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Theorem spthonepeq 24794
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 24792 . 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 24739 . . . . 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 24776 . . . . . 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 1014 . . . . 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 708 . . . 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 24726 . . . . . . . 8  |-  ( F ( V Walks  E ) P  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
7 lencl 12552 . . . . . . . . 9  |-  ( F  e. Word  dom  E  ->  (
# `  F )  e.  NN0 )
87anim1i 566 . . . . . . . 8  |-  ( ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( ( # `  F )  e.  NN0  /\  P : ( 0 ... ( # `  F
) ) --> V ) )
9 df-f1 5575 . . . . . . . . . . . 12  |-  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  <->  ( P : ( 0 ... ( # `  F
) ) --> V  /\  Fun  `' P ) )
10 eqeq2 2469 . . . . . . . . . . . . . 14  |-  ( A  =  B  ->  (
( P `  0
)  =  A  <->  ( P `  0 )  =  B ) )
11 eqtr3 2482 . . . . . . . . . . . . . . . 16  |-  ( ( ( P `  ( # `
 F ) )  =  B  /\  ( P `  0 )  =  B )  ->  ( P `  ( # `  F
) )  =  ( P `  0 ) )
12 elnn0uz 11119 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  F )  e.  NN0  <->  ( # `  F
)  e.  ( ZZ>= ` 
0 ) )
13 eluzfz2 11697 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
# `  F )  e.  ( ZZ>= `  0 )  ->  ( # `  F
)  e.  ( 0 ... ( # `  F
) ) )
1412, 13sylbi 195 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  e.  ( 0 ... ( # `  F
) ) )
15 0elfz 11777 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
# `  F )  e.  NN0  ->  0  e.  ( 0 ... ( # `
 F ) ) )
1614, 15jca 530 . . . . . . . . . . . . . . . . . . 19  |-  ( (
# `  F )  e.  NN0  ->  ( ( # `
 F )  e.  ( 0 ... ( # `
 F ) )  /\  0  e.  ( 0 ... ( # `  F ) ) ) )
17 f1veqaeq 6143 . . . . . . . . . . . . . . . . . . 19  |-  ( ( P : ( 0 ... ( # `  F
) ) -1-1-> V  /\  ( ( # `  F
)  e.  ( 0 ... ( # `  F
) )  /\  0  e.  ( 0 ... ( # `
 F ) ) ) )  ->  (
( P `  ( # `
 F ) )  =  ( P ` 
0 )  ->  ( # `
 F )  =  0 ) )
1816, 17sylan2 472 . . . . . . . . . . . . . . . . . 18  |-  ( ( P : ( 0 ... ( # `  F
) ) -1-1-> V  /\  ( # `  F )  e.  NN0 )  -> 
( ( P `  ( # `  F ) )  =  ( P `
 0 )  -> 
( # `  F )  =  0 ) )
1918ex 432 . . . . . . . . . . . . . . . . 17  |-  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  -> 
( ( # `  F
)  e.  NN0  ->  ( ( P `  ( # `
 F ) )  =  ( P ` 
0 )  ->  ( # `
 F )  =  0 ) ) )
2019com13 80 . . . . . . . . . . . . . . . 16  |-  ( ( P `  ( # `  F ) )  =  ( P `  0
)  ->  ( ( # `
 F )  e. 
NN0  ->  ( P :
( 0 ... ( # `
 F ) )
-1-1-> V  ->  ( # `  F
)  =  0 ) ) )
2111, 20syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( P `  ( # `
 F ) )  =  B  /\  ( P `  0 )  =  B )  ->  (
( # `  F )  e.  NN0  ->  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  -> 
( # `  F )  =  0 ) ) )
2221expcom 433 . . . . . . . . . . . . . 14  |-  ( ( P `  0 )  =  B  ->  (
( P `  ( # `
 F ) )  =  B  ->  (
( # `  F )  e.  NN0  ->  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  -> 
( # `  F )  =  0 ) ) ) )
2310, 22syl6bi 228 . . . . . . . . . . . . 13  |-  ( A  =  B  ->  (
( P `  0
)  =  A  -> 
( ( P `  ( # `  F ) )  =  B  -> 
( ( # `  F
)  e.  NN0  ->  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  -> 
( # `  F )  =  0 ) ) ) ) )
2423com15 93 . . . . . . . . . . . 12  |-  ( P : ( 0 ... ( # `  F
) ) -1-1-> V  -> 
( ( P ` 
0 )  =  A  ->  ( ( P `
 ( # `  F
) )  =  B  ->  ( ( # `  F )  e.  NN0  ->  ( A  =  B  ->  ( # `  F
)  =  0 ) ) ) ) )
259, 24sylbir 213 . . . . . . . . . . 11  |-  ( ( P : ( 0 ... ( # `  F
) ) --> V  /\  Fun  `' P )  ->  (
( P `  0
)  =  A  -> 
( ( P `  ( # `  F ) )  =  B  -> 
( ( # `  F
)  e.  NN0  ->  ( A  =  B  -> 
( # `  F )  =  0 ) ) ) ) )
2625expcom 433 . . . . . . . . . 10  |-  ( Fun  `' P  ->  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( P ` 
0 )  =  A  ->  ( ( P `
 ( # `  F
) )  =  B  ->  ( ( # `  F )  e.  NN0  ->  ( A  =  B  ->  ( # `  F
)  =  0 ) ) ) ) ) )
2726com15 93 . . . . . . . . 9  |-  ( (
# `  F )  e.  NN0  ->  ( P : ( 0 ... ( # `  F
) ) --> V  -> 
( ( P ` 
0 )  =  A  ->  ( ( P `
 ( # `  F
) )  =  B  ->  ( Fun  `' P  ->  ( A  =  B  ->  ( # `  F
)  =  0 ) ) ) ) ) )
2827imp 427 . . . . . . . 8  |-  ( ( ( # `  F
)  e.  NN0  /\  P : ( 0 ... ( # `  F
) ) --> V )  ->  ( ( P `
 0 )  =  A  ->  ( ( P `  ( # `  F
) )  =  B  ->  ( Fun  `' P  ->  ( A  =  B  ->  ( # `  F
)  =  0 ) ) ) ) )
296, 8, 283syl 20 . . . . . . 7  |-  ( F ( V Walks  E ) P  ->  ( ( P `  0 )  =  A  ->  ( ( P `  ( # `  F ) )  =  B  ->  ( Fun  `' P  ->  ( A  =  B  ->  ( # `  F )  =  0 ) ) ) ) )
30293imp1 1207 . . . . . 6  |-  ( ( ( F ( V Walks 
E ) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  /\  Fun  `' P )  ->  ( A  =  B  ->  (
# `  F )  =  0 ) )
31 fveq2 5848 . . . . . . . . . . . 12  |-  ( (
# `  F )  =  0  ->  ( P `  ( # `  F
) )  =  ( P `  0 ) )
3231eqeq1d 2456 . . . . . . . . . . 11  |-  ( (
# `  F )  =  0  ->  (
( P `  ( # `
 F ) )  =  B  <->  ( P `  0 )  =  B ) )
3332anbi2d 701 . . . . . . . . . 10  |-  ( (
# `  F )  =  0  ->  (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B )  <-> 
( ( P ` 
0 )  =  A  /\  ( P ` 
0 )  =  B ) ) )
34 eqtr2 2481 . . . . . . . . . 10  |-  ( ( ( P `  0
)  =  A  /\  ( P `  0 )  =  B )  ->  A  =  B )
3533, 34syl6bi 228 . . . . . . . . 9  |-  ( (
# `  F )  =  0  ->  (
( ( P ` 
0 )  =  A  /\  ( P `  ( # `  F ) )  =  B )  ->  A  =  B ) )
3635com12 31 . . . . . . . 8  |-  ( ( ( P `  0
)  =  A  /\  ( P `  ( # `  F ) )  =  B )  ->  (
( # `  F )  =  0  ->  A  =  B ) )
37363adant1 1012 . . . . . . 7  |-  ( ( F ( V Walks  E
) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  ->  ( ( # `
 F )  =  0  ->  A  =  B ) )
3837adantr 463 . . . . . 6  |-  ( ( ( F ( V Walks 
E ) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  /\  Fun  `' P )  ->  (
( # `  F )  =  0  ->  A  =  B ) )
3930, 38impbid 191 . . . . 5  |-  ( ( ( F ( V Walks 
E ) P  /\  ( P `  0 )  =  A  /\  ( P `  ( # `  F
) )  =  B )  /\  Fun  `' P )  ->  ( A  =  B  <->  ( # `  F
)  =  0 ) )
4039adantrl 713 . . . 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 228 . . 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 427 . 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 16 1  |-  ( F ( A ( V SPathOn  E ) B ) P  ->  ( A  =  B  <->  ( # `  F
)  =  0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106   class class class wbr 4439   `'ccnv 4987   dom cdm 4988   Fun wfun 5564   -->wf 5566   -1-1->wf1 5567   ` cfv 5570  (class class class)co 6270   0cc0 9481   NN0cn0 10791   ZZ>=cuz 11082   ...cfz 11675   #chash 12390  Word cword 12521   Walks cwalk 24703   Trails ctrail 24704   SPaths cspath 24706   WalkOn cwlkon 24707   SPathOn cspthon 24710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12391  df-word 12529  df-wlk 24713  df-trail 24714  df-pth 24715  df-spth 24716  df-wlkon 24719  df-spthon 24722
This theorem is referenced by: (None)
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