |-  ( (
# `  F )  =  0  ->  (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B )  <-> 
( ( P ` 
0 )  =  A  /\  ( P ` 
0 )  =  B ) ) )
34 eqtr2 2449 . . . . . . . . . 10  |-  ( ( ( P `  0
)  =  A  /\  ( P `  0 )  =  B )  ->  A  =  B )
3533, 34syl6bi 231 . . . . . . . . 9  |-  ( (
# `  F )  =  0  ->  (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B )  ->  A  =  B ) )
3635com12 32 . . . . . . . 8  |-  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B )  ->  (
( # `  F )  =  0  ->  A  =  B ) )
37363adant1 1023 . . . . . . 7  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  ( ( # `
 F )  =  0  ->  A  =  B ) )
3837adantr 466 . . . . . 6  |-  ( ( ( F ( V Walks 
E ) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  /\  Fun  `' P )  ->  (
( # `  F )  =  0  ->  A  =  B ) )
3930, 38impbid 193 . . . . 5  |-  ( ( ( F ( V Walks 
E ) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  /\  Fun  `' P )  ->  ( A  =  B  <->  ( # `  F
)  =  0 ) )
4039adantrl 720 . . . 4  |-  ( ( ( F ( V Walks 
E ) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  /\  ( F ( V Trails  E ) P  /\  Fun  `' P ) )  -> 
( A  =  B  <-> 
( # `  F )  =  0 ) )
415, 40syl6bi 231 . . 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 ) ) )
4241imp 430 . 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 ) )
431, 42syl 17 1  |-  ( F ( A ( V SPathOn  E ) B ) P  ->  ( A  =  B  <->  ( # `  F
)  =  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   _Vcvv 3081   class class class wbr 4420   `'ccnv 4849   dom cdm 4850   Fun wfun 5592   -->wf 5594   -1-1->wf1 5595   ` cfv 5598  (class class class)co 6302   0cc0 9540   NN0cn0 10870   ZZ>=cuz 11160   ...cfz 11785   #chash 12515  Word cword 12649   Walks cwalk 25212   Trails ctrail 25213   SPaths cspath 25215   WalkOn cwlkon 25216   SPathOn cspthon 25219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-pm 7480  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8375  df-cda 8599  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-nn 10611  df-2 10669  df-n0 10871  df-z 10939  df-uz 11161  df-fz 11786  df-fzo 11917  df-hash 12516  df-word 12657  df-wlk 25222  df-trail 25223  df-pth 25224  df-spth 25225  df-wlkon 25228  df-spthon 25231
This theorem is referenced by: (None)
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