MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pthdadjvtx Structured version   Visualization version   Unicode version
Theorem pthdadjvtx 26626
Description: The adjacent vertices of a path of length at least 2 are distinct. (Contributed by AV, 5-Feb-2021.)
Assertion
Ref Expression
pthdadjvtx |- ( ( F (Paths ` G ) P /\ 1 < ( # ` F ) /\ I e. ( 0..^ ( # ` F ) ) ) -> ( P ` I ) =/= ( P ` ( I + 1 ) ) )
Proof of Theorem pthdadjvtx
StepHypRef Expression
1 elfzo0l 12558 . . 3 |- ( I e. ( 0..^ ( # ` F ) ) -> ( I = 0 \/ I e. ( 1..^ ( # ` F ) ) ) )
2 simpr 477 . . . . . . . . 9 |- ( ( 1 < ( # ` F ) /\ F (Paths ` G ) P ) -> F (Paths ` G ) P )
3 pthiswlk 26623 . . . . . . . . . . 11 |- ( F (Paths ` G ) P -> F (Walks ` G ) P )
4 wlkcl 26511 . . . . . . . . . . 11 |- ( F (Walks ` G ) P -> ( # ` F ) e. NN0 )
5 1zzd 11408 . . . . . . . . . . . . . 14 |- ( ( ( # ` F ) e. NN0 /\ 1 < ( # ` F ) ) -> 1 e. ZZ )
6 nn0z 11400 . . . . . . . . . . . . . . 15 |- ( ( # ` F ) e. NN0 -> ( # ` F ) e. ZZ )
76adantr 481 . . . . . . . . . . . . . 14 |- ( ( ( # ` F ) e. NN0 /\ 1 < ( # ` F ) ) -> ( # ` F ) e. ZZ )
8 simpr 477 . . . . . . . . . . . . . 14 |- ( ( ( # ` F ) e. NN0 /\ 1 < ( # ` F ) ) -> 1 < ( # ` F ) )
9 fzolb 12476 . . . . . . . . . . . . . 14 |- ( 1 e. ( 1..^ ( # ` F ) ) <-> ( 1 e. ZZ /\ ( # ` F ) e. ZZ /\ 1 < ( # ` F ) ) )
105, 7, 8, 9syl3anbrc 1246 . . . . . . . . . . . . 13 |- ( ( ( # ` F ) e. NN0 /\ 1 < ( # ` F ) ) -> 1 e. ( 1..^ ( # ` F ) ) )
11 0elfz 12436 . . . . . . . . . . . . . 14 |- ( ( # ` F ) e. NN0 -> 0 e. ( 0 ... ( # ` F ) ) )
1211adantr 481 . . . . . . . . . . . . 13 |- ( ( ( # ` F ) e. NN0 /\ 1 < ( # ` F ) ) -> 0 e. ( 0 ... ( # ` F ) ) )
13 ax-1ne0 10005 . . . . . . . . . . . . . 14 |- 1 =/= 0
1413a1i 11 . . . . . . . . . . . . 13 |- ( ( ( # ` F ) e. NN0 /\ 1 < ( # ` F ) ) -> 1 =/= 0 )
1510, 12, 143jca 1242 . . . . . . . . . . . 12 |- ( ( ( # ` F ) e. NN0 /\ 1 < ( # ` F ) ) -> ( 1 e. ( 1..^ ( # ` F ) ) /\ 0 e. ( 0 ... ( # ` F ) ) /\ 1 =/= 0 ) )
1615ex 450 . . . . . . . . . . 11 |- ( ( # ` F ) e. NN0 -> ( 1 < ( # ` F ) -> ( 1 e. ( 1..^ ( # ` F ) ) /\ 0 e. ( 0 ... ( # ` F ) ) /\ 1 =/= 0 ) ) )
173, 4, 163syl 18 . . . . . . . . . 10 |- ( F (Paths ` G ) P -> ( 1 < ( # ` F ) -> ( 1 e. ( 1..^ ( # ` F ) ) /\ 0 e. ( 0 ... ( # ` F ) ) /\ 1 =/= 0 ) ) )
1817impcom 446 . . . . . . . . 9 |- ( ( 1 < ( # ` F ) /\ F (Paths ` G ) P ) -> ( 1 e. ( 1..^ ( # ` F ) ) /\ 0 e. ( 0 ... ( # ` F ) ) /\ 1 =/= 0 ) )
19 pthdivtx 26625 . . . . . . . . 9 |- ( ( F (Paths ` G ) P /\ ( 1 e. ( 1..^ ( # ` F ) ) /\ 0 e. ( 0 ... ( # ` F ) ) /\ 1 =/= 0 ) ) -> ( P ` 1 ) =/= ( P ` 0 ) )
202, 18, 19syl2anc 693 . . . . . . . 8 |- ( ( 1 < ( # ` F ) /\ F (Paths ` G ) P ) -> ( P ` 1 ) =/= ( P ` 0 ) )
2120necomd 2849 . . . . . . 7 |- ( ( 1 < ( # ` F ) /\ F (Paths ` G ) P ) -> ( P ` 0 ) =/= ( P ` 1 ) )
22213adant1 1079 . . . . . 6 |- ( ( I = 0 /\ 1 < ( # ` F ) /\ F (Paths ` G ) P ) -> ( P ` 0 ) =/= ( P ` 1 ) )
23 fveq2 6191 . . . . . . . 8 |- ( I = 0 -> ( P ` I ) = ( P ` 0 ) )
24 oveq1 6657 . . . . . . . . . 10 |- ( I = 0 -> ( I + 1 ) = ( 0 + 1 ) )
25 0p1e1 11132 . . . . . . . . . 10 |- ( 0 + 1 ) = 1
2624, 25syl6eq 2672 . . . . . . . . 9 |- ( I = 0 -> ( I + 1 ) = 1 )
2726fveq2d 6195 . . . . . . . 8 |- ( I = 0 -> ( P ` ( I + 1 ) ) = ( P ` 1 ) )
2823, 27neeq12d 2855 . . . . . . 7 |- ( I = 0 -> ( ( P ` I ) =/= ( P ` ( I + 1 ) ) <-> ( P ` 0 ) =/= ( P ` 1 ) ) )
29283ad2ant1 1082 . . . . . 6 |- ( ( I = 0 /\ 1 < ( # ` F ) /\ F (Paths ` G ) P ) -> ( ( P ` I ) =/= ( P ` ( I + 1 ) ) <-> ( P ` 0 ) =/= ( P ` 1 ) ) )
3022, 29mpbird 247 . . . . 5 |- ( ( I = 0 /\ 1 < ( # ` F ) /\ F (Paths ` G ) P ) -> ( P ` I ) =/= ( P ` ( I + 1 ) ) )
31303exp 1264 . . . 4 |- ( I = 0 -> ( 1 < ( # ` F ) -> ( F (Paths ` G ) P -> ( P ` I ) =/= ( P ` ( I + 1 ) ) ) ) )
32 simp3 1063 . . . . . 6 |- ( ( I e. ( 1..^ ( # ` F ) ) /\ 1 < ( # ` F ) /\ F (Paths ` G ) P ) -> F (Paths ` G ) P )
33 id 22 . . . . . . . 8 |- ( I e. ( 1..^ ( # ` F ) ) -> I e. ( 1..^ ( # ` F ) ) )
34 fzo0ss1 12498 . . . . . . . . . 10 |- ( 1..^ ( # ` F ) ) C_ ( 0..^ ( # ` F ) )
3534sseli 3599 . . . . . . . . 9 |- ( I e. ( 1..^ ( # ` F ) ) -> I e. ( 0..^ ( # ` F ) ) )
36 fzofzp1 12565 . . . . . . . . 9 |- ( I e. ( 0..^ ( # ` F ) ) -> ( I + 1 ) e. ( 0 ... ( # ` F ) ) )
3735, 36syl 17 . . . . . . . 8 |- ( I e. ( 1..^ ( # ` F ) ) -> ( I + 1 ) e. ( 0 ... ( # ` F ) ) )
38 elfzoelz 12470 . . . . . . . . . . 11 |- ( I e. ( 1..^ ( # ` F ) ) -> I e. ZZ )
3938zcnd 11483 . . . . . . . . . 10 |- ( I e. ( 1..^ ( # ` F ) ) -> I e. CC )
40 1cnd 10056 . . . . . . . . . 10 |- ( I e. ( 1..^ ( # ` F ) ) -> 1 e. CC )
4113a1i 11 . . . . . . . . . 10 |- ( I e. ( 1..^ ( # ` F ) ) -> 1 =/= 0 )
4239, 40, 413jca 1242 . . . . . . . . 9 |- ( I e. ( 1..^ ( # ` F ) ) -> ( I e. CC /\ 1 e. CC /\ 1 =/= 0 ) )
43 addn0nid 10451 . . . . . . . . . 10 |- ( ( I e. CC /\ 1 e. CC /\ 1 =/= 0 ) -> ( I + 1 ) =/= I )
4443necomd 2849 . . . . . . . . 9 |- ( ( I e. CC /\ 1 e. CC /\ 1 =/= 0 ) -> I =/= ( I + 1 ) )
4542, 44syl 17 . . . . . . . 8 |- ( I e. ( 1..^ ( # ` F ) ) -> I =/= ( I + 1 ) )
4633, 37, 453jca 1242 . . . . . . 7 |- ( I e. ( 1..^ ( # ` F ) ) -> ( I e. ( 1..^ ( # ` F ) ) /\ ( I + 1 ) e. ( 0 ... ( # ` F ) ) /\ I =/= ( I + 1 ) ) )
47463ad2ant1 1082 . . . . . 6 |- ( ( I e. ( 1..^ ( # ` F ) ) /\ 1 < ( # ` F ) /\ F (Paths ` G ) P ) -> ( I e. ( 1..^ ( # ` F ) ) /\ ( I + 1 ) e. ( 0 ... ( # ` F ) ) /\ I =/= ( I + 1 ) ) )
48 pthdivtx 26625 . . . . . 6 |- ( ( F (Paths ` G ) P /\ ( I e. ( 1..^ ( # ` F ) ) /\ ( I + 1 ) e. ( 0 ... ( # ` F ) ) /\ I =/= ( I + 1 ) ) ) -> ( P ` I ) =/= ( P ` ( I + 1 ) ) )
4932, 47, 48syl2anc 693 . . . . 5 |- ( ( I e. ( 1..^ ( # ` F ) ) /\ 1 < ( # ` F ) /\ F (Paths ` G ) P ) -> ( P ` I ) =/= ( P ` ( I + 1 ) ) )
50493exp 1264 . . . 4 |- ( I e. ( 1..^ ( # ` F ) ) -> ( 1 < ( # ` F ) -> ( F (Paths ` G ) P -> ( P ` I ) =/= ( P ` ( I + 1 ) ) ) ) )
5131, 50jaoi 394 . . 3 |- ( ( I = 0 \/ I e. ( 1..^ ( # ` F ) ) ) -> ( 1 < ( # ` F ) -> ( F (Paths ` G ) P -> ( P ` I ) =/= ( P ` ( I + 1 ) ) ) ) )
521, 51syl 17 . 2 |- ( I e. ( 0..^ ( # ` F ) ) -> ( 1 < ( # ` F ) -> ( F (Paths ` G ) P -> ( P ` I ) =/= ( P ` ( I + 1 ) ) ) ) )
53523imp31 1257 1 |- ( ( F (Paths ` G ) P /\ 1 < ( # ` F ) /\ I e. ( 0..^ ( # ` F ) ) ) -> ( P ` I ) =/= ( P ` ( I + 1 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   NN0cn0 11292   ZZcz 11377   ...cfz 12326  ..^cfzo 12465   #chash 13117  Walkscwlks 26492  Pathscpths 26608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-wlks 26495  df-trls 26589  df-pths 26612
This theorem is referenced by:  2pthnloop  26627  upgr3v3e3cycl  27040  upgr4cycl4dv4e  27045
  Copyright terms: Public domain W3C validator