Theorem wlkres 16603

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. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 5-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.)

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 prefix N )
wlkres.q	|- Q = ( P |` ( 0 ... N ) )

Assertion

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

Proof of Theorem wlkres

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

Step	Hyp	Ref	Expression
1		wlkres.d	. . . . 5 |- ( ph -> F (Walks ` G ) P )
2		wlkres.i	. . . . . 6 |- I = (iEdg ` G )
3	2	wlkf 16554	. . . . 5 |- ( F (Walks ` G ) P -> F e. Word dom I )
4	1, 3	syl 14	. . . 4 |- ( ph -> F e. Word dom I )
5		wlkres.n	. . . . 5 |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )
6		elfzonn0 10581	. . . . 5 |- ( N e. ( 0..^ (♯ ` F ) ) -> N e. NN0 )
7	5, 6	syl 14	. . . 4 |- ( ph -> N e. NN0 )
8		pfxwrdsymbg 11445	. . . 4 |- ( ( F e. Word dom I /\ N e. NN0 ) -> ( F prefix N ) e. Word ( F " ( 0..^ N ) ) )
9	4, 7, 8	syl2anc 415	. . 3 |- ( ph -> ( F prefix N ) e. Word ( F " ( 0..^ N ) ) )
10		wlkres.h	. . . 4 |- H = ( F prefix N )
11	10	a1i 9	. . 3 |- ( ph -> H = ( F prefix N ) )
12		wlkres.e	. . . . . 6 |- ( ph -> (iEdg ` S ) = ( I |` ( F " ( 0..^ N ) ) ) )
13	12	dmeqd 4981	. . . . 5 |- ( ph -> dom (iEdg ` S ) = dom ( I |` ( F " ( 0..^ N ) ) ) )
14		wrdf 11293	. . . . . . 7 |- ( F e. Word dom I -> F : ( 0..^ (♯ ` F ) ) --> dom I )
15		fimass 5548	. . . . . . 7 |- ( F : ( 0..^ (♯ ` F ) ) --> dom I -> ( F " ( 0..^ N ) ) C_ dom I )
16	4, 14, 15	3syl 17	. . . . . 6 |- ( ph -> ( F " ( 0..^ N ) ) C_ dom I )
17		ssdmres 5083	. . . . . 6 |- ( ( F " ( 0..^ N ) ) C_ dom I <-> dom ( I |` ( F " ( 0..^ N ) ) ) = ( F " ( 0..^ N ) ) )
18	16, 17	sylib 122	. . . . 5 |- ( ph -> dom ( I |` ( F " ( 0..^ N ) ) ) = ( F " ( 0..^ N ) ) )
19	13, 18	eqtrd 2271	. . . 4 |- ( ph -> dom (iEdg ` S ) = ( F " ( 0..^ N ) ) )
20		wrdeq 11309	. . . 4 |- ( dom (iEdg ` S ) = ( F " ( 0..^ N ) ) -> Word dom (iEdg ` S ) = Word ( F " ( 0..^ N ) ) )
21	19, 20	syl 14	. . 3 |- ( ph -> Word dom (iEdg ` S ) = Word ( F " ( 0..^ N ) ) )
22	9, 11, 21	3eltr4d 2322	. 2 |- ( ph -> H e. Word dom (iEdg ` S ) )
23		wlkres.v	. . . . . . . 8 |- V = (Vtx ` G )
24	23	wlkp 16558	. . . . . . 7 |- ( F (Walks ` G ) P -> P : ( 0 ... (♯ ` F ) ) --> V )
25	1, 24	syl 14	. . . . . 6 |- ( ph -> P : ( 0 ... (♯ ` F ) ) --> V )
26		wlkres.s	. . . . . . 7 |- ( ph -> (Vtx ` S ) = V )
27	26	feq3d 5520	. . . . . 6 |- ( ph -> ( P : ( 0 ... (♯ ` F ) ) --> (Vtx ` S ) <-> P : ( 0 ... (♯ ` F ) ) --> V ) )
28	25, 27	mpbird 167	. . . . 5 |- ( ph -> P : ( 0 ... (♯ ` F ) ) --> (Vtx ` S ) )
29		fzossfz 10556	. . . . . . 7 |- ( 0..^ (♯ ` F ) ) C_ ( 0 ... (♯ ` F ) )
30	29, 5	sselid 3246	. . . . . 6 |- ( ph -> N e. ( 0 ... (♯ ` F ) ) )
31		elfzuz3 10408	. . . . . 6 |- ( N e. ( 0 ... (♯ ` F ) ) -> (♯ ` F ) e. ( ZZ>= ` N ) )
32		fzss2 10453	. . . . . 6 |- ( (♯ ` F ) e. ( ZZ>= ` N ) -> ( 0 ... N ) C_ ( 0 ... (♯ ` F ) ) )
33	30, 31, 32	3syl 17	. . . . 5 |- ( ph -> ( 0 ... N ) C_ ( 0 ... (♯ ` F ) ) )
34	28, 33	fssresd 5564	. . . 4 |- ( ph -> ( P |` ( 0 ... N ) ) : ( 0 ... N ) --> (Vtx ` S ) )
35	10	fveq2i 5696	. . . . . . 7 |- (♯ ` H ) = (♯ ` ( F prefix N ) )
36		pfxlen 11440	. . . . . . . 8 |- ( ( F e. Word dom I /\ N e. ( 0 ... (♯ ` F ) ) ) -> (♯ ` ( F prefix N ) ) = N )
37	4, 30, 36	syl2anc 415	. . . . . . 7 |- ( ph -> (♯ ` ( F prefix N ) ) = N )
38	35, 37	eqtrid 2283	. . . . . 6 |- ( ph -> (♯ ` H ) = N )
39	38	oveq2d 6095	. . . . 5 |- ( ph -> ( 0 ... (♯ ` H ) ) = ( 0 ... N ) )
40	39	feq2d 5519	. . . 4 |- ( ph -> ( ( P |` ( 0 ... N ) ) : ( 0 ... (♯ ` H ) ) --> (Vtx ` S ) <-> ( P |` ( 0 ... N ) ) : ( 0 ... N ) --> (Vtx ` S ) ) )
41	34, 40	mpbird 167	. . 3 |- ( ph -> ( P |` ( 0 ... N ) ) : ( 0 ... (♯ ` H ) ) --> (Vtx ` S ) )
42		wlkres.q	. . . 4 |- Q = ( P |` ( 0 ... N ) )
43	42	feq1i 5524	. . 3 |- ( Q : ( 0 ... (♯ ` H ) ) --> (Vtx ` S ) <-> ( P |` ( 0 ... N ) ) : ( 0 ... (♯ ` H ) ) --> (Vtx ` S ) )
44	41, 43	sylibr 134	. 2 |- ( ph -> Q : ( 0 ... (♯ ` H ) ) --> (Vtx ` S ) )
45	23, 2	wlkprop 16551	. . . . . 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 ) ) ) ) )
46	1, 45	syl 14	. . . . 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 ) ) ) ) )
47	46	adantr 276	. . . 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 ) ) ) ) )
48	38	oveq2d 6095	. . . . . . . . . . 11 |- ( ph -> ( 0..^ (♯ ` H ) ) = ( 0..^ N ) )
49	48	eleq2d 2308	. . . . . . . . . 10 |- ( ph -> ( x e. ( 0..^ (♯ ` H ) ) <-> x e. ( 0..^ N ) ) )
50	42	fveq1i 5694	. . . . . . . . . . . . 13 |- ( Q ` x ) = ( ( P |` ( 0 ... N ) ) ` x )
51		fzossfz 10556	. . . . . . . . . . . . . . . 16 |- ( 0..^ N ) C_ ( 0 ... N )
52	51	a1i 9	. . . . . . . . . . . . . . 15 |- ( ph -> ( 0..^ N ) C_ ( 0 ... N ) )
53	52	sselda 3248	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( 0..^ N ) ) -> x e. ( 0 ... N ) )
54	53	fvresd 5718	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( P |` ( 0 ... N ) ) ` x ) = ( P ` x ) )
55	50, 54	eqtr2id 2284	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( P ` x ) = ( Q ` x ) )
56	42	fveq1i 5694	. . . . . . . . . . . . 13 |- ( Q ` ( x + 1 ) ) = ( ( P |` ( 0 ... N ) ) ` ( x + 1 ) )
57		fzofzp1 10628	. . . . . . . . . . . . . . 15 |- ( x e. ( 0..^ N ) -> ( x + 1 ) e. ( 0 ... N ) )
58	57	adantl 277	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( x + 1 ) e. ( 0 ... N ) )
59	58	fvresd 5718	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( P |` ( 0 ... N ) ) ` ( x + 1 ) ) = ( P ` ( x + 1 ) ) )
60	56, 59	eqtr2id 2284	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) )
61	55, 60	jca 306	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) )
62	61	ex 115	. . . . . . . . . 10 |- ( ph -> ( x e. ( 0..^ N ) -> ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) ) )
63	49, 62	sylbid 150	. . . . . . . . 9 |- ( ph -> ( x e. ( 0..^ (♯ ` H ) ) -> ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) ) )
64	63	imp 124	. . . . . . . 8 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) )
65	4	ancli 323	. . . . . . . . . . . . . 14 |- ( ph -> ( ph /\ F e. Word dom I ) )
66	14	ffund 5535	. . . . . . . . . . . . . . . . 17 |- ( F e. Word dom I -> Fun F )
67	66	adantl 277	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ F e. Word dom I ) -> Fun F )
68	67	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ F e. Word dom I ) /\ x e. ( 0..^ N ) ) -> Fun F )
69		fdm 5537	. . . . . . . . . . . . . . . . . 18 |- ( F : ( 0..^ (♯ ` F ) ) --> dom I -> dom F = ( 0..^ (♯ ` F ) ) )
70		elfzouz2 10552	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. ( 0..^ (♯ ` F ) ) -> (♯ ` F ) e. ( ZZ>= ` N ) )
71		fzoss2 10564	. . . . . . . . . . . . . . . . . . . 20 |- ( (♯ ` F ) e. ( ZZ>= ` N ) -> ( 0..^ N ) C_ ( 0..^ (♯ ` F ) ) )
72	5, 70, 71	3syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( 0..^ N ) C_ ( 0..^ (♯ ` F ) ) )
73		sseq2 3272	. . . . . . . . . . . . . . . . . . 19 |- ( dom F = ( 0..^ (♯ ` F ) ) -> ( ( 0..^ N ) C_ dom F <-> ( 0..^ N ) C_ ( 0..^ (♯ ` F ) ) ) )
74	72, 73	imbitrrid 156	. . . . . . . . . . . . . . . . . 18 |- ( dom F = ( 0..^ (♯ ` F ) ) -> ( ph -> ( 0..^ N ) C_ dom F ) )
75	14, 69, 74	3syl 17	. . . . . . . . . . . . . . . . 17 |- ( F e. Word dom I -> ( ph -> ( 0..^ N ) C_ dom F ) )
76	75	impcom 125	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ F e. Word dom I ) -> ( 0..^ N ) C_ dom F )
77	76	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ F e. Word dom I ) /\ x e. ( 0..^ N ) ) -> ( 0..^ N ) C_ dom F )
78		simpr 110	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ F e. Word dom I ) /\ x e. ( 0..^ N ) ) -> x e. ( 0..^ N ) )
79	68, 77, 78	resfvresima 5950	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ F e. Word dom I ) /\ x e. ( 0..^ N ) ) -> ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) = ( I ` ( F ` x ) ) )
80	65, 79	sylan 283	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) = ( I ` ( F ` x ) ) )
81	80	eqcomd 2244	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( I ` ( F ` x ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) )
82	81	ex 115	. . . . . . . . . . 11 |- ( ph -> ( x e. ( 0..^ N ) -> ( I ` ( F ` x ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) ) )
83	49, 82	sylbid 150	. . . . . . . . . 10 |- ( ph -> ( x e. ( 0..^ (♯ ` H ) ) -> ( I ` ( F ` x ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) ) )
84	83	imp 124	. . . . . . . . 9 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( I ` ( F ` x ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) )
85	12	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> (iEdg ` S ) = ( I |` ( F " ( 0..^ N ) ) ) )
86	10	fveq1i 5694	. . . . . . . . . . 11 |- ( H ` x ) = ( ( F prefix N ) ` x )
87	4	adantr 276	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> F e. Word dom I )
88	30	adantr 276	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> N e. ( 0 ... (♯ ` F ) ) )
89		pfxres 11436	. . . . . . . . . . . . 13 |- ( ( F e. Word dom I /\ N e. ( 0 ... (♯ ` F ) ) ) -> ( F prefix N ) = ( F |` ( 0..^ N ) ) )
90	87, 88, 89	syl2anc 415	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( F prefix N ) = ( F |` ( 0..^ N ) ) )
91	90	fveq1d 5695	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( ( F prefix N ) ` x ) = ( ( F |` ( 0..^ N ) ) ` x ) )
92	86, 91	eqtrid 2283	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( H ` x ) = ( ( F |` ( 0..^ N ) ) ` x ) )
93	85, 92	fveq12d 5700	. . . . . . . . 9 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( (iEdg ` S ) ` ( H ` x ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) )
94	84, 93	eqtr4d 2274	. . . . . . . 8 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) )
95	64, 94	jca 306	. . . . . . 7 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) )
96	5, 70	syl 14	. . . . . . . . . . 11 |- ( ph -> (♯ ` F ) e. ( ZZ>= ` N ) )
97	38	fveq2d 5697	. . . . . . . . . . 11 |- ( ph -> ( ZZ>= ` (♯ ` H ) ) = ( ZZ>= ` N ) )
98	96, 97	eleqtrrd 2318	. . . . . . . . . 10 |- ( ph -> (♯ ` F ) e. ( ZZ>= ` (♯ ` H ) ) )
99		fzoss2 10564	. . . . . . . . . 10 |- ( (♯ ` F ) e. ( ZZ>= ` (♯ ` H ) ) -> ( 0..^ (♯ ` H ) ) C_ ( 0..^ (♯ ` F ) ) )
100	98, 99	syl 14	. . . . . . . . 9 |- ( ph -> ( 0..^ (♯ ` H ) ) C_ ( 0..^ (♯ ` F ) ) )
101	100	sselda 3248	. . . . . . . 8 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> x e. ( 0..^ (♯ ` F ) ) )
102		wkslem1 16544	. . . . . . . . 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 ) ) ) ) )
103	102	rspcv 2925	. . . . . . . 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 ) ) ) ) )
104	101, 103	syl 14	. . . . . . 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 ) ) ) ) )
105		eqeq12 2251	. . . . . . . . . 10 |- ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) -> ( ( P ` x ) = ( P ` ( x + 1 ) ) <-> ( Q ` x ) = ( Q ` ( x + 1 ) ) ) )
106	105	adantr 276	. . . . . . . . 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 ) ) ) )
107		simpr 110	. . . . . . . . . 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 ) ) )
108		sneq 3719	. . . . . . . . . . . 12 |- ( ( P ` x ) = ( Q ` x ) -> { ( P ` x ) } = { ( Q ` x ) } )
109	108	adantr 276	. . . . . . . . . . 11 |- ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) -> { ( P ` x ) } = { ( Q ` x ) } )
110	109	adantr 276	. . . . . . . . . 10 |- ( ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) -> { ( P ` x ) } = { ( Q ` x ) } )
111	107, 110	eqeq12d 2253	. . . . . . . . 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 ) } ) )
112		preq12 3789	. . . . . . . . . . 11 |- ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) -> { ( P ` x ) , ( P ` ( x + 1 ) ) } = { ( Q ` x ) , ( Q ` ( x + 1 ) ) } )
113	112	adantr 276	. . . . . . . . . 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 ) ) } )
114	113, 107	sseq12d 3279	. . . . . . . . 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 ) ) ) )
115	106, 111, 114	ifpbi123d 1005	. . . . . . . 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 ) ) ) ) )
116	115	biimpd 144	. . . . . . 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 ) ) ) ) )
117	95, 104, 116	sylsyld 58	. . . . . 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 ) ) ) ) )
118	117	com12 30	. . . . 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 ) ) ) ) )
119	118	3ad2ant3 1051	. . . 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 ) ) ) ) )
120	47, 119	mpcom 36	. . 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 ) ) ) )
121	120	ralrimiva 2623	. 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 ) ) ) )
122	23, 2, 1, 5, 26	wlkreslem 16602	. . 3 |- ( ph -> S e. _V )
123		eqid 2238	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
124		eqid 2238	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
125	123, 124	iswlkg 16553	. . 3 |- ( S 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 ) ) ) ) ) )
126	122, 125	syl 14	. 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 ) ) ) ) ) )
127	22, 44, 121, 126	mpbir3and 1211	1 |- ( ph -> H (Walks ` S ) Q )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  if-wif 990   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   C_ wss 3220   {csn 3708   {cpr 3709   class class class wbr 4128   dom cdm 4772   |` cres 4774   "cima 4775   Fun wfun 5369   -->wf 5371   ` cfv 5375  (class class class)co 6079   0cc0 8173   1c1 8174   + caddc 8176   NN0cn0 9546   ZZ>=cuz 9904   ...cfz 10394  ..^cfzo 10532  ♯chash 11197  Word cword 11287   prefix cpfx 11427  Vtxcvtx 16236  iEdgciedg 16237  Walkscwlks 16541

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 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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-ifp 991  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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-1o 6681  df-er 6801  df-map 6918  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-z 9628  df-dec 9761  df-uz 9905  df-fz 10395  df-fzo 10533  df-ihash 11198  df-word 11288  df-substr 11401  df-pfx 11428  df-ndx 13338  df-slot 13339  df-base 13341  df-edgf 16229  df-vtx 16238  df-iedg 16239  df-wlks 16542

This theorem is referenced by:  trlres  16614  eupthres  16681