Theorem clwwlknonccat 16554

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 2240	. . . . . 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 2267	. . . 4 |- ( ( ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) /\ ( B e. ( N ClWWalksN G ) /\ ( B ` 0 ) = X ) ) -> ( A ` 0 ) = ( B ` 0 ) )
11		clwwlknccat 16544	. . . 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 2234	. . . . . . . 8 |- (Vtx ` G ) = (Vtx ` G )
14	13	clwwlknwrd 16535	. . . . . . 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 16535	. . . . . . 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 16533	. . . . . . . 8 |- ( A e. ( M ClWWalksN G ) -> M e. NN )
21		clwwlknlen 16532	. . . . . . . 8 |- ( A e. ( M ClWWalksN G ) -> (♯ ` A ) = M )
22		nngt0 9279	. . . . . . . . 9 |- ( M e. NN -> 0 < M )
23		breq2 4118	. . . . . . . . 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 11316	. . . . 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 2267	. . 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 16551	. . 3 |- ( A e. ( X (ClWWalksNOn ` G ) M ) <-> ( A e. ( M ClWWalksN G ) /\ ( A ` 0 ) = X ) )
33		isclwwlknon 16551	. . 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 16551	. 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 2205   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   0cc0 8143   + caddc 8146   < clt 8324   NNcn 9254  ♯chash 11163  Word cword 11249   ++ cconcat 11303  Vtxcvtx 16133   ClWWalksN cclwwlkn 16524  ClWWalksNOncclwwlknon 16547

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-map 6897  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-rp 10005  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-lsw 11295  df-concat 11304  df-ndx 13299  df-slot 13300  df-base 13302  df-vtx 16135  df-clwwlk 16513  df-clwwlkn 16525  df-clwwlknon 16548

This theorem is referenced by: (None)