Theorem wlkp1 26578

Description: Append one path segment (edge) E from vertex ( P ` N ) to a vertex C to a walk <. F , P >. to become a walk <. H , Q >. of the supergraph S obtained by adding the new edge to the graph G. Formerly proven directly for Eulerian paths (for pseudographs), see eupthp1 27076. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 6-Mar-2021.) (Prove shortened by AV, 18-Apr-2021.)

Hypotheses

Ref	Expression
wlkp1.v	|- V = (Vtx ` G )
wlkp1.i	|- I = (iEdg ` G )
wlkp1.f	|- ( ph -> Fun I )
wlkp1.a	|- ( ph -> I e. Fin )
wlkp1.b	|- ( ph -> B e. _V )
wlkp1.c	|- ( ph -> C e. V )
wlkp1.d	|- ( ph -> -. B e. dom I )
wlkp1.w	|- ( ph -> F (Walks ` G ) P )
wlkp1.n	|- N = ( # ` F )
wlkp1.e	|- ( ph -> E e. (Edg ` G ) )
wlkp1.x	|- ( ph -> { ( P ` N ) , C } C_ E )
wlkp1.u	|- ( ph -> (iEdg ` S ) = ( I u. { <. B , E >. } ) )
wlkp1.h	|- H = ( F u. { <. N , B >. } )
wlkp1.q	|- Q = ( P u. { <. ( N + 1 ) , C >. } )
wlkp1.s	|- ( ph -> (Vtx ` S ) = V )
wlkp1.l	|- ( ( ph /\ C = ( P ` N ) ) -> E = { C } )

Assertion

Ref	Expression
wlkp1	|- ( ph -> H (Walks ` S ) Q )

Proof of Theorem wlkp1

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkp1.w	. . . . . 6 |- ( ph -> F (Walks ` G ) P )
2		wlkp1.i	. . . . . . 7 |- I = (iEdg ` G )
3	2	wlkf 26510	. . . . . 6 |- ( F (Walks ` G ) P -> F e. Word dom I )
4		wrdf 13310	. . . . . . 7 |- ( F e. Word dom I -> F : ( 0..^ ( # ` F ) ) --> dom I )
5		wlkp1.n	. . . . . . . . . 10 |- N = ( # ` F )
6	5	eqcomi 2631	. . . . . . . . 9 |- ( # ` F ) = N
7	6	oveq2i 6661	. . . . . . . 8 |- ( 0..^ ( # ` F ) ) = ( 0..^ N )
8	7	feq2i 6037	. . . . . . 7 |- ( F : ( 0..^ ( # ` F ) ) --> dom I <-> F : ( 0..^ N ) --> dom I )
9	4, 8	sylib 208	. . . . . 6 |- ( F e. Word dom I -> F : ( 0..^ N ) --> dom I )
10	1, 3, 9	3syl 18	. . . . 5 |- ( ph -> F : ( 0..^ N ) --> dom I )
11		fvex 6201	. . . . . . . 8 |- ( # ` F ) e. _V
12	5, 11	eqeltri 2697	. . . . . . 7 |- N e. _V
13	12	a1i 11	. . . . . 6 |- ( ph -> N e. _V )
14		wlkp1.b	. . . . . . . 8 |- ( ph -> B e. _V )
15		snidg 4206	. . . . . . . 8 |- ( B e. _V -> B e. { B } )
16	14, 15	syl 17	. . . . . . 7 |- ( ph -> B e. { B } )
17		wlkp1.e	. . . . . . . 8 |- ( ph -> E e. (Edg ` G ) )
18		dmsnopg 5606	. . . . . . . 8 |- ( E e. (Edg ` G ) -> dom { <. B , E >. } = { B } )
19	17, 18	syl 17	. . . . . . 7 |- ( ph -> dom { <. B , E >. } = { B } )
20	16, 19	eleqtrrd 2704	. . . . . 6 |- ( ph -> B e. dom { <. B , E >. } )
21	13, 20	fsnd 6179	. . . . 5 |- ( ph -> { <. N , B >. } : { N } --> dom { <. B , E >. } )
22		fzodisjsn 12505	. . . . . 6 |- ( ( 0..^ N ) i^i { N } ) = (/)
23	22	a1i 11	. . . . 5 |- ( ph -> ( ( 0..^ N ) i^i { N } ) = (/) )
24		fun 6066	. . . . 5 |- ( ( ( F : ( 0..^ N ) --> dom I /\ { <. N , B >. } : { N } --> dom { <. B , E >. } ) /\ ( ( 0..^ N ) i^i { N } ) = (/) ) -> ( F u. { <. N , B >. } ) : ( ( 0..^ N ) u. { N } ) --> ( dom I u. dom { <. B , E >. } ) )
25	10, 21, 23, 24	syl21anc 1325	. . . 4 |- ( ph -> ( F u. { <. N , B >. } ) : ( ( 0..^ N ) u. { N } ) --> ( dom I u. dom { <. B , E >. } ) )
26		wlkp1.h	. . . . . 6 |- H = ( F u. { <. N , B >. } )
27	26	a1i 11	. . . . 5 |- ( ph -> H = ( F u. { <. N , B >. } ) )
28		wlkp1.v	. . . . . . . 8 |- V = (Vtx ` G )
29		wlkp1.f	. . . . . . . 8 |- ( ph -> Fun I )
30		wlkp1.a	. . . . . . . 8 |- ( ph -> I e. Fin )
31		wlkp1.c	. . . . . . . 8 |- ( ph -> C e. V )
32		wlkp1.d	. . . . . . . 8 |- ( ph -> -. B e. dom I )
33		wlkp1.x	. . . . . . . 8 |- ( ph -> { ( P ` N ) , C } C_ E )
34		wlkp1.u	. . . . . . . 8 |- ( ph -> (iEdg ` S ) = ( I u. { <. B , E >. } ) )
35	28, 2, 29, 30, 14, 31, 32, 1, 5, 17, 33, 34, 26	wlkp1lem2 26571	. . . . . . 7 |- ( ph -> ( # ` H ) = ( N + 1 ) )
36	35	oveq2d 6666	. . . . . 6 |- ( ph -> ( 0..^ ( # ` H ) ) = ( 0..^ ( N + 1 ) ) )
37		wlkcl 26511	. . . . . . . 8 |- ( F (Walks ` G ) P -> ( # ` F ) e. NN0 )
38		eleq1 2689	. . . . . . . . . . 11 |- ( ( # ` F ) = N -> ( ( # ` F ) e. NN0 <-> N e. NN0 ) )
39	38	eqcoms 2630	. . . . . . . . . 10 |- ( N = ( # ` F ) -> ( ( # ` F ) e. NN0 <-> N e. NN0 ) )
40		elnn0uz 11725	. . . . . . . . . . 11 |- ( N e. NN0 <-> N e. ( ZZ>= ` 0 ) )
41	40	biimpi 206	. . . . . . . . . 10 |- ( N e. NN0 -> N e. ( ZZ>= ` 0 ) )
42	39, 41	syl6bi 243	. . . . . . . . 9 |- ( N = ( # ` F ) -> ( ( # ` F ) e. NN0 -> N e. ( ZZ>= ` 0 ) ) )
43	5, 42	ax-mp 5	. . . . . . . 8 |- ( ( # ` F ) e. NN0 -> N e. ( ZZ>= ` 0 ) )
44	1, 37, 43	3syl 18	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` 0 ) )
45		fzosplitsn 12576	. . . . . . 7 |- ( N e. ( ZZ>= ` 0 ) -> ( 0..^ ( N + 1 ) ) = ( ( 0..^ N ) u. { N } ) )
46	44, 45	syl 17	. . . . . 6 |- ( ph -> ( 0..^ ( N + 1 ) ) = ( ( 0..^ N ) u. { N } ) )
47	36, 46	eqtrd 2656	. . . . 5 |- ( ph -> ( 0..^ ( # ` H ) ) = ( ( 0..^ N ) u. { N } ) )
48	34	dmeqd 5326	. . . . . 6 |- ( ph -> dom (iEdg ` S ) = dom ( I u. { <. B , E >. } ) )
49		dmun 5331	. . . . . 6 |- dom ( I u. { <. B , E >. } ) = ( dom I u. dom { <. B , E >. } )
50	48, 49	syl6eq 2672	. . . . 5 |- ( ph -> dom (iEdg ` S ) = ( dom I u. dom { <. B , E >. } ) )
51	27, 47, 50	feq123d 6034	. . . 4 |- ( ph -> ( H : ( 0..^ ( # ` H ) ) --> dom (iEdg ` S ) <-> ( F u. { <. N , B >. } ) : ( ( 0..^ N ) u. { N } ) --> ( dom I u. dom { <. B , E >. } ) ) )
52	25, 51	mpbird 247	. . 3 |- ( ph -> H : ( 0..^ ( # ` H ) ) --> dom (iEdg ` S ) )
53		iswrdb 13311	. . 3 |- ( H e. Word dom (iEdg ` S ) <-> H : ( 0..^ ( # ` H ) ) --> dom (iEdg ` S ) )
54	52, 53	sylibr 224	. 2 |- ( ph -> H e. Word dom (iEdg ` S ) )
55	28	wlkp 26512	. . . . . . 7 |- ( F (Walks ` G ) P -> P : ( 0 ... ( # ` F ) ) --> V )
56	1, 55	syl 17	. . . . . 6 |- ( ph -> P : ( 0 ... ( # ` F ) ) --> V )
57	5	oveq2i 6661	. . . . . . 7 |- ( 0 ... N ) = ( 0 ... ( # ` F ) )
58	57	feq2i 6037	. . . . . 6 |- ( P : ( 0 ... N ) --> V <-> P : ( 0 ... ( # ` F ) ) --> V )
59	56, 58	sylibr 224	. . . . 5 |- ( ph -> P : ( 0 ... N ) --> V )
60		ovexd 6680	. . . . . 6 |- ( ph -> ( N + 1 ) e. _V )
61	60, 31	fsnd 6179	. . . . 5 |- ( ph -> { <. ( N + 1 ) , C >. } : { ( N + 1 ) } --> V )
62		fzp1disj 12399	. . . . . 6 |- ( ( 0 ... N ) i^i { ( N + 1 ) } ) = (/)
63	62	a1i 11	. . . . 5 |- ( ph -> ( ( 0 ... N ) i^i { ( N + 1 ) } ) = (/) )
64		fun 6066	. . . . 5 |- ( ( ( P : ( 0 ... N ) --> V /\ { <. ( N + 1 ) , C >. } : { ( N + 1 ) } --> V ) /\ ( ( 0 ... N ) i^i { ( N + 1 ) } ) = (/) ) -> ( P u. { <. ( N + 1 ) , C >. } ) : ( ( 0 ... N ) u. { ( N + 1 ) } ) --> ( V u. V ) )
65	59, 61, 63, 64	syl21anc 1325	. . . 4 |- ( ph -> ( P u. { <. ( N + 1 ) , C >. } ) : ( ( 0 ... N ) u. { ( N + 1 ) } ) --> ( V u. V ) )
66		fzsuc 12388	. . . . . 6 |- ( N e. ( ZZ>= ` 0 ) -> ( 0 ... ( N + 1 ) ) = ( ( 0 ... N ) u. { ( N + 1 ) } ) )
67	44, 66	syl 17	. . . . 5 |- ( ph -> ( 0 ... ( N + 1 ) ) = ( ( 0 ... N ) u. { ( N + 1 ) } ) )
68		unidm 3756	. . . . . . 7 |- ( V u. V ) = V
69	68	eqcomi 2631	. . . . . 6 |- V = ( V u. V )
70	69	a1i 11	. . . . 5 |- ( ph -> V = ( V u. V ) )
71	67, 70	feq23d 6040	. . . 4 |- ( ph -> ( ( P u. { <. ( N + 1 ) , C >. } ) : ( 0 ... ( N + 1 ) ) --> V <-> ( P u. { <. ( N + 1 ) , C >. } ) : ( ( 0 ... N ) u. { ( N + 1 ) } ) --> ( V u. V ) ) )
72	65, 71	mpbird 247	. . 3 |- ( ph -> ( P u. { <. ( N + 1 ) , C >. } ) : ( 0 ... ( N + 1 ) ) --> V )
73		wlkp1.q	. . . . 5 |- Q = ( P u. { <. ( N + 1 ) , C >. } )
74	73	a1i 11	. . . 4 |- ( ph -> Q = ( P u. { <. ( N + 1 ) , C >. } ) )
75	35	oveq2d 6666	. . . 4 |- ( ph -> ( 0 ... ( # ` H ) ) = ( 0 ... ( N + 1 ) ) )
76		wlkp1.s	. . . 4 |- ( ph -> (Vtx ` S ) = V )
77	74, 75, 76	feq123d 6034	. . 3 |- ( ph -> ( Q : ( 0 ... ( # ` H ) ) --> (Vtx ` S ) <-> ( P u. { <. ( N + 1 ) , C >. } ) : ( 0 ... ( N + 1 ) ) --> V ) )
78	72, 77	mpbird 247	. 2 |- ( ph -> Q : ( 0 ... ( # ` H ) ) --> (Vtx ` S ) )
79		wlkp1.l	. . 3 |- ( ( ph /\ C = ( P ` N ) ) -> E = { C } )
80	28, 2, 29, 30, 14, 31, 32, 1, 5, 17, 33, 34, 26, 73, 76, 79	wlkp1lem8 26577	. 2 |- ( ph -> A. k e. ( 0..^ ( # ` H ) )if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) )
81	28, 2, 29, 30, 14, 31, 32, 1, 5, 17, 33, 34, 26, 73, 76	wlkp1lem4 26573	. . 3 |- ( ph -> ( S e. _V /\ H e. _V /\ Q e. _V ) )
82		eqid 2622	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
83		eqid 2622	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
84	82, 83	iswlk 26506	. . 3 |- ( ( S e. _V /\ H e. _V /\ Q e. _V ) -> ( H (Walks ` S ) Q <-> ( H e. Word dom (iEdg ` S ) /\ Q : ( 0 ... ( # ` H ) ) --> (Vtx ` S ) /\ A. k e. ( 0..^ ( # ` H ) )if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) ) ) )
85	81, 84	syl 17	. 2 |- ( ph -> ( H (Walks ` S ) Q <-> ( H e. Word dom (iEdg ` S ) /\ Q : ( 0 ... ( # ` H ) ) --> (Vtx ` S ) /\ A. k e. ( 0..^ ( # ` H ) )if- ( ( Q ` k ) = ( Q ` ( k + 1 ) ) , ( (iEdg ` S ) ` ( H ` k ) ) = { ( Q ` k ) } , { ( Q ` k ) , ( Q ` ( k + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` k ) ) ) ) ) )
86	54, 78, 80, 85	mpbir3and 1245	1 |- ( ph -> H (Walks ` S ) Q )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384  if-wif 1012   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   <.cop 4183   class class class wbr 4653   dom cdm 5114   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939  Walkscwlks 26492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-wlks 26495

This theorem is referenced by:  eupthp1  27076