Theorem clwwlknonccat 16428

Description: The concatenation of two words representing closed walks on a vertex X represents a closed walk on vertex X. The resulting walk is a "double loop", starting at vertex X, coming back to X by the first walk, following the second walk and finally coming back to X again. (Contributed by AV, 24-Apr-2022.)

Assertion

Ref	Expression
clwwlknonccat	|- ( ( A e. ( X (ClWWalksNOn ` G ) M ) /\ B e. ( X (ClWWalksNOn ` G ) N ) ) -> ( A ++ B ) e. ( X (ClWWalksNOn ` G ) ( M + N ) ) )

Proof of Theorem clwwlknonccat

Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 |- ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) -> A e. ( M ClWWalksN G ) )
2	1	adantr 276	. . . 4 |- ( ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) /\ ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) ) -> A e. ( M ClWWalksN G ) )
3		simpl 109	. . . . 5 |- ( ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) -> B e. ( N ClWWalksN G ) )
4	3	adantl 277	. . . 4 |- ( ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) /\ ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) ) -> B e. ( N ClWWalksN G ) )
5		simpr 110	. . . . . 6 |- ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) -> ( A ` 0 ) = X )
6	5	adantr 276	. . . . 5 |- ( ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) /\ ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) ) -> ( A ` 0 ) = X )
7		simpr 110	. . . . . . 7 |- ( ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) -> ( B ` 0 ) = X )
8	7	eqcomd 2238	. . . . . 6 |- ( ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) -> X = ( B ` 0 ) )
9	8	adantl 277	. . . . 5 |- ( ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) /\ ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) ) -> X = ( B ` 0 ) )
10	6, 9	eqtrd 2265	. . . 4 |- ( ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) /\ ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) ) -> ( A ` 0 ) = ( B ` 0 ) )
11		clwwlknccat 16418	. . . 4 |- ( ( A e. ( M ClWWalksN G ) /\ B e. ( N ClWWalksN G ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( A ++ B ) e. ( ( M + N ) ClWWalksN G ) )
12	2, 4, 10, 11	syl3anc 1274	. . 3 |- ( ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) /\ ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) ) -> ( A ++ B ) e. ( ( M + N ) ClWWalksN G ) )
13		eqid 2232	. . . . . . . 8 |- (Vtx ` G ) = (Vtx ` G )
14	13	clwwlknwrd 16409	. . . . . . 7 |- ( A e. ( M ClWWalksN G ) -> A e. Word (Vtx ` G ) )
15	14	adantr 276	. . . . . 6 |- ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) -> A e. Word (Vtx ` G ) )
16	15	adantr 276	. . . . 5 |- ( ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) /\ ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) ) -> A e. Word (Vtx ` G ) )
17	13	clwwlknwrd 16409	. . . . . . 7 |- ( B e. ( N ClWWalksN G ) -> B e. Word (Vtx ` G ) )
18	17	adantr 276	. . . . . 6 |- ( ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) -> B e. Word (Vtx ` G ) )
19	18	adantl 277	. . . . 5 |- ( ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) /\ ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) ) -> B e. Word (Vtx ` G ) )
20		clwwlknnn 16407	. . . . . . . 8 |- ( A e. ( M ClWWalksN G ) -> M e. NN )
21		clwwlknlen 16406	. . . . . . . 8 |- ( A e. ( M ClWWalksN G ) -> (♯ ` A ) = M )
22		nngt0 9262	. . . . . . . . 9 |- ( M e. NN -> 0 < M )
23		breq2 4113	. . . . . . . . 9 |- ( (♯ ` A ) = M -> ( 0 < (♯ ` A ) <-> 0 < M ) )
24	22, 23	syl5ibrcom 157	. . . . . . . 8 |- ( M e. NN -> ( (♯ ` A ) = M -> 0 < (♯ ` A ) ) )
25	20, 21, 24	sylc 62	. . . . . . 7 |- ( A e. ( M ClWWalksN G ) -> 0 < (♯ ` A ) )
26	25	adantr 276	. . . . . 6 |- ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) -> 0 < (♯ ` A ) )
27	26	adantr 276	. . . . 5 |- ( ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) /\ ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) ) -> 0 < (♯ ` A ) )
28		ccatfv0 11291	. . . . 5 |- ( ( A e. Word (Vtx ` G ) /\ B e. Word (Vtx ` G ) /\ 0 < (♯ ` A ) ) -> ( ( A ++ B ) ` 0 ) = ( A ` 0 ) )
29	16, 19, 27, 28	syl3anc 1274	. . . 4 |- ( ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) /\ ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) ) -> ( ( A ++ B ) ` 0 ) = ( A ` 0 ) )
30	29, 6	eqtrd 2265	. . 3 |- ( ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) /\ ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) ) -> ( ( A ++ B ) ` 0 ) = X )
31	12, 30	jca 306	. 2 |- ( ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) /\ ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) ) -> ( ( A ++ B ) e. ( ( M + N ) ClWWalksN G ) /\ ( ( A ++ B ) ` 0 ) = X ) )
32		isclwwlknon 16425	. . 3 |- ( A e. ( X (ClWWalksNOn ` G ) M ) <-> ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) )
33		isclwwlknon 16425	. . 3 |- ( B e. ( X (ClWWalksNOn ` G ) N ) <-> ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) )
34	32, 33	anbi12i 460	. 2 |- ( ( A e. ( X (ClWWalksNOn ` G ) M ) /\ B e. ( X (ClWWalksNOn ` G ) N ) ) <-> ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) /\ ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) ) )
35		isclwwlknon 16425	. 2 |- ( ( A ++ B ) e. ( X (ClWWalksNOn ` G ) ( M + N ) ) <-> ( ( A ++ B ) e. ( ( M + N ) ClWWalksN G ) /\ ( ( A ++ B ) ` 0 ) = X ) )
36	31, 34, 35	3imtr4i 201	1 |- ( ( A e. ( X (ClWWalksNOn ` G ) M ) /\ B e. ( X (ClWWalksNOn ` G ) N ) ) -> ( A ++ B ) e. ( X (ClWWalksNOn ` G ) ( M + N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   0cc0 8127   + caddc 8130   < clt 8308   NNcn 9237  ♯chash 11138  Word cword 11224   ++ cconcat 11278  Vtxcvtx 16007   ClWWalksN cclwwlkn 16398  ClWWalksNOncclwwlknon 16421

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-map 6884  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-rp 9987  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-lsw 11270  df-concat 11279  df-ndx 13215  df-slot 13216  df-base 13218  df-vtx 16009  df-clwwlk 16387  df-clwwlkn 16399  df-clwwlknon 16422

This theorem is referenced by: (None)