Theorem clwwlknccat 16869

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

Assertion

Ref	Expression
clwwlknccat	|- ( ( A e. ( M ClWWalksN G ) /\ B e. ( N ClWWalksN G ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( A ++ B ) e. ( ( M + N ) ClWWalksN G ) )

Proof of Theorem clwwlknccat

Step	Hyp	Ref	Expression
1		isclwwlkni 16853	. . 3 |- ( A e. ( M ClWWalksN G ) -> ( A e. (ClWWalks ` G ) /\ (♯ ` A ) = M ) )
2		isclwwlkni 16853	. . 3 |- ( B e. ( N ClWWalksN G ) -> ( B e. (ClWWalks ` G ) /\ (♯ ` B ) = N ) )
3		id 19	. . 3 |- ( ( A ` 0 ) = ( B ` 0 ) -> ( A ` 0 ) = ( B ` 0 ) )
4		simpl 109	. . . 4 |- ( ( A e. (ClWWalks ` G ) /\ (♯ ` A ) = M ) -> A e. (ClWWalks ` G ) )
5		simpl 109	. . . 4 |- ( ( B e. (ClWWalks ` G ) /\ (♯ ` B ) = N ) -> B e. (ClWWalks ` G ) )
6		clwwlkccat 16847	. . . 4 |- ( ( A e. (ClWWalks ` G ) /\ B e. (ClWWalks ` G ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( A ++ B ) e. (ClWWalks ` G ) )
7	4, 5, 3, 6	syl3an 1320	. . 3 |- ( ( ( A e. (ClWWalks ` G ) /\ (♯ ` A ) = M ) /\ ( B e. (ClWWalks ` G ) /\ (♯ ` B ) = N ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( A ++ B ) e. (ClWWalks ` G ) )
8	1, 2, 3, 7	syl3an 1320	. 2 |- ( ( A e. ( M ClWWalksN G ) /\ B e. ( N ClWWalksN G ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( A ++ B ) e. (ClWWalks ` G ) )
9		eqid 2238	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
10	9	clwwlknwrd 16860	. . . . 5 |- ( A e. ( M ClWWalksN G ) -> A e. Word (Vtx ` G ) )
11	9	clwwlknwrd 16860	. . . . 5 |- ( B e. ( N ClWWalksN G ) -> B e. Word (Vtx ` G ) )
12		ccatlen 11385	. . . . 5 |- ( ( A e. Word (Vtx ` G ) /\ B e. Word (Vtx ` G ) ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
13	10, 11, 12	syl2an 289	. . . 4 |- ( ( A e. ( M ClWWalksN G ) /\ B e. ( N ClWWalksN G ) ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
14		clwwlknlen 16857	. . . . 5 |- ( A e. ( M ClWWalksN G ) -> (♯ ` A ) = M )
15		clwwlknlen 16857	. . . . 5 |- ( B e. ( N ClWWalksN G ) -> (♯ ` B ) = N )
16	14, 15	oveqan12d 6107	. . . 4 |- ( ( A e. ( M ClWWalksN G ) /\ B e. ( N ClWWalksN G ) ) -> ( (♯ ` A ) + (♯ ` B ) ) = ( M + N ) )
17	13, 16	eqtrd 2271	. . 3 |- ( ( A e. ( M ClWWalksN G ) /\ B e. ( N ClWWalksN G ) ) -> (♯ ` ( A ++ B ) ) = ( M + N ) )
18	17	3adant3 1048	. 2 |- ( ( A e. ( M ClWWalksN G ) /\ B e. ( N ClWWalksN G ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> (♯ ` ( A ++ B ) ) = ( M + N ) )
19		clwwlknnn 16858	. . . . . 6 |- ( A e. ( M ClWWalksN G ) -> M e. NN )
20	19	nnnn0d 9631	. . . . 5 |- ( A e. ( M ClWWalksN G ) -> M e. NN0 )
21	20	3ad2ant1 1049	. . . 4 |- ( ( A e. ( M ClWWalksN G ) /\ B e. ( N ClWWalksN G ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> M e. NN0 )
22		clwwlknnn 16858	. . . . . 6 |- ( B e. ( N ClWWalksN G ) -> N e. NN )
23	22	nnnn0d 9631	. . . . 5 |- ( B e. ( N ClWWalksN G ) -> N e. NN0 )
24	23	3ad2ant2 1050	. . . 4 |- ( ( A e. ( M ClWWalksN G ) /\ B e. ( N ClWWalksN G ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> N e. NN0 )
25	21, 24	nn0addcld 9635	. . 3 |- ( ( A e. ( M ClWWalksN G ) /\ B e. ( N ClWWalksN G ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( M + N ) e. NN0 )
26		isclwwlkng 16852	. . 3 |- ( ( M + N ) e. NN0 -> ( ( A ++ B ) e. ( ( M + N ) ClWWalksN G ) <-> ( ( A ++ B ) e. (ClWWalks ` G ) /\ (♯ ` ( A ++ B ) ) = ( M + N ) ) ) )
27	25, 26	syl 14	. 2 |- ( ( A e. ( M ClWWalksN G ) /\ B e. ( N ClWWalksN G ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( ( A ++ B ) e. ( ( M + N ) ClWWalksN G ) <-> ( ( A ++ B ) e. (ClWWalks ` G ) /\ (♯ ` ( A ++ B ) ) = ( M + N ) ) ) )
28	8, 18, 27	mpbir2and 957	1 |- ( ( A e. ( M ClWWalksN G ) /\ B e. ( N ClWWalksN G ) /\ ( A ` 0 ) = ( B ` 0 ) ) -> ( A ++ B ) e. ( ( M + N ) ClWWalksN G ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5380  (class class class)co 6088   0cc0 8185   + caddc 8188   NN0cn0 9574  ♯chash 11236  Word cword 11326   ++ cconcat 11380  Vtxcvtx 16458  ClWWalkscclwwlk 16837   ClWWalksN cclwwlkn 16849

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-map 6928  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-rp 10072  df-fz 10429  df-fzo 10567  df-ihash 11237  df-word 11327  df-lsw 11372  df-concat 11381  df-ndx 13413  df-slot 13414  df-base 13416  df-vtx 16460  df-clwwlk 16838  df-clwwlkn 16850

This theorem is used by:  clwwlknonccat  16879