Theorem wlkres 26567

Description: The restriction <. H , Q >. of a walk <. F , P >. to an initial segment of the walk (of length N) forms a walk on the subgraph S consisting of the edges in the initial segment. Formerly proven directly for Eulerian paths, see eupthres 27075. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 5-Mar-2021.)

Hypotheses

Ref	Expression
wlkres.v	|- V = (Vtx ` G )
wlkres.i	|- I = (iEdg ` G )
wlkres.d	|- ( ph -> F (Walks ` G ) P )
wlkres.n	|- ( ph -> N e. ( 0..^ ( # ` F ) ) )
wlkres.s	|- ( ph -> (Vtx ` S ) = V )
wlkres.e	|- ( ph -> (iEdg ` S ) = ( I |` ( F " ( 0..^ N ) ) ) )
wlkres.h	|- H = ( F |` ( 0..^ N ) )
wlkres.q	|- Q = ( P |` ( 0 ... N ) )

Assertion

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

Proof of Theorem wlkres

Dummy variables k i x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkres.h	. . 3 |- H = ( F |` ( 0..^ N ) )
2		wlkres.d	. . . . . . . 8 |- ( ph -> F (Walks ` G ) P )
3		wlkres.i	. . . . . . . . 9 |- I = (iEdg ` G )
4	3	wlkf 26510	. . . . . . . 8 |- ( F (Walks ` G ) P -> F e. Word dom I )
5		wrdfn 13319	. . . . . . . 8 |- ( F e. Word dom I -> F Fn ( 0..^ ( # ` F ) ) )
6	2, 4, 5	3syl 18	. . . . . . 7 |- ( ph -> F Fn ( 0..^ ( # ` F ) ) )
7		wlkres.n	. . . . . . . 8 |- ( ph -> N e. ( 0..^ ( # ` F ) ) )
8		elfzouz2 12484	. . . . . . . 8 |- ( N e. ( 0..^ ( # ` F ) ) -> ( # ` F ) e. ( ZZ>= ` N ) )
9		fzoss2 12496	. . . . . . . 8 |- ( ( # ` F ) e. ( ZZ>= ` N ) -> ( 0..^ N ) C_ ( 0..^ ( # ` F ) ) )
10	7, 8, 9	3syl 18	. . . . . . 7 |- ( ph -> ( 0..^ N ) C_ ( 0..^ ( # ` F ) ) )
11		fnssres 6004	. . . . . . 7 |- ( ( F Fn ( 0..^ ( # ` F ) ) /\ ( 0..^ N ) C_ ( 0..^ ( # ` F ) ) ) -> ( F |` ( 0..^ N ) ) Fn ( 0..^ N ) )
12	6, 10, 11	syl2anc 693	. . . . . 6 |- ( ph -> ( F |` ( 0..^ N ) ) Fn ( 0..^ N ) )
13		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 0..^ N ) ) -> x e. ( 0..^ N ) )
14		fveq2 6191	. . . . . . . . . . . . . 14 |- ( i = x -> ( F ` i ) = ( F ` x ) )
15	14	eqeq1d 2624	. . . . . . . . . . . . 13 |- ( i = x -> ( ( F ` i ) = ( ( F |` ( 0..^ N ) ) ` x ) <-> ( F ` x ) = ( ( F |` ( 0..^ N ) ) ` x ) ) )
16	15	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ph /\ x e. ( 0..^ N ) ) /\ i = x ) -> ( ( F ` i ) = ( ( F |` ( 0..^ N ) ) ` x ) <-> ( F ` x ) = ( ( F |` ( 0..^ N ) ) ` x ) ) )
17		fvres 6207	. . . . . . . . . . . . . 14 |- ( x e. ( 0..^ N ) -> ( ( F |` ( 0..^ N ) ) ` x ) = ( F ` x ) )
18	17	adantl 482	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( F |` ( 0..^ N ) ) ` x ) = ( F ` x ) )
19	18	eqcomd 2628	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( F ` x ) = ( ( F |` ( 0..^ N ) ) ` x ) )
20	13, 16, 19	rspcedvd 3317	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( 0..^ N ) ) -> E. i e. ( 0..^ N ) ( F ` i ) = ( ( F |` ( 0..^ N ) ) ` x ) )
21	6	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 0..^ N ) ) -> F Fn ( 0..^ ( # ` F ) ) )
22	10	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( 0..^ N ) C_ ( 0..^ ( # ` F ) ) )
23	21, 22	fvelimabd 6254	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( ( F |` ( 0..^ N ) ) ` x ) e. ( F " ( 0..^ N ) ) <-> E. i e. ( 0..^ N ) ( F ` i ) = ( ( F |` ( 0..^ N ) ) ` x ) ) )
24	20, 23	mpbird 247	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( F |` ( 0..^ N ) ) ` x ) e. ( F " ( 0..^ N ) ) )
25	2, 4	syl 17	. . . . . . . . . . . . 13 |- ( ph -> F e. Word dom I )
26		wrdf 13310	. . . . . . . . . . . . . 14 |- ( F e. Word dom I -> F : ( 0..^ ( # ` F ) ) --> dom I )
27		ffvelrn 6357	. . . . . . . . . . . . . . . . 17 |- ( ( F : ( 0..^ ( # ` F ) ) --> dom I /\ x e. ( 0..^ ( # ` F ) ) ) -> ( F ` x ) e. dom I )
28	27	expcom 451	. . . . . . . . . . . . . . . 16 |- ( x e. ( 0..^ ( # ` F ) ) -> ( F : ( 0..^ ( # ` F ) ) --> dom I -> ( F ` x ) e. dom I ) )
29	10	sselda 3603	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ x e. ( 0..^ N ) ) -> x e. ( 0..^ ( # ` F ) ) )
30	28, 29	syl11 33	. . . . . . . . . . . . . . 15 |- ( F : ( 0..^ ( # ` F ) ) --> dom I -> ( ( ph /\ x e. ( 0..^ N ) ) -> ( F ` x ) e. dom I ) )
31	30	expd 452	. . . . . . . . . . . . . 14 |- ( F : ( 0..^ ( # ` F ) ) --> dom I -> ( ph -> ( x e. ( 0..^ N ) -> ( F ` x ) e. dom I ) ) )
32	26, 31	syl 17	. . . . . . . . . . . . 13 |- ( F e. Word dom I -> ( ph -> ( x e. ( 0..^ N ) -> ( F ` x ) e. dom I ) ) )
33	25, 32	mpcom 38	. . . . . . . . . . . 12 |- ( ph -> ( x e. ( 0..^ N ) -> ( F ` x ) e. dom I ) )
34	33	imp 445	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( F ` x ) e. dom I )
35	18, 34	eqeltrd 2701	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( F |` ( 0..^ N ) ) ` x ) e. dom I )
36	24, 35	elind 3798	. . . . . . . . 9 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( F |` ( 0..^ N ) ) ` x ) e. ( ( F " ( 0..^ N ) ) i^i dom I ) )
37		dmres 5419	. . . . . . . . 9 |- dom ( I |` ( F " ( 0..^ N ) ) ) = ( ( F " ( 0..^ N ) ) i^i dom I )
38	36, 37	syl6eleqr 2712	. . . . . . . 8 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( F |` ( 0..^ N ) ) ` x ) e. dom ( I |` ( F " ( 0..^ N ) ) ) )
39		wlkres.e	. . . . . . . . . . 11 |- ( ph -> (iEdg ` S ) = ( I |` ( F " ( 0..^ N ) ) ) )
40	39	dmeqd 5326	. . . . . . . . . 10 |- ( ph -> dom (iEdg ` S ) = dom ( I |` ( F " ( 0..^ N ) ) ) )
41	40	eleq2d 2687	. . . . . . . . 9 |- ( ph -> ( ( ( F |` ( 0..^ N ) ) ` x ) e. dom (iEdg ` S ) <-> ( ( F |` ( 0..^ N ) ) ` x ) e. dom ( I |` ( F " ( 0..^ N ) ) ) ) )
42	41	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( ( F |` ( 0..^ N ) ) ` x ) e. dom (iEdg ` S ) <-> ( ( F |` ( 0..^ N ) ) ` x ) e. dom ( I |` ( F " ( 0..^ N ) ) ) ) )
43	38, 42	mpbird 247	. . . . . . 7 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( F |` ( 0..^ N ) ) ` x ) e. dom (iEdg ` S ) )
44	43	ralrimiva 2966	. . . . . 6 |- ( ph -> A. x e. ( 0..^ N ) ( ( F |` ( 0..^ N ) ) ` x ) e. dom (iEdg ` S ) )
45		ffnfv 6388	. . . . . 6 |- ( ( F |` ( 0..^ N ) ) : ( 0..^ N ) --> dom (iEdg ` S ) <-> ( ( F |` ( 0..^ N ) ) Fn ( 0..^ N ) /\ A. x e. ( 0..^ N ) ( ( F |` ( 0..^ N ) ) ` x ) e. dom (iEdg ` S ) ) )
46	12, 44, 45	sylanbrc 698	. . . . 5 |- ( ph -> ( F |` ( 0..^ N ) ) : ( 0..^ N ) --> dom (iEdg ` S ) )
47		fzossfz 12488	. . . . . . . . 9 |- ( 0..^ ( # ` F ) ) C_ ( 0 ... ( # ` F ) )
48	47, 7	sseldi 3601	. . . . . . . 8 |- ( ph -> N e. ( 0 ... ( # ` F ) ) )
49		wlkreslem0 26565	. . . . . . . 8 |- ( ( F e. Word dom I /\ N e. ( 0 ... ( # ` F ) ) ) -> ( # ` ( F |` ( 0..^ N ) ) ) = N )
50	25, 48, 49	syl2anc 693	. . . . . . 7 |- ( ph -> ( # ` ( F |` ( 0..^ N ) ) ) = N )
51	50	oveq2d 6666	. . . . . 6 |- ( ph -> ( 0..^ ( # ` ( F |` ( 0..^ N ) ) ) ) = ( 0..^ N ) )
52	51	feq2d 6031	. . . . 5 |- ( ph -> ( ( F |` ( 0..^ N ) ) : ( 0..^ ( # ` ( F |` ( 0..^ N ) ) ) ) --> dom (iEdg ` S ) <-> ( F |` ( 0..^ N ) ) : ( 0..^ N ) --> dom (iEdg ` S ) ) )
53	46, 52	mpbird 247	. . . 4 |- ( ph -> ( F |` ( 0..^ N ) ) : ( 0..^ ( # ` ( F |` ( 0..^ N ) ) ) ) --> dom (iEdg ` S ) )
54		iswrdb 13311	. . . 4 |- ( ( F |` ( 0..^ N ) ) e. Word dom (iEdg ` S ) <-> ( F |` ( 0..^ N ) ) : ( 0..^ ( # ` ( F |` ( 0..^ N ) ) ) ) --> dom (iEdg ` S ) )
55	53, 54	sylibr 224	. . 3 |- ( ph -> ( F |` ( 0..^ N ) ) e. Word dom (iEdg ` S ) )
56	1, 55	syl5eqel 2705	. 2 |- ( ph -> H e. Word dom (iEdg ` S ) )
57		wlkres.v	. . . . . . . 8 |- V = (Vtx ` G )
58	57	wlkp 26512	. . . . . . 7 |- ( F (Walks ` G ) P -> P : ( 0 ... ( # ` F ) ) --> V )
59	2, 58	syl 17	. . . . . 6 |- ( ph -> P : ( 0 ... ( # ` F ) ) --> V )
60		wlkres.s	. . . . . . 7 |- ( ph -> (Vtx ` S ) = V )
61	60	feq3d 6032	. . . . . 6 |- ( ph -> ( P : ( 0 ... ( # ` F ) ) --> (Vtx ` S ) <-> P : ( 0 ... ( # ` F ) ) --> V ) )
62	59, 61	mpbird 247	. . . . 5 |- ( ph -> P : ( 0 ... ( # ` F ) ) --> (Vtx ` S ) )
63		elfzuz3 12339	. . . . . 6 |- ( N e. ( 0 ... ( # ` F ) ) -> ( # ` F ) e. ( ZZ>= ` N ) )
64		fzss2 12381	. . . . . 6 |- ( ( # ` F ) e. ( ZZ>= ` N ) -> ( 0 ... N ) C_ ( 0 ... ( # ` F ) ) )
65	48, 63, 64	3syl 18	. . . . 5 |- ( ph -> ( 0 ... N ) C_ ( 0 ... ( # ` F ) ) )
66	62, 65	fssresd 6071	. . . 4 |- ( ph -> ( P |` ( 0 ... N ) ) : ( 0 ... N ) --> (Vtx ` S ) )
67	1	fveq2i 6194	. . . . . . 7 |- ( # ` H ) = ( # ` ( F |` ( 0..^ N ) ) )
68	67, 50	syl5eq 2668	. . . . . 6 |- ( ph -> ( # ` H ) = N )
69	68	oveq2d 6666	. . . . 5 |- ( ph -> ( 0 ... ( # ` H ) ) = ( 0 ... N ) )
70	69	feq2d 6031	. . . 4 |- ( ph -> ( ( P |` ( 0 ... N ) ) : ( 0 ... ( # ` H ) ) --> (Vtx ` S ) <-> ( P |` ( 0 ... N ) ) : ( 0 ... N ) --> (Vtx ` S ) ) )
71	66, 70	mpbird 247	. . 3 |- ( ph -> ( P |` ( 0 ... N ) ) : ( 0 ... ( # ` H ) ) --> (Vtx ` S ) )
72		wlkres.q	. . . 4 |- Q = ( P |` ( 0 ... N ) )
73	72	feq1i 6036	. . 3 |- ( Q : ( 0 ... ( # ` H ) ) --> (Vtx ` S ) <-> ( P |` ( 0 ... N ) ) : ( 0 ... ( # ` H ) ) --> (Vtx ` S ) )
74	71, 73	sylibr 224	. 2 |- ( ph -> Q : ( 0 ... ( # ` H ) ) --> (Vtx ` S ) )
75		wlkv 26508	. . . . . . 7 |- ( F (Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
76	57, 3	iswlk 26506	. . . . . . . 8 |- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F (Walks ` G ) P <-> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) ) )
77	76	biimpd 219	. . . . . . 7 |- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F (Walks ` G ) P -> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) ) )
78	75, 77	mpcom 38	. . . . . 6 |- ( F (Walks ` G ) P -> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )
79	2, 78	syl 17	. . . . 5 |- ( ph -> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )
80	79	adantr 481	. . . 4 |- ( ( ph /\ x e. ( 0..^ ( # ` H ) ) ) -> ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )
81	68	oveq2d 6666	. . . . . . . . . . 11 |- ( ph -> ( 0..^ ( # ` H ) ) = ( 0..^ N ) )
82	81	eleq2d 2687	. . . . . . . . . 10 |- ( ph -> ( x e. ( 0..^ ( # ` H ) ) <-> x e. ( 0..^ N ) ) )
83	72	fveq1i 6192	. . . . . . . . . . . . 13 |- ( Q ` x ) = ( ( P |` ( 0 ... N ) ) ` x )
84		fzossfz 12488	. . . . . . . . . . . . . . . 16 |- ( 0..^ N ) C_ ( 0 ... N )
85	84	a1i 11	. . . . . . . . . . . . . . 15 |- ( ph -> ( 0..^ N ) C_ ( 0 ... N ) )
86	85	sselda 3603	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( 0..^ N ) ) -> x e. ( 0 ... N ) )
87	86	fvresd 6208	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( P |` ( 0 ... N ) ) ` x ) = ( P ` x ) )
88	83, 87	syl5req 2669	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( P ` x ) = ( Q ` x ) )
89	72	fveq1i 6192	. . . . . . . . . . . . 13 |- ( Q ` ( x + 1 ) ) = ( ( P |` ( 0 ... N ) ) ` ( x + 1 ) )
90		fzofzp1 12565	. . . . . . . . . . . . . . 15 |- ( x e. ( 0..^ N ) -> ( x + 1 ) e. ( 0 ... N ) )
91	90	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( x + 1 ) e. ( 0 ... N ) )
92	91	fvresd 6208	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( P |` ( 0 ... N ) ) ` ( x + 1 ) ) = ( P ` ( x + 1 ) ) )
93	89, 92	syl5req 2669	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) )
94	88, 93	jca 554	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) )
95	94	ex 450	. . . . . . . . . 10 |- ( ph -> ( x e. ( 0..^ N ) -> ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) ) )
96	82, 95	sylbid 230	. . . . . . . . 9 |- ( ph -> ( x e. ( 0..^ ( # ` H ) ) -> ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) ) )
97	96	imp 445	. . . . . . . 8 |- ( ( ph /\ x e. ( 0..^ ( # ` H ) ) ) -> ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) )
98	25	ancli 574	. . . . . . . . . . . . . 14 |- ( ph -> ( ph /\ F e. Word dom I ) )
99	26	ffund 6049	. . . . . . . . . . . . . . . . 17 |- ( F e. Word dom I -> Fun F )
100	99	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ F e. Word dom I ) -> Fun F )
101	100	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ F e. Word dom I ) /\ x e. ( 0..^ N ) ) -> Fun F )
102		fdm 6051	. . . . . . . . . . . . . . . . . 18 |- ( F : ( 0..^ ( # ` F ) ) --> dom I -> dom F = ( 0..^ ( # ` F ) ) )
103		sseq2 3627	. . . . . . . . . . . . . . . . . . 19 |- ( dom F = ( 0..^ ( # ` F ) ) -> ( ( 0..^ N ) C_ dom F <-> ( 0..^ N ) C_ ( 0..^ ( # ` F ) ) ) )
104	10, 103	syl5ibr 236	. . . . . . . . . . . . . . . . . 18 |- ( dom F = ( 0..^ ( # ` F ) ) -> ( ph -> ( 0..^ N ) C_ dom F ) )
105	26, 102, 104	3syl 18	. . . . . . . . . . . . . . . . 17 |- ( F e. Word dom I -> ( ph -> ( 0..^ N ) C_ dom F ) )
106	105	impcom 446	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ F e. Word dom I ) -> ( 0..^ N ) C_ dom F )
107	106	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ F e. Word dom I ) /\ x e. ( 0..^ N ) ) -> ( 0..^ N ) C_ dom F )
108		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ F e. Word dom I ) /\ x e. ( 0..^ N ) ) -> x e. ( 0..^ N ) )
109	101, 107, 108	resfvresima 6494	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ F e. Word dom I ) /\ x e. ( 0..^ N ) ) -> ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) = ( I ` ( F ` x ) ) )
110	98, 109	sylan 488	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) = ( I ` ( F ` x ) ) )
111	110	eqcomd 2628	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( I ` ( F ` x ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) )
112	111	ex 450	. . . . . . . . . . 11 |- ( ph -> ( x e. ( 0..^ N ) -> ( I ` ( F ` x ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) ) )
113	82, 112	sylbid 230	. . . . . . . . . 10 |- ( ph -> ( x e. ( 0..^ ( # ` H ) ) -> ( I ` ( F ` x ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) ) )
114	113	imp 445	. . . . . . . . 9 |- ( ( ph /\ x e. ( 0..^ ( # ` H ) ) ) -> ( I ` ( F ` x ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) )
115	39	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 0..^ ( # ` H ) ) ) -> (iEdg ` S ) = ( I |` ( F " ( 0..^ N ) ) ) )
116	1	fveq1i 6192	. . . . . . . . . . 11 |- ( H ` x ) = ( ( F |` ( 0..^ N ) ) ` x )
117	116	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 0..^ ( # ` H ) ) ) -> ( H ` x ) = ( ( F |` ( 0..^ N ) ) ` x ) )
118	115, 117	fveq12d 6197	. . . . . . . . 9 |- ( ( ph /\ x e. ( 0..^ ( # ` H ) ) ) -> ( (iEdg ` S ) ` ( H ` x ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) )
119	114, 118	eqtr4d 2659	. . . . . . . 8 |- ( ( ph /\ x e. ( 0..^ ( # ` H ) ) ) -> ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) )
120	97, 119	jca 554	. . . . . . 7 |- ( ( ph /\ x e. ( 0..^ ( # ` H ) ) ) -> ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) )
121	7, 8	syl 17	. . . . . . . . . . 11 |- ( ph -> ( # ` F ) e. ( ZZ>= ` N ) )
122	1	a1i 11	. . . . . . . . . . . . . 14 |- ( ph -> H = ( F |` ( 0..^ N ) ) )
123	122	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ph -> ( # ` H ) = ( # ` ( F |` ( 0..^ N ) ) ) )
124	123, 50	eqtrd 2656	. . . . . . . . . . . 12 |- ( ph -> ( # ` H ) = N )
125	124	fveq2d 6195	. . . . . . . . . . 11 |- ( ph -> ( ZZ>= ` ( # ` H ) ) = ( ZZ>= ` N ) )
126	121, 125	eleqtrrd 2704	. . . . . . . . . 10 |- ( ph -> ( # ` F ) e. ( ZZ>= ` ( # ` H ) ) )
127		fzoss2 12496	. . . . . . . . . 10 |- ( ( # ` F ) e. ( ZZ>= ` ( # ` H ) ) -> ( 0..^ ( # ` H ) ) C_ ( 0..^ ( # ` F ) ) )
128	126, 127	syl 17	. . . . . . . . 9 |- ( ph -> ( 0..^ ( # ` H ) ) C_ ( 0..^ ( # ` F ) ) )
129	128	sselda 3603	. . . . . . . 8 |- ( ( ph /\ x e. ( 0..^ ( # ` H ) ) ) -> x e. ( 0..^ ( # ` F ) ) )
130		wkslem1 26503	. . . . . . . . 9 |- ( k = x -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) <-> if- ( ( P ` x ) = ( P ` ( x + 1 ) ) , ( I ` ( F ` x ) ) = { ( P ` x ) } , { ( P ` x ) , ( P ` ( x + 1 ) ) } C_ ( I ` ( F ` x ) ) ) ) )
131	130	rspcv 3305	. . . . . . . 8 |- ( x e. ( 0..^ ( # ` F ) ) -> ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> if- ( ( P ` x ) = ( P ` ( x + 1 ) ) , ( I ` ( F ` x ) ) = { ( P ` x ) } , { ( P ` x ) , ( P ` ( x + 1 ) ) } C_ ( I ` ( F ` x ) ) ) ) )
132	129, 131	syl 17	. . . . . . 7 |- ( ( ph /\ x e. ( 0..^ ( # ` H ) ) ) -> ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> if- ( ( P ` x ) = ( P ` ( x + 1 ) ) , ( I ` ( F ` x ) ) = { ( P ` x ) } , { ( P ` x ) , ( P ` ( x + 1 ) ) } C_ ( I ` ( F ` x ) ) ) ) )
133		eqeq12 2635	. . . . . . . . . 10 |- ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) -> ( ( P ` x ) = ( P ` ( x + 1 ) ) <-> ( Q ` x ) = ( Q ` ( x + 1 ) ) ) )
134	133	adantr 481	. . . . . . . . 9 |- ( ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) -> ( ( P ` x ) = ( P ` ( x + 1 ) ) <-> ( Q ` x ) = ( Q ` ( x + 1 ) ) ) )
135		simpr 477	. . . . . . . . . 10 |- ( ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) -> ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) )
136		sneq 4187	. . . . . . . . . . . 12 |- ( ( P ` x ) = ( Q ` x ) -> { ( P ` x ) } = { ( Q ` x ) } )
137	136	adantr 481	. . . . . . . . . . 11 |- ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) -> { ( P ` x ) } = { ( Q ` x ) } )
138	137	adantr 481	. . . . . . . . . 10 |- ( ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) -> { ( P ` x ) } = { ( Q ` x ) } )
139	135, 138	eqeq12d 2637	. . . . . . . . 9 |- ( ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) -> ( ( I ` ( F ` x ) ) = { ( P ` x ) } <-> ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } ) )
140		preq12 4270	. . . . . . . . . . 11 |- ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) -> { ( P ` x ) , ( P ` ( x + 1 ) ) } = { ( Q ` x ) , ( Q ` ( x + 1 ) ) } )
141	140	adantr 481	. . . . . . . . . 10 |- ( ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) -> { ( P ` x ) , ( P ` ( x + 1 ) ) } = { ( Q ` x ) , ( Q ` ( x + 1 ) ) } )
142	141, 135	sseq12d 3634	. . . . . . . . 9 |- ( ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) -> ( { ( P ` x ) , ( P ` ( x + 1 ) ) } C_ ( I ` ( F ` x ) ) <-> { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) )
143	134, 139, 142	ifpbi123d 1027	. . . . . . . 8 |- ( ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) -> (if- ( ( P ` x ) = ( P ` ( x + 1 ) ) , ( I ` ( F ` x ) ) = { ( P ` x ) } , { ( P ` x ) , ( P ` ( x + 1 ) ) } C_ ( I ` ( F ` x ) ) ) <-> if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) ) )
144	143	biimpd 219	. . . . . . 7 |- ( ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) -> (if- ( ( P ` x ) = ( P ` ( x + 1 ) ) , ( I ` ( F ` x ) ) = { ( P ` x ) } , { ( P ` x ) , ( P ` ( x + 1 ) ) } C_ ( I ` ( F ` x ) ) ) -> if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) ) )
145	120, 132, 144	sylsyld 61	. . . . . 6 |- ( ( ph /\ x e. ( 0..^ ( # ` H ) ) ) -> ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) ) )
146	145	com12 32	. . . . 5 |- ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> ( ( ph /\ x e. ( 0..^ ( # ` H ) ) ) -> if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) ) )
147	146	3ad2ant3 1084	. . . 4 |- ( ( F e. Word dom I /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> ( ( ph /\ x e. ( 0..^ ( # ` H ) ) ) -> if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) ) )
148	80, 147	mpcom 38	. . 3 |- ( ( ph /\ x e. ( 0..^ ( # ` H ) ) ) -> if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) )
149	148	ralrimiva 2966	. 2 |- ( ph -> A. x e. ( 0..^ ( # ` H ) )if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) )
150	57, 3, 2, 7, 60, 39, 1, 72	wlkreslem 26566	. . 3 |- ( ph -> ( S e. _V /\ H e. _V /\ Q e. _V ) )
151		eqid 2622	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
152		eqid 2622	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
153	151, 152	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. x e. ( 0..^ ( # ` H ) )if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) ) ) )
154	150, 153	syl 17	. 2 |- ( ph -> ( H (Walks ` S ) Q <-> ( H e. Word dom (iEdg ` S ) /\ Q : ( 0 ... ( # ` H ) ) --> (Vtx ` S ) /\ A. x e. ( 0..^ ( # ` H ) )if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) ) ) )
155	56, 74, 149, 154	mpbir3and 1245	1 |- ( ph -> H (Walks ` S ) Q )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384  if-wif 1012   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {csn 4177   {cpr 4179   class class class wbr 4653   dom cdm 5114   |` cres 5116   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291  Vtxcvtx 25874  iEdgciedg 25875  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-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-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-substr 13303  df-wlks 26495

This theorem is referenced by:  trlres  26597  eupthres  27075