Theorem usgra2wlkspthlem1 30205

Description: Lemma 1 for usgra2wlkspth 30207. (Contributed by Alexander van der Vekens, 2-Mar-2018.)

Assertion

Ref	Expression
usgra2wlkspthlem1	|- ( ( F e. Word dom E /\ E : dom E -1-1-> ran E /\ ( # ` F ) = 2 ) -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> Fun `' F ) )

Distinct variable groups:   i, E   i, F   P, i

Allowed substitution hints:   A( i)   B( i)

Proof of Theorem usgra2wlkspthlem1

Step	Hyp	Ref	Expression
1		wrdf 12236	. . 3 |- ( F e. Word dom E -> F : ( 0..^ ( # ` F ) ) --> dom E )
2		simpl 454	. . . . . . . . . . . . 13 |- ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) -> F : ( 0..^ ( # ` F ) ) --> dom E )
3	2	adantr 462	. . . . . . . . . . . 12 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> F : ( 0..^ ( # ` F ) ) --> dom E )
4		oveq2 6098	. . . . . . . . . . . . . . . . 17 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = ( 0..^ 2 ) )
5		fzo0to2pr 11610	. . . . . . . . . . . . . . . . 17 |- ( 0..^ 2 ) = { 0 , 1 }
6	4, 5	syl6req 2490	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) = 2 -> { 0 , 1 } = ( 0..^ ( # ` F ) ) )
7	6	adantr 462	. . . . . . . . . . . . . . 15 |- ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> { 0 , 1 } = ( 0..^ ( # ` F ) ) )
8	7	adantl 463	. . . . . . . . . . . . . 14 |- ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) -> { 0 , 1 } = ( 0..^ ( # ` F ) ) )
9	8	adantr 462	. . . . . . . . . . . . 13 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> { 0 , 1 } = ( 0..^ ( # ` F ) ) )
10	9	feq2d 5544	. . . . . . . . . . . 12 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> ( F : { 0 , 1 } --> dom E <-> F : ( 0..^ ( # ` F ) ) --> dom E ) )
11	3, 10	mpbird 232	. . . . . . . . . . 11 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> F : { 0 , 1 } --> dom E )
12		preq1 3951	. . . . . . . . . . . . . . . . . . . 20 |- ( ( P ` 0 ) = A -> { ( P ` 0 ) , ( P ` 1 ) } = { A , ( P ` 1 ) } )
13	12	eqeq2d 2452	. . . . . . . . . . . . . . . . . . 19 |- ( ( P ` 0 ) = A -> ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } <-> ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } ) )
14		preq2 3952	. . . . . . . . . . . . . . . . . . . 20 |- ( ( P ` 2 ) = B -> { ( P ` 1 ) , ( P ` 2 ) } = { ( P ` 1 ) , B } )
15	14	eqeq2d 2452	. . . . . . . . . . . . . . . . . . 19 |- ( ( P ` 2 ) = B -> ( ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } <-> ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) )
16	13, 15	bi2anan9 863	. . . . . . . . . . . . . . . . . 18 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) <-> ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) )
17	16	3adant3 1003	. . . . . . . . . . . . . . . . 17 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) -> ( ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) <-> ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) )
18		fveq2 5688	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( F ` 0 ) = ( F ` 1 ) -> ( E ` ( F ` 0 ) ) = ( E ` ( F ` 1 ) ) )
19	18	eqeq1d 2449	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( F ` 0 ) = ( F ` 1 ) -> ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } <-> ( E ` ( F ` 1 ) ) = { A , ( P ` 1 ) } ) )
20	19	anbi1d 699	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( F ` 0 ) = ( F ` 1 ) -> ( ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) <-> ( ( E ` ( F ` 1 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) )
21	20	anbi2d 698	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( F ` 0 ) = ( F ` 1 ) -> ( ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) <-> ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( E ` ( F ` 1 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) ) )
22		eqtr2 2459	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( E ` ( F ` 1 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) -> { A , ( P ` 1 ) } = { ( P ` 1 ) , B } )
23		fvex 5698	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( P ` 0 ) e. _V
24	23	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( P ` 0 ) = A -> ( P ` 0 ) e. _V )
25		eleq1 2501	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( A = ( P ` 0 ) -> ( A e. _V <-> ( P ` 0 ) e. _V ) )
26	25	eqcoms 2444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( P ` 0 ) = A -> ( A e. _V <-> ( P ` 0 ) e. _V ) )
27	24, 26	mpbird 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( P ` 0 ) = A -> A e. _V )
28		fvex 5698	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( P ` 1 ) e. _V
29	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( P ` 0 ) = A -> ( P ` 1 ) e. _V )
30	27, 29	jca 529	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( P ` 0 ) = A -> ( A e. _V /\ ( P ` 1 ) e. _V ) )
31	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( P ` 2 ) = B -> ( P ` 1 ) e. _V )
32		fvex 5698	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( P ` 2 ) e. _V
33	32	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( P ` 2 ) = B -> ( P ` 2 ) e. _V )
34		eleq1 2501	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( B = ( P ` 2 ) -> ( B e. _V <-> ( P ` 2 ) e. _V ) )
35	34	eqcoms 2444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( P ` 2 ) = B -> ( B e. _V <-> ( P ` 2 ) e. _V ) )
36	33, 35	mpbird 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( P ` 2 ) = B -> B e. _V )
37	31, 36	jca 529	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( P ` 2 ) = B -> ( ( P ` 1 ) e. _V /\ B e. _V ) )
38	30, 37	anim12i 563	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( ( A e. _V /\ ( P ` 1 ) e. _V ) /\ ( ( P ` 1 ) e. _V /\ B e. _V ) ) )
39	38	adantr 462	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) -> ( ( A e. _V /\ ( P ` 1 ) e. _V ) /\ ( ( P ` 1 ) e. _V /\ B e. _V ) ) )
40		preq12bg 4048	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( ( A e. _V /\ ( P ` 1 ) e. _V ) /\ ( ( P ` 1 ) e. _V /\ B e. _V ) ) -> ( { A , ( P ` 1 ) } = { ( P ` 1 ) , B } <-> ( ( A = ( P ` 1 ) /\ ( P ` 1 ) = B ) \/ ( A = B /\ ( P ` 1 ) = ( P ` 1 ) ) ) ) )
41	39, 40	syl 16	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) -> ( { A , ( P ` 1 ) } = { ( P ` 1 ) , B } <-> ( ( A = ( P ` 1 ) /\ ( P ` 1 ) = B ) \/ ( A = B /\ ( P ` 1 ) = ( P ` 1 ) ) ) ) )
42		eqtr 2458	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( A = ( P ` 1 ) /\ ( P ` 1 ) = B ) -> A = B )
43		simpl 454	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( A = B /\ ( P ` 1 ) = ( P ` 1 ) ) -> A = B )
44	42, 43	jaoi 379	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( A = ( P ` 1 ) /\ ( P ` 1 ) = B ) \/ ( A = B /\ ( P ` 1 ) = ( P ` 1 ) ) ) -> A = B )
45	41, 44	syl6bi 228	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) -> ( { A , ( P ` 1 ) } = { ( P ` 1 ) , B } -> A = B ) )
46	22, 45	syl5com 30	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( E ` ( F ` 1 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) -> ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) -> A = B ) )
47	46	impcom 430	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( E ` ( F ` 1 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) -> A = B )
48	21, 47	syl6bi 228	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( F ` 0 ) = ( F ` 1 ) -> ( ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) -> A = B ) )
49	48	com12 31	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) -> ( ( F ` 0 ) = ( F ` 1 ) -> A = B ) )
50	49	necon3d 2644	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) -> ( A =/= B -> ( F ` 0 ) =/= ( F ` 1 ) ) )
51	50	a1d 25	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) ) -> ( F : ( 0..^ 2 ) --> dom E -> ( A =/= B -> ( F ` 0 ) =/= ( F ` 1 ) ) ) )
52	51	exp31 601	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) -> ( F : ( 0..^ 2 ) --> dom E -> ( A =/= B -> ( F ` 0 ) =/= ( F ` 1 ) ) ) ) ) )
53	52	com25 91	. . . . . . . . . . . . . . . . . 18 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( A =/= B -> ( ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) -> ( F : ( 0..^ 2 ) --> dom E -> ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) ) ) )
54	53	3impia 1179	. . . . . . . . . . . . . . . . 17 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) -> ( ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) -> ( F : ( 0..^ 2 ) --> dom E -> ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) ) )
55	17, 54	sylbid 215	. . . . . . . . . . . . . . . 16 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) -> ( ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) -> ( F : ( 0..^ 2 ) --> dom E -> ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) ) )
56	55	imp 429	. . . . . . . . . . . . . . 15 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( F : ( 0..^ 2 ) --> dom E -> ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) )
57	56	com13 80	. . . . . . . . . . . . . 14 |- ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( F : ( 0..^ 2 ) --> dom E -> ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) )
58	4	feq2d 5544	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) = 2 -> ( F : ( 0..^ ( # ` F ) ) --> dom E <-> F : ( 0..^ 2 ) --> dom E ) )
59		fveq2 5688	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 2 -> ( P ` ( # ` F ) ) = ( P ` 2 ) )
60	59	eqeq1d 2449	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` F ) = 2 -> ( ( P ` ( # ` F ) ) = B <-> ( P ` 2 ) = B ) )
61	60	3anbi2d 1289	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` F ) = 2 -> ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) <-> ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) ) )
62	4, 5	syl6eq 2489	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = { 0 , 1 } )
63	62	raleqdv 2921	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` F ) = 2 -> ( A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> A. i e. { 0 , 1 } ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) )
64		c0ex 9376	. . . . . . . . . . . . . . . . . . . 20 |- 0 e. _V
65		1ex 9377	. . . . . . . . . . . . . . . . . . . 20 |- 1 e. _V
66		fveq2 5688	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 0 -> ( F ` i ) = ( F ` 0 ) )
67	66	fveq2d 5692	. . . . . . . . . . . . . . . . . . . . 21 |- ( i = 0 -> ( E ` ( F ` i ) ) = ( E ` ( F ` 0 ) ) )
68		fveq2 5688	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 0 -> ( P ` i ) = ( P ` 0 ) )
69		oveq1 6097	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( i = 0 -> ( i + 1 ) = ( 0 + 1 ) )
70	69	fveq2d 5692	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( i = 0 -> ( P ` ( i + 1 ) ) = ( P ` ( 0 + 1 ) ) )
71		0p1e1 10429	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 0 + 1 ) = 1
72	71	fveq2i 5691	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( P ` ( 0 + 1 ) ) = ( P ` 1 )
73	70, 72	syl6eq 2489	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 0 -> ( P ` ( i + 1 ) ) = ( P ` 1 ) )
74	68, 73	preq12d 3959	. . . . . . . . . . . . . . . . . . . . 21 |- ( i = 0 -> { ( P ` i ) , ( P ` ( i + 1 ) ) } = { ( P ` 0 ) , ( P ` 1 ) } )
75	67, 74	eqeq12d 2455	. . . . . . . . . . . . . . . . . . . 20 |- ( i = 0 -> ( ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) )
76		fveq2 5688	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 1 -> ( F ` i ) = ( F ` 1 ) )
77	76	fveq2d 5692	. . . . . . . . . . . . . . . . . . . . 21 |- ( i = 1 -> ( E ` ( F ` i ) ) = ( E ` ( F ` 1 ) ) )
78		fveq2 5688	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 1 -> ( P ` i ) = ( P ` 1 ) )
79		oveq1 6097	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( i = 1 -> ( i + 1 ) = ( 1 + 1 ) )
80	79	fveq2d 5692	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( i = 1 -> ( P ` ( i + 1 ) ) = ( P ` ( 1 + 1 ) ) )
81		1p1e2 10431	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 1 + 1 ) = 2
82	81	fveq2i 5691	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( P ` ( 1 + 1 ) ) = ( P ` 2 )
83	80, 82	syl6eq 2489	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 1 -> ( P ` ( i + 1 ) ) = ( P ` 2 ) )
84	78, 83	preq12d 3959	. . . . . . . . . . . . . . . . . . . . 21 |- ( i = 1 -> { ( P ` i ) , ( P ` ( i + 1 ) ) } = { ( P ` 1 ) , ( P ` 2 ) } )
85	77, 84	eqeq12d 2455	. . . . . . . . . . . . . . . . . . . 20 |- ( i = 1 -> ( ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) )
86	64, 65, 75, 85	ralpr 3926	. . . . . . . . . . . . . . . . . . 19 |- ( A. i e. { 0 , 1 } ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) )
87	63, 86	syl6bb 261	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` F ) = 2 -> ( A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) )
88	61, 87	anbi12d 705	. . . . . . . . . . . . . . . . 17 |- ( ( # ` F ) = 2 -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) <-> ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) ) )
89	88	imbi1d 317	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) = 2 -> ( ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( F ` 0 ) =/= ( F ` 1 ) ) <-> ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) )
90	58, 89	imbi12d 320	. . . . . . . . . . . . . . 15 |- ( ( # ` F ) = 2 -> ( ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) <-> ( F : ( 0..^ 2 ) --> dom E -> ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) ) )
91	90	adantr 462	. . . . . . . . . . . . . 14 |- ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) <-> ( F : ( 0..^ 2 ) --> dom E -> ( ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) /\ ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) ) )
92	57, 91	mpbird 232	. . . . . . . . . . . . 13 |- ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( F ` 0 ) =/= ( F ` 1 ) ) ) )
93	92	impcom 430	. . . . . . . . . . . 12 |- ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> ( F ` 0 ) =/= ( F ` 1 ) ) )
94	93	imp 429	. . . . . . . . . . 11 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> ( F ` 0 ) =/= ( F ` 1 ) )
95	11, 94	jca 529	. . . . . . . . . 10 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> ( F : { 0 , 1 } --> dom E /\ ( F ` 0 ) =/= ( F ` 1 ) ) )
96	64, 65	pm3.2i 452	. . . . . . . . . . . 12 |- ( 0 e. _V /\ 1 e. _V )
97		0ne1 10385	. . . . . . . . . . . 12 |- 0 =/= 1
98	96, 97	pm3.2i 452	. . . . . . . . . . 11 |- ( ( 0 e. _V /\ 1 e. _V ) /\ 0 =/= 1 )
99		eqid 2441	. . . . . . . . . . . 12 |- { 0 , 1 } = { 0 , 1 }
100	99	f12dfv 30055	. . . . . . . . . . 11 |- ( ( ( 0 e. _V /\ 1 e. _V ) /\ 0 =/= 1 ) -> ( F : { 0 , 1 } -1-1-> dom E <-> ( F : { 0 , 1 } --> dom E /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) )
101	98, 100	mp1i 12	. . . . . . . . . 10 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> ( F : { 0 , 1 } -1-1-> dom E <-> ( F : { 0 , 1 } --> dom E /\ ( F ` 0 ) =/= ( F ` 1 ) ) ) )
102	95, 101	mpbird 232	. . . . . . . . 9 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> F : { 0 , 1 } -1-1-> dom E )
103		df-f1 5420	. . . . . . . . 9 |- ( F : { 0 , 1 } -1-1-> dom E <-> ( F : { 0 , 1 } --> dom E /\ Fun `' F ) )
104	102, 103	sylib 196	. . . . . . . 8 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> ( F : { 0 , 1 } --> dom E /\ Fun `' F ) )
105	104	simprd 460	. . . . . . 7 |- ( ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) /\ ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) ) -> Fun `' F )
106	105	ex 434	. . . . . 6 |- ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> Fun `' F ) )
107	106	expcom 435	. . . . 5 |- ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> Fun `' F ) ) )
108	107	ex 434	. . . 4 |- ( ( # ` F ) = 2 -> ( E : dom E -1-1-> ran E -> ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> Fun `' F ) ) ) )
109	108	com13 80	. . 3 |- ( F : ( 0..^ ( # ` F ) ) --> dom E -> ( E : dom E -1-1-> ran E -> ( ( # ` F ) = 2 -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> Fun `' F ) ) ) )
110	1, 109	syl 16	. 2 |- ( F e. Word dom E -> ( E : dom E -1-1-> ran E -> ( ( # ` F ) = 2 -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> Fun `' F ) ) ) )
111	110	3imp 1176	1 |- ( ( F e. Word dom E /\ E : dom E -1-1-> ran E /\ ( # ` F ) = 2 ) -> ( ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) /\ A. i e. ( 0..^ ( # ` F ) ) ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } ) -> Fun `' F ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 960   = wceq 1364   e. wcel 1761   =/= wne 2604   A.wral 2713   _Vcvv 2970   {cpr 3876   `'ccnv 4835   dom cdm 4836   ran crn 4837   Fun wfun 5409   -->wf 5411   -1-1->wf1 5412   ` cfv 5415  (class class class)co 6090   0cc0 9278   1c1 9279   + caddc 9281   2c2 10367  ..^cfzo 11544   #chash 12099  Word cword 12217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225

This theorem is referenced by:  usgra2wlkspth  30207