Theorem umgr2cwwk2dif 16579

Description: If a word represents a closed walk of length at least 2 in a multigraph, the first two symbols of the word must be different. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 30-Apr-2021.)

Assertion

Ref	Expression
umgr2cwwk2dif	|- ( ( G e. UMGraph /\ N e. ( ZZ>= ` 2 ) /\ W e. ( N ClWWalksN G ) ) -> ( W ` 1 ) =/= ( W ` 0 ) )

Proof of Theorem umgr2cwwk2dif

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2238	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2238	. . . 4 |- (Edg ` G ) = (Edg ` G )
3	1, 2	clwwlknp 16572	. . 3 |- ( W e. ( N ClWWalksN G ) -> ( ( W e. Word (Vtx ` G ) /\ (♯ ` W ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) )
4		simpr 110	. . . . 5 |- ( ( ( ( ( W e. Word (Vtx ` G ) /\ (♯ ` W ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ N e. ( ZZ>= ` 2 ) ) /\ G e. UMGraph ) -> G e. UMGraph )
5		uz2m1nn 9984	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 2 ) -> ( N - 1 ) e. NN )
6		lbfzo0 10570	. . . . . . . . . . 11 |- ( 0 e. ( 0..^ ( N - 1 ) ) <-> ( N - 1 ) e. NN )
7	5, 6	sylibr 134	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` 2 ) -> 0 e. ( 0..^ ( N - 1 ) ) )
8		fveq2 5690	. . . . . . . . . . . . 13 |- ( i = 0 -> ( W ` i ) = ( W ` 0 ) )
9	8	adantl 277	. . . . . . . . . . . 12 |- ( ( N e. ( ZZ>= ` 2 ) /\ i = 0 ) -> ( W ` i ) = ( W ` 0 ) )
10		oveq1 6082	. . . . . . . . . . . . . . 15 |- ( i = 0 -> ( i + 1 ) = ( 0 + 1 ) )
11	10	adantl 277	. . . . . . . . . . . . . 14 |- ( ( N e. ( ZZ>= ` 2 ) /\ i = 0 ) -> ( i + 1 ) = ( 0 + 1 ) )
12		0p1e1 9397	. . . . . . . . . . . . . 14 |- ( 0 + 1 ) = 1
13	11, 12	eqtrdi 2287	. . . . . . . . . . . . 13 |- ( ( N e. ( ZZ>= ` 2 ) /\ i = 0 ) -> ( i + 1 ) = 1 )
14	13	fveq2d 5694	. . . . . . . . . . . 12 |- ( ( N e. ( ZZ>= ` 2 ) /\ i = 0 ) -> ( W ` ( i + 1 ) ) = ( W ` 1 ) )
15	9, 14	preq12d 3792	. . . . . . . . . . 11 |- ( ( N e. ( ZZ>= ` 2 ) /\ i = 0 ) -> { ( W ` i ) , ( W ` ( i + 1 ) ) } = { ( W ` 0 ) , ( W ` 1 ) } )
16	15	eleq1d 2307	. . . . . . . . . 10 |- ( ( N e. ( ZZ>= ` 2 ) /\ i = 0 ) -> ( { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) <-> { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )
17	7, 16	rspcdv 2932	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> ( A. i e. ( 0..^ ( N - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) -> { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )
18	17	com12 30	. . . . . . . 8 |- ( A. i e. ( 0..^ ( N - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) -> ( N e. ( ZZ>= ` 2 ) -> { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )
19	18	3ad2ant2 1050	. . . . . . 7 |- ( ( ( W e. Word (Vtx ` G ) /\ (♯ ` W ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) -> ( N e. ( ZZ>= ` 2 ) -> { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) )
20	19	imp 124	. . . . . 6 |- ( ( ( ( W e. Word (Vtx ` G ) /\ (♯ ` W ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ N e. ( ZZ>= ` 2 ) ) -> { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) )
21	20	adantr 276	. . . . 5 |- ( ( ( ( ( W e. Word (Vtx ` G ) /\ (♯ ` W ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ N e. ( ZZ>= ` 2 ) ) /\ G e. UMGraph ) -> { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) )
22	2	umgredgne 16305	. . . . . 6 |- ( ( G e. UMGraph /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) -> ( W ` 0 ) =/= ( W ` 1 ) )
23	22	necomd 2506	. . . . 5 |- ( ( G e. UMGraph /\ { ( W ` 0 ) , ( W ` 1 ) } e. (Edg ` G ) ) -> ( W ` 1 ) =/= ( W ` 0 ) )
24	4, 21, 23	syl2anc 415	. . . 4 |- ( ( ( ( ( W e. Word (Vtx ` G ) /\ (♯ ` W ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) /\ N e. ( ZZ>= ` 2 ) ) /\ G e. UMGraph ) -> ( W ` 1 ) =/= ( W ` 0 ) )
25	24	exp31 364	. . 3 |- ( ( ( W e. Word (Vtx ` G ) /\ (♯ ` W ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` W ) , ( W ` 0 ) } e. (Edg ` G ) ) -> ( N e. ( ZZ>= ` 2 ) -> ( G e. UMGraph -> ( W ` 1 ) =/= ( W ` 0 ) ) ) )
26	3, 25	syl 14	. 2 |- ( W e. ( N ClWWalksN G ) -> ( N e. ( ZZ>= ` 2 ) -> ( G e. UMGraph -> ( W ` 1 ) =/= ( W ` 0 ) ) ) )
27	26	3imp31 1227	1 |- ( ( G e. UMGraph /\ N e. ( ZZ>= ` 2 ) /\ W e. ( N ClWWalksN G ) ) -> ( W ` 1 ) =/= ( W ` 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   {cpr 3706   ` cfv 5372  (class class class)co 6075   0cc0 8169   1c1 8170   + caddc 8172   - cmin 8487   NNcn 9283   2c2 9334   ZZ>=cuz 9900  ..^cfzo 10527  ♯chash 11192  Word cword 11282  lastSclsw 11327  Vtxcvtx 16167  Edgcedg 16212  UMGraphcumgr 16247   ClWWalksN cclwwlkn 16558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-2o 6678  df-er 6797  df-map 6914  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-z 9624  df-dec 9757  df-uz 9901  df-fz 10391  df-fzo 10528  df-ihash 11193  df-word 11283  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-edg 16213  df-umgren 16249  df-clwwlk 16547  df-clwwlkn 16559

This theorem is referenced by:  umgr2cwwkdifex  16580