Theorem eupth2eucrct 27077

Description: Append one path segment to an Eulerian path <. F , P >. which may not be an (Eulerian) circuit to become an Eulerian circuit <. H , Q >. of the supergraph S obtained by adding the new edge to the graph G. (Contributed by AV, 11-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)

Hypotheses

Ref	Expression
eupthp1.v	|- V = (Vtx ` G )
eupthp1.i	|- I = (iEdg ` G )
eupthp1.f	|- ( ph -> Fun I )
eupthp1.a	|- ( ph -> I e. Fin )
eupthp1.b	|- ( ph -> B e. _V )
eupthp1.c	|- ( ph -> C e. V )
eupthp1.d	|- ( ph -> -. B e. dom I )
eupthp1.p	|- ( ph -> F (EulerPaths ` G ) P )
eupthp1.n	|- N = ( # ` F )
eupthp1.e	|- ( ph -> E e. (Edg ` G ) )
eupthp1.x	|- ( ph -> { ( P ` N ) , C } C_ E )
eupthp1.u	|- (iEdg ` S ) = ( I u. { <. B , E >. } )
eupthp1.h	|- H = ( F u. { <. N , B >. } )
eupthp1.q	|- Q = ( P u. { <. ( N + 1 ) , C >. } )
eupthp1.s	|- (Vtx ` S ) = V
eupthp1.l	|- ( ( ph /\ C = ( P ` N ) ) -> E = { C } )
eupth2eucrct.c	|- ( ph -> C = ( P ` 0 ) )

Assertion

Ref	Expression
eupth2eucrct	|- ( ph -> ( H (EulerPaths ` S ) Q /\ H (Circuits ` S ) Q ) )

Proof of Theorem eupth2eucrct

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eupthp1.v	. . 3 |- V = (Vtx ` G )
2		eupthp1.i	. . 3 |- I = (iEdg ` G )
3		eupthp1.f	. . 3 |- ( ph -> Fun I )
4		eupthp1.a	. . 3 |- ( ph -> I e. Fin )
5		eupthp1.b	. . 3 |- ( ph -> B e. _V )
6		eupthp1.c	. . 3 |- ( ph -> C e. V )
7		eupthp1.d	. . 3 |- ( ph -> -. B e. dom I )
8		eupthp1.p	. . 3 |- ( ph -> F (EulerPaths ` G ) P )
9		eupthp1.n	. . 3 |- N = ( # ` F )
10		eupthp1.e	. . 3 |- ( ph -> E e. (Edg ` G ) )
11		eupthp1.x	. . 3 |- ( ph -> { ( P ` N ) , C } C_ E )
12		eupthp1.u	. . 3 |- (iEdg ` S ) = ( I u. { <. B , E >. } )
13		eupthp1.h	. . 3 |- H = ( F u. { <. N , B >. } )
14		eupthp1.q	. . 3 |- Q = ( P u. { <. ( N + 1 ) , C >. } )
15		eupthp1.s	. . 3 |- (Vtx ` S ) = V
16		eupthp1.l	. . 3 |- ( ( ph /\ C = ( P ` N ) ) -> E = { C } )
17	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16	eupthp1 27076	. 2 |- ( ph -> H (EulerPaths ` S ) Q )
18		simpr 477	. . 3 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> H (EulerPaths ` S ) Q )
19		eupthistrl 27071	. . . . 5 |- ( H (EulerPaths ` S ) Q -> H (Trails ` S ) Q )
20	19	adantl 482	. . . 4 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> H (Trails ` S ) Q )
21		fveq2 6191	. . . . . . . 8 |- ( k = 0 -> ( Q ` k ) = ( Q ` 0 ) )
22		fveq2 6191	. . . . . . . 8 |- ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
23	21, 22	eqeq12d 2637	. . . . . . 7 |- ( k = 0 -> ( ( Q ` k ) = ( P ` k ) <-> ( Q ` 0 ) = ( P ` 0 ) ) )
24		eupthiswlk 27072	. . . . . . . . 9 |- ( F (EulerPaths ` G ) P -> F (Walks ` G ) P )
25	8, 24	syl 17	. . . . . . . 8 |- ( ph -> F (Walks ` G ) P )
26	12	a1i 11	. . . . . . . 8 |- ( ph -> (iEdg ` S ) = ( I u. { <. B , E >. } ) )
27	15	a1i 11	. . . . . . . 8 |- ( ph -> (Vtx ` S ) = V )
28	1, 2, 3, 4, 5, 6, 7, 25, 9, 10, 11, 26, 13, 14, 27	wlkp1lem5 26574	. . . . . . 7 |- ( ph -> A. k e. ( 0 ... N ) ( Q ` k ) = ( P ` k ) )
29	2	wlkf 26510	. . . . . . . . 9 |- ( F (Walks ` G ) P -> F e. Word dom I )
30	24, 29	syl 17	. . . . . . . 8 |- ( F (EulerPaths ` G ) P -> F e. Word dom I )
31		lencl 13324	. . . . . . . . 9 |- ( F e. Word dom I -> ( # ` F ) e. NN0 )
32	9	eleq1i 2692	. . . . . . . . . 10 |- ( N e. NN0 <-> ( # ` F ) e. NN0 )
33		0elfz 12436	. . . . . . . . . 10 |- ( N e. NN0 -> 0 e. ( 0 ... N ) )
34	32, 33	sylbir 225	. . . . . . . . 9 |- ( ( # ` F ) e. NN0 -> 0 e. ( 0 ... N ) )
35	31, 34	syl 17	. . . . . . . 8 |- ( F e. Word dom I -> 0 e. ( 0 ... N ) )
36	8, 30, 35	3syl 18	. . . . . . 7 |- ( ph -> 0 e. ( 0 ... N ) )
37	23, 28, 36	rspcdva 3316	. . . . . 6 |- ( ph -> ( Q ` 0 ) = ( P ` 0 ) )
38	37	adantr 481	. . . . 5 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> ( Q ` 0 ) = ( P ` 0 ) )
39		eupth2eucrct.c	. . . . . . 7 |- ( ph -> C = ( P ` 0 ) )
40	39	eqcomd 2628	. . . . . 6 |- ( ph -> ( P ` 0 ) = C )
41	40	adantr 481	. . . . 5 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> ( P ` 0 ) = C )
42	14	a1i 11	. . . . . . 7 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> Q = ( P u. { <. ( N + 1 ) , C >. } ) )
43	13	fveq2i 6194	. . . . . . . . 9 |- ( # ` H ) = ( # ` ( F u. { <. N , B >. } ) )
44	43	a1i 11	. . . . . . . 8 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> ( # ` H ) = ( # ` ( F u. { <. N , B >. } ) ) )
45		wrdfin 13323	. . . . . . . . . . . 12 |- ( F e. Word dom I -> F e. Fin )
46	29, 45	syl 17	. . . . . . . . . . 11 |- ( F (Walks ` G ) P -> F e. Fin )
47	8, 24, 46	3syl 18	. . . . . . . . . 10 |- ( ph -> F e. Fin )
48	47	adantr 481	. . . . . . . . 9 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> F e. Fin )
49		snfi 8038	. . . . . . . . . 10 |- { <. N , B >. } e. Fin
50	49	a1i 11	. . . . . . . . 9 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> { <. N , B >. } e. Fin )
51		wrddm 13312	. . . . . . . . . . . . 13 |- ( F e. Word dom I -> dom F = ( 0..^ ( # ` F ) ) )
52	8, 30, 51	3syl 18	. . . . . . . . . . . 12 |- ( ph -> dom F = ( 0..^ ( # ` F ) ) )
53		fzonel 12483	. . . . . . . . . . . . . . . 16 |- -. ( # ` F ) e. ( 0..^ ( # ` F ) )
54	53	a1i 11	. . . . . . . . . . . . . . 15 |- ( ph -> -. ( # ` F ) e. ( 0..^ ( # ` F ) ) )
55	9	eleq1i 2692	. . . . . . . . . . . . . . 15 |- ( N e. ( 0..^ ( # ` F ) ) <-> ( # ` F ) e. ( 0..^ ( # ` F ) ) )
56	54, 55	sylnibr 319	. . . . . . . . . . . . . 14 |- ( ph -> -. N e. ( 0..^ ( # ` F ) ) )
57		eleq2 2690	. . . . . . . . . . . . . . 15 |- ( dom F = ( 0..^ ( # ` F ) ) -> ( N e. dom F <-> N e. ( 0..^ ( # ` F ) ) ) )
58	57	notbid 308	. . . . . . . . . . . . . 14 |- ( dom F = ( 0..^ ( # ` F ) ) -> ( -. N e. dom F <-> -. N e. ( 0..^ ( # ` F ) ) ) )
59	56, 58	syl5ibrcom 237	. . . . . . . . . . . . 13 |- ( ph -> ( dom F = ( 0..^ ( # ` F ) ) -> -. N e. dom F ) )
60		fvex 6201	. . . . . . . . . . . . . . . 16 |- ( # ` F ) e. _V
61	9, 60	eqeltri 2697	. . . . . . . . . . . . . . 15 |- N e. _V
62	61	a1i 11	. . . . . . . . . . . . . 14 |- ( ph -> N e. _V )
63	62, 5	opeldmd 5327	. . . . . . . . . . . . 13 |- ( ph -> ( <. N , B >. e. F -> N e. dom F ) )
64	59, 63	nsyld 154	. . . . . . . . . . . 12 |- ( ph -> ( dom F = ( 0..^ ( # ` F ) ) -> -. <. N , B >. e. F ) )
65	52, 64	mpd 15	. . . . . . . . . . 11 |- ( ph -> -. <. N , B >. e. F )
66	65	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> -. <. N , B >. e. F )
67		disjsn 4246	. . . . . . . . . 10 |- ( ( F i^i { <. N , B >. } ) = (/) <-> -. <. N , B >. e. F )
68	66, 67	sylibr 224	. . . . . . . . 9 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> ( F i^i { <. N , B >. } ) = (/) )
69		hashun 13171	. . . . . . . . 9 |- ( ( F e. Fin /\ { <. N , B >. } e. Fin /\ ( F i^i { <. N , B >. } ) = (/) ) -> ( # ` ( F u. { <. N , B >. } ) ) = ( ( # ` F ) + ( # ` { <. N , B >. } ) ) )
70	48, 50, 68, 69	syl3anc 1326	. . . . . . . 8 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> ( # ` ( F u. { <. N , B >. } ) ) = ( ( # ` F ) + ( # ` { <. N , B >. } ) ) )
71	9	eqcomi 2631	. . . . . . . . . 10 |- ( # ` F ) = N
72		opex 4932	. . . . . . . . . . 11 |- <. N , B >. e. _V
73		hashsng 13159	. . . . . . . . . . 11 |- ( <. N , B >. e. _V -> ( # ` { <. N , B >. } ) = 1 )
74	72, 73	ax-mp 5	. . . . . . . . . 10 |- ( # ` { <. N , B >. } ) = 1
75	71, 74	oveq12i 6662	. . . . . . . . 9 |- ( ( # ` F ) + ( # ` { <. N , B >. } ) ) = ( N + 1 )
76	75	a1i 11	. . . . . . . 8 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> ( ( # ` F ) + ( # ` { <. N , B >. } ) ) = ( N + 1 ) )
77	44, 70, 76	3eqtrd 2660	. . . . . . 7 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> ( # ` H ) = ( N + 1 ) )
78	42, 77	fveq12d 6197	. . . . . 6 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> ( Q ` ( # ` H ) ) = ( ( P u. { <. ( N + 1 ) , C >. } ) ` ( N + 1 ) ) )
79		ovexd 6680	. . . . . . . . 9 |- ( ph -> ( N + 1 ) e. _V )
80	1, 2, 3, 4, 5, 6, 7, 25, 9	wlkp1lem1 26570	. . . . . . . . 9 |- ( ph -> -. ( N + 1 ) e. dom P )
81	79, 6, 80	3jca 1242	. . . . . . . 8 |- ( ph -> ( ( N + 1 ) e. _V /\ C e. V /\ -. ( N + 1 ) e. dom P ) )
82	81	adantr 481	. . . . . . 7 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> ( ( N + 1 ) e. _V /\ C e. V /\ -. ( N + 1 ) e. dom P ) )
83		fsnunfv 6453	. . . . . . 7 |- ( ( ( N + 1 ) e. _V /\ C e. V /\ -. ( N + 1 ) e. dom P ) -> ( ( P u. { <. ( N + 1 ) , C >. } ) ` ( N + 1 ) ) = C )
84	82, 83	syl 17	. . . . . 6 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> ( ( P u. { <. ( N + 1 ) , C >. } ) ` ( N + 1 ) ) = C )
85	78, 84	eqtr2d 2657	. . . . 5 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> C = ( Q ` ( # ` H ) ) )
86	38, 41, 85	3eqtrd 2660	. . . 4 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> ( Q ` 0 ) = ( Q ` ( # ` H ) ) )
87		iscrct 26685	. . . 4 |- ( H (Circuits ` S ) Q <-> ( H (Trails ` S ) Q /\ ( Q ` 0 ) = ( Q ` ( # ` H ) ) ) )
88	20, 86, 87	sylanbrc 698	. . 3 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> H (Circuits ` S ) Q )
89	18, 88	jca 554	. 2 |- ( ( ph /\ H (EulerPaths ` S ) Q ) -> ( H (EulerPaths ` S ) Q /\ H (Circuits ` S ) Q ) )
90	17, 89	mpdan 702	1 |- ( ph -> ( H (EulerPaths ` S ) Q /\ H (Circuits ` S ) Q ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _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   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   1c1 9937   + caddc 9939   NN0cn0 11292   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939  Walkscwlks 26492  Trailsctrls 26587  Circuitsccrcts 26679  EulerPathsceupth 27057

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  df-trls 26589  df-crcts 26681  df-eupth 27058

This theorem is referenced by: (None)