Theorem clwwlknonccat 16218

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 2235	. . . . . 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 2262	. . . 4 |- ( ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) /\ ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) ) -> ( A ` 0 ) = ( B ` 0 ) )
11		clwwlknccat 16208	. . . 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 1271	. . 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 2229	. . . . . . . 8 |- (Vtx ` G ) = (Vtx ` G )
14	13	clwwlknwrd 16199	. . . . . . 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 16199	. . . . . . 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 16197	. . . . . . . 8 |- ( A e. ( M ClWWalksN G ) -> M e. NN )
21		clwwlknlen 16196	. . . . . . . 8 |- ( A e. ( M ClWWalksN G ) -> (♯ ` A ) = M )
22		nngt0 9156	. . . . . . . . 9 |- ( M e. NN -> 0 < M )
23		breq2 4088	. . . . . . . . 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 11167	. . . . 5 |- ( ( A e. Word (Vtx ` G ) /\ B e. Word (Vtx ` G ) /\ 0 < (♯ ` A ) ) -> ( ( A ++ B ) ` 0 ) = ( A ` 0 ) )
29	16, 19, 27, 28	syl3anc 1271	. . . 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 2262	. . 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 16215	. . 3 |- ( A e. ( X (ClWWalksNOn ` G ) M ) <-> ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) )
33		isclwwlknon 16215	. . 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 16215	. 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 1395   e. wcel 2200   class class class wbr 4084   ` cfv 5322  (class class class)co 6011   0cc0 8020   + caddc 8023   < clt 8202   NNcn 9131  ♯chash 11025  Word cword 11100   ++ cconcat 11154  Vtxcvtx 15850   ClWWalksN cclwwlkn 16188  ClWWalksNOncclwwlknon 16211

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-mulrcl 8119  ax-addcom 8120  ax-mulcom 8121  ax-addass 8122  ax-mulass 8123  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-1rid 8127  ax-0id 8128  ax-rnegex 8129  ax-precex 8130  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136  ax-pre-mulgt0 8137

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-iord 4459  df-on 4461  df-ilim 4462  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-frec 6550  df-1o 6575  df-er 6695  df-map 6812  df-en 6903  df-dom 6904  df-fin 6905  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-reap 8743  df-ap 8750  df-inn 9132  df-n0 9391  df-z 9468  df-uz 9744  df-rp 9877  df-fz 10232  df-fzo 10366  df-ihash 11026  df-word 11101  df-lsw 11146  df-concat 11155  df-ndx 13072  df-slot 13073  df-base 13075  df-vtx 15852  df-clwwlk 16177  df-clwwlkn 16189  df-clwwlknon 16212

This theorem is referenced by: (None)