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Theorem isclwwlknx 16571
Description: Characterization of a word representing a closed walk of a fixed length, definition of ClWWalks expanded. (Contributed by AV, 25-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)
Hypotheses
Ref Expression
isclwwlknx.v |- V = (Vtx ` G )
isclwwlknx.e |- E = (Edg ` G )
Assertion
Ref Expression
isclwwlknx |- ( N e. NN -> ( W e. ( N ClWWalksN G ) <-> ( ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ (♯ ` W ) = N ) ) )
Distinct variable groups:   i, G   i, W
Allowed substitution hints:   E( i)   N( i)   V( i)
Proof of Theorem isclwwlknx
StepHypRef Expression
1 eleq1 2301 . . . . . . . . . 10 |- ( (♯ ` W ) = N -> ( (♯ ` W ) e. NN <-> N e. NN ) )
2 len0nnbi 11317 . . . . . . . . . . 11 |- ( W e. Word V -> ( W =/= (/) <-> (♯ ` W ) e. NN ) )
32biimprcd 160 . . . . . . . . . 10 |- ( (♯ ` W ) e. NN -> ( W e. Word V -> W =/= (/) ) )
41, 3biimtrrdi 164 . . . . . . . . 9 |- ( (♯ ` W ) = N -> ( N e. NN -> ( W e. Word V -> W =/= (/) ) ) )
54impcom 125 . . . . . . . 8 |- ( ( N e. NN /\ (♯ ` W ) = N ) -> ( W e. Word V -> W =/= (/) ) )
65imp 124 . . . . . . 7 |- ( ( ( N e. NN /\ (♯ ` W ) = N ) /\ W e. Word V ) -> W =/= (/) )
76biantrurd 305 . . . . . 6 |- ( ( ( N e. NN /\ (♯ ` W ) = N ) /\ W e. Word V ) -> ( ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) <-> ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) )
87bicomd 141 . . . . 5 |- ( ( ( N e. NN /\ (♯ ` W ) = N ) /\ W e. Word V ) -> ( ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) <-> ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) )
98pm5.32da 456 . . . 4 |- ( ( N e. NN /\ (♯ ` W ) = N ) -> ( ( W e. Word V /\ ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) <-> ( W e. Word V /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) )
109ex 115 . . 3 |- ( N e. NN -> ( (♯ ` W ) = N -> ( ( W e. Word V /\ ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) <-> ( W e. Word V /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) ) )
1110pm5.32rd 455 . 2 |- ( N e. NN -> ( ( ( W e. Word V /\ ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) /\ (♯ ` W ) = N ) <-> ( ( W e. Word V /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) /\ (♯ ` W ) = N ) ) )
12 nnnn0 9549 . . . 4 |- ( N e. NN -> N e. NN0 )
13 isclwwlkng 16561 . . . 4 |- ( N e. NN0 -> ( W e. ( N ClWWalksN G ) <-> ( W e. (ClWWalks ` G ) /\ (♯ ` W ) = N ) ) )
1412, 13syl 14 . . 3 |- ( N e. NN -> ( W e. ( N ClWWalksN G ) <-> ( W e. (ClWWalks ` G ) /\ (♯ ` W ) = N ) ) )
15 isclwwlknx.v . . . . . 6 |- V = (Vtx ` G )
16 isclwwlknx.e . . . . . 6 |- E = (Edg ` G )
1715, 16isclwwlk 16549 . . . . 5 |- ( W e. (ClWWalks ` G ) <-> ( ( W e. Word V /\ W =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) )
18 3anass 1013 . . . . 5 |- ( ( ( W e. Word V /\ W =/= (/) ) /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) <-> ( ( W e. Word V /\ W =/= (/) ) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) )
19 anass 405 . . . . 5 |- ( ( ( W e. Word V /\ W =/= (/) ) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) <-> ( W e. Word V /\ ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) )
2017, 18, 193bitri 206 . . . 4 |- ( W e. (ClWWalks ` G ) <-> ( W e. Word V /\ ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) )
2120anbi1i 462 . . 3 |- ( ( W e. (ClWWalks ` G ) /\ (♯ ` W ) = N ) <-> ( ( W e. Word V /\ ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) /\ (♯ ` W ) = N ) )
2214, 21bitrdi 196 . 2 |- ( N e. NN -> ( W e. ( N ClWWalksN G ) <-> ( ( W e. Word V /\ ( W =/= (/) /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) ) /\ (♯ ` W ) = N ) ) )
23 3anass 1013 . . . 4 |- ( ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) <-> ( W e. Word V /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) )
2423anbi1i 462 . . 3 |- ( ( ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ (♯ ` W ) = N ) <-> ( ( W e. Word V /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) /\ (♯ ` W ) = N ) )
2524a1i 9 . 2 |- ( N e. NN -> ( ( ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ (♯ ` W ) = N ) <-> ( ( W e. Word V /\ ( A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) ) /\ (♯ ` W ) = N ) ) )
2611, 22, 253bitr4d 220 1 |- ( N e. NN -> ( W e. ( N ClWWalksN G ) <-> ( ( W e. Word V /\ A. i e. ( 0..^ ( (♯ ` W ) - 1 ) ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E /\ { (lastS ` W ) , ( W ` 0 ) } e. E ) /\ (♯ ` W ) = N ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   (/)c0 3520   {cpr 3706   ` cfv 5372  (class class class)co 6075   0cc0 8169   1c1 8170   + caddc 8172   - cmin 8487   NNcn 9283   NN0cn0 9542  ..^cfzo 10527  ♯chash 11192  Word cword 11282  lastSclsw 11327  Vtxcvtx 16167  Edgcedg 16212  ClWWalkscclwwlk 16546   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-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-n0 9543  df-z 9624  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-vtx 16169  df-clwwlk 16547  df-clwwlkn 16559
This theorem is referenced by:  clwwlknp  16572  clwwlkn1  16573  clwwlkn2  16576  clwwlkext2edg  16577  clwwlknonex2  16594
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