Theorem spthonepeq 26648

Description: The endpoints of a simple path between two vertices are equal iff the path is of length 0. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 18-Jan-2021.) (Proof shortened by AV, 31-Oct-2021.)

Assertion

Ref	Expression
spthonepeq	|- ( F ( A (SPathsOn ` G ) B ) P -> ( A = B <-> ( # ` F ) = 0 ) )

Proof of Theorem spthonepeq

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- (Vtx ` G ) = (Vtx ` G )
2	1	spthonprop 26641	. 2 |- ( F ( A (SPathsOn ` G ) B ) P -> ( ( G e. _V /\ A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F ( A (TrailsOn ` G ) B ) P /\ F (SPaths ` G ) P ) ) )
3	1	istrlson 26603	. . . . . 6 |- ( ( ( A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( A (TrailsOn ` G ) B ) P <-> ( F ( A (WalksOn ` G ) B ) P /\ F (Trails ` G ) P ) ) )
4	3	3adantl1 1217	. . . . 5 |- ( ( ( G e. _V /\ A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( A (TrailsOn ` G ) B ) P <-> ( F ( A (WalksOn ` G ) B ) P /\ F (Trails ` G ) P ) ) )
5		isspth 26620	. . . . . 6 |- ( F (SPaths ` G ) P <-> ( F (Trails ` G ) P /\ Fun `' P ) )
6	5	a1i 11	. . . . 5 |- ( ( ( G e. _V /\ A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) ) -> ( F (SPaths ` G ) P <-> ( F (Trails ` G ) P /\ Fun `' P ) ) )
7	4, 6	anbi12d 747	. . . 4 |- ( ( ( G e. _V /\ A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) ) -> ( ( F ( A (TrailsOn ` G ) B ) P /\ F (SPaths ` G ) P ) <-> ( ( F ( A (WalksOn ` G ) B ) P /\ F (Trails ` G ) P ) /\ ( F (Trails ` G ) P /\ Fun `' P ) ) ) )
8	1	wlkonprop 26554	. . . . . . . 8 |- ( F ( A (WalksOn ` G ) B ) P -> ( ( G e. _V /\ A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) )
9		wlkcl 26511	. . . . . . . . . . . . 13 |- ( F (Walks ` G ) P -> ( # ` F ) e. NN0 )
10	1	wlkp 26512	. . . . . . . . . . . . 13 |- ( F (Walks ` G ) P -> P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) )
11		df-f1 5893	. . . . . . . . . . . . . . . 16 |- ( P : ( 0 ... ( # ` F ) ) -1-1-> (Vtx ` G ) <-> ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ Fun `' P ) )
12		eqeq2 2633	. . . . . . . . . . . . . . . . . 18 |- ( A = B -> ( ( P ` 0 ) = A <-> ( P ` 0 ) = B ) )
13		eqtr3 2643	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( P ` ( # ` F ) ) = B /\ ( P ` 0 ) = B ) -> ( P ` ( # ` F ) ) = ( P ` 0 ) )
14		elnn0uz 11725	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( # ` F ) e. NN0 <-> ( # ` F ) e. ( ZZ>= ` 0 ) )
15		eluzfz2 12349	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( # ` F ) e. ( ZZ>= ` 0 ) -> ( # ` F ) e. ( 0 ... ( # ` F ) ) )
16	14, 15	sylbi 207	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( # ` F ) e. NN0 -> ( # ` F ) e. ( 0 ... ( # ` F ) ) )
17		0elfz 12436	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( # ` F ) e. NN0 -> 0 e. ( 0 ... ( # ` F ) ) )
18	16, 17	jca 554	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( # ` F ) e. NN0 -> ( ( # ` F ) e. ( 0 ... ( # ` F ) ) /\ 0 e. ( 0 ... ( # ` F ) ) ) )
19		f1veqaeq 6514	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( P : ( 0 ... ( # ` F ) ) -1-1-> (Vtx ` G ) /\ ( ( # ` F ) e. ( 0 ... ( # ` F ) ) /\ 0 e. ( 0 ... ( # ` F ) ) ) ) -> ( ( P ` ( # ` F ) ) = ( P ` 0 ) -> ( # ` F ) = 0 ) )
20	18, 19	sylan2 491	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( P : ( 0 ... ( # ` F ) ) -1-1-> (Vtx ` G ) /\ ( # ` F ) e. NN0 ) -> ( ( P ` ( # ` F ) ) = ( P ` 0 ) -> ( # ` F ) = 0 ) )
21	20	ex 450	. . . . . . . . . . . . . . . . . . . . 21 |- ( P : ( 0 ... ( # ` F ) ) -1-1-> (Vtx ` G ) -> ( ( # ` F ) e. NN0 -> ( ( P ` ( # ` F ) ) = ( P ` 0 ) -> ( # ` F ) = 0 ) ) )
22	21	com13 88	. . . . . . . . . . . . . . . . . . . 20 |- ( ( P ` ( # ` F ) ) = ( P ` 0 ) -> ( ( # ` F ) e. NN0 -> ( P : ( 0 ... ( # ` F ) ) -1-1-> (Vtx ` G ) -> ( # ` F ) = 0 ) ) )
23	13, 22	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( P ` ( # ` F ) ) = B /\ ( P ` 0 ) = B ) -> ( ( # ` F ) e. NN0 -> ( P : ( 0 ... ( # ` F ) ) -1-1-> (Vtx ` G ) -> ( # ` F ) = 0 ) ) )
24	23	expcom 451	. . . . . . . . . . . . . . . . . 18 |- ( ( P ` 0 ) = B -> ( ( P ` ( # ` F ) ) = B -> ( ( # ` F ) e. NN0 -> ( P : ( 0 ... ( # ` F ) ) -1-1-> (Vtx ` G ) -> ( # ` F ) = 0 ) ) ) )
25	12, 24	syl6bi 243	. . . . . . . . . . . . . . . . 17 |- ( A = B -> ( ( P ` 0 ) = A -> ( ( P ` ( # ` F ) ) = B -> ( ( # ` F ) e. NN0 -> ( P : ( 0 ... ( # ` F ) ) -1-1-> (Vtx ` G ) -> ( # ` F ) = 0 ) ) ) ) )
26	25	com15 101	. . . . . . . . . . . . . . . 16 |- ( P : ( 0 ... ( # ` F ) ) -1-1-> (Vtx ` G ) -> ( ( P ` 0 ) = A -> ( ( P ` ( # ` F ) ) = B -> ( ( # ` F ) e. NN0 -> ( A = B -> ( # ` F ) = 0 ) ) ) ) )
27	11, 26	sylbir 225	. . . . . . . . . . . . . . 15 |- ( ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) /\ Fun `' P ) -> ( ( P ` 0 ) = A -> ( ( P ` ( # ` F ) ) = B -> ( ( # ` F ) e. NN0 -> ( A = B -> ( # ` F ) = 0 ) ) ) ) )
28	27	expcom 451	. . . . . . . . . . . . . 14 |- ( Fun `' P -> ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> ( ( P ` 0 ) = A -> ( ( P ` ( # ` F ) ) = B -> ( ( # ` F ) e. NN0 -> ( A = B -> ( # ` F ) = 0 ) ) ) ) ) )
29	28	com15 101	. . . . . . . . . . . . 13 |- ( ( # ` F ) e. NN0 -> ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` G ) -> ( ( P ` 0 ) = A -> ( ( P ` ( # ` F ) ) = B -> ( Fun `' P -> ( A = B -> ( # ` F ) = 0 ) ) ) ) ) )
30	9, 10, 29	sylc 65	. . . . . . . . . . . 12 |- ( F (Walks ` G ) P -> ( ( P ` 0 ) = A -> ( ( P ` ( # ` F ) ) = B -> ( Fun `' P -> ( A = B -> ( # ` F ) = 0 ) ) ) ) )
31	30	3imp1 1280	. . . . . . . . . . 11 |- ( ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) /\ Fun `' P ) -> ( A = B -> ( # ` F ) = 0 ) )
32		fveq2 6191	. . . . . . . . . . . . . . . . 17 |- ( ( # ` F ) = 0 -> ( P ` ( # ` F ) ) = ( P ` 0 ) )
33	32	eqeq1d 2624	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) = 0 -> ( ( P ` ( # ` F ) ) = B <-> ( P ` 0 ) = B ) )
34	33	anbi2d 740	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) = 0 -> ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) <-> ( ( P ` 0 ) = A /\ ( P ` 0 ) = B ) ) )
35		eqtr2 2642	. . . . . . . . . . . . . . 15 |- ( ( ( P ` 0 ) = A /\ ( P ` 0 ) = B ) -> A = B )
36	34, 35	syl6bi 243	. . . . . . . . . . . . . 14 |- ( ( # ` F ) = 0 -> ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> A = B ) )
37	36	com12 32	. . . . . . . . . . . . 13 |- ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( ( # ` F ) = 0 -> A = B ) )
38	37	3adant1 1079	. . . . . . . . . . . 12 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( ( # ` F ) = 0 -> A = B ) )
39	38	adantr 481	. . . . . . . . . . 11 |- ( ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) /\ Fun `' P ) -> ( ( # ` F ) = 0 -> A = B ) )
40	31, 39	impbid 202	. . . . . . . . . 10 |- ( ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) /\ Fun `' P ) -> ( A = B <-> ( # ` F ) = 0 ) )
41	40	ex 450	. . . . . . . . 9 |- ( ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) -> ( Fun `' P -> ( A = B <-> ( # ` F ) = 0 ) ) )
42	41	3ad2ant3 1084	. . . . . . . 8 |- ( ( ( G e. _V /\ A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F (Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) -> ( Fun `' P -> ( A = B <-> ( # ` F ) = 0 ) ) )
43	8, 42	syl 17	. . . . . . 7 |- ( F ( A (WalksOn ` G ) B ) P -> ( Fun `' P -> ( A = B <-> ( # ` F ) = 0 ) ) )
44	43	adantld 483	. . . . . 6 |- ( F ( A (WalksOn ` G ) B ) P -> ( ( F (Trails ` G ) P /\ Fun `' P ) -> ( A = B <-> ( # ` F ) = 0 ) ) )
45	44	adantr 481	. . . . 5 |- ( ( F ( A (WalksOn ` G ) B ) P /\ F (Trails ` G ) P ) -> ( ( F (Trails ` G ) P /\ Fun `' P ) -> ( A = B <-> ( # ` F ) = 0 ) ) )
46	45	imp 445	. . . 4 |- ( ( ( F ( A (WalksOn ` G ) B ) P /\ F (Trails ` G ) P ) /\ ( F (Trails ` G ) P /\ Fun `' P ) ) -> ( A = B <-> ( # ` F ) = 0 ) )
47	7, 46	syl6bi 243	. . 3 |- ( ( ( G e. _V /\ A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) ) -> ( ( F ( A (TrailsOn ` G ) B ) P /\ F (SPaths ` G ) P ) -> ( A = B <-> ( # ` F ) = 0 ) ) )
48	47	3impia 1261	. 2 |- ( ( ( G e. _V /\ A e. (Vtx ` G ) /\ B e. (Vtx ` G ) ) /\ ( F e. _V /\ P e. _V ) /\ ( F ( A (TrailsOn ` G ) B ) P /\ F (SPaths ` G ) P ) ) -> ( A = B <-> ( # ` F ) = 0 ) )
49	2, 48	syl 17	1 |- ( F ( A (SPathsOn ` G ) B ) P -> ( A = B <-> ( # ` F ) = 0 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   `'ccnv 5113   Fun wfun 5882   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   0cc0 9936   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326   #chash 13117  Vtxcvtx 25874  Walkscwlks 26492  WalksOncwlkson 26493  Trailsctrls 26587  TrailsOnctrlson 26588  SPathscspths 26609  SPathsOncspthson 26611

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-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-spths 26613  df-spthson 26615

This theorem is referenced by:  wspthsnonn0vne  26813