Theorem usgra2wlkspthlem1 30308

Description: Lemma 1 for usgra2wlkspth 30310. (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 12252	. . 3 |- ( F e. Word dom E -> F : ( 0..^ ( # ` F ) ) --> dom E )
2		simpl 457	. . . . . . . . . . . . 13 |- ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) -> F : ( 0..^ ( # ` F ) ) --> dom E )
3	2	adantr 465	. . . . . . . . . . . 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 6111	. . . . . . . . . . . . . . . . 17 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = ( 0..^ 2 ) )
5		fzo0to2pr 11626	. . . . . . . . . . . . . . . . 17 |- ( 0..^ 2 ) = { 0 , 1 }
6	4, 5	syl6req 2492	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) = 2 -> { 0 , 1 } = ( 0..^ ( # ` F ) ) )
7	6	adantr 465	. . . . . . . . . . . . . . 15 |- ( ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) -> { 0 , 1 } = ( 0..^ ( # ` F ) ) )
8	7	adantl 466	. . . . . . . . . . . . . 14 |- ( ( F : ( 0..^ ( # ` F ) ) --> dom E /\ ( ( # ` F ) = 2 /\ E : dom E -1-1-> ran E ) ) -> { 0 , 1 } = ( 0..^ ( # ` F ) ) )
9	8	adantr 465	. . . . . . . . . . . . 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 5559	. . . . . . . . . . . 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 3966	. . . . . . . . . . . . . . . . . . . 20 |- ( ( P ` 0 ) = A -> { ( P ` 0 ) , ( P ` 1 ) } = { A , ( P ` 1 ) } )
13	12	eqeq2d 2454	. . . . . . . . . . . . . . . . . . 19 |- ( ( P ` 0 ) = A -> ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } <-> ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } ) )
14		preq2 3967	. . . . . . . . . . . . . . . . . . . 20 |- ( ( P ` 2 ) = B -> { ( P ` 1 ) , ( P ` 2 ) } = { ( P ` 1 ) , B } )
15	14	eqeq2d 2454	. . . . . . . . . . . . . . . . . . 19 |- ( ( P ` 2 ) = B -> ( ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } <-> ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) )
16	13, 15	bi2anan9 868	. . . . . . . . . . . . . . . . . 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 1008	. . . . . . . . . . . . . . . . 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 5703	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( F ` 0 ) = ( F ` 1 ) -> ( E ` ( F ` 0 ) ) = ( E ` ( F ` 1 ) ) )
19	18	eqeq1d 2451	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( F ` 0 ) = ( F ` 1 ) -> ( ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } <-> ( E ` ( F ` 1 ) ) = { A , ( P ` 1 ) } ) )
20	19	anbi1d 704	. . . . . . . . . . . . . . . . . . . . . . . . 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 703	. . . . . . . . . . . . . . . . . . . . . . . 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 2461	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( E ` ( F ` 1 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) -> { A , ( P ` 1 ) } = { ( P ` 1 ) , B } )
23		fvex 5713	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( P ` 0 ) e. _V
24	23	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( P ` 0 ) = A -> ( P ` 0 ) e. _V )
25		eleq1 2503	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( A = ( P ` 0 ) -> ( A e. _V <-> ( P ` 0 ) e. _V ) )
26	25	eqcoms 2446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5713	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( P ` 1 ) e. _V
29	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( P ` 0 ) = A -> ( P ` 1 ) e. _V )
30	27, 29	jca 532	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5713	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( P ` 2 ) e. _V
33	32	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( P ` 2 ) = B -> ( P ` 2 ) e. _V )
34		eleq1 2503	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( B = ( P ` 2 ) -> ( B e. _V <-> ( P ` 2 ) e. _V ) )
35	34	eqcoms 2446	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 532	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( P ` 2 ) = B -> ( ( P ` 1 ) e. _V /\ B e. _V ) )
38	30, 37	anim12i 566	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( ( P ` 0 ) = A /\ ( P ` 2 ) = B ) -> ( ( A e. _V /\ ( P ` 1 ) e. _V ) /\ ( ( P ` 1 ) e. _V /\ B e. _V ) ) )
39	38	adantr 465	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4063	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2460	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( A = ( P ` 1 ) /\ ( P ` 1 ) = B ) -> A = B )
43		simpl 457	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2658	. . . . . . . . . . . . . . . . . . . . 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 604	. . . . . . . . . . . . . . . . . . 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 1184	. . . . . . . . . . . . . . . . 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 5559	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) = 2 -> ( F : ( 0..^ ( # ` F ) ) --> dom E <-> F : ( 0..^ 2 ) --> dom E ) )
59		fveq2 5703	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 2 -> ( P ` ( # ` F ) ) = ( P ` 2 ) )
60	59	eqeq1d 2451	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` F ) = 2 -> ( ( P ` ( # ` F ) ) = B <-> ( P ` 2 ) = B ) )
61	60	3anbi2d 1294	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` F ) = 2 -> ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) <-> ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) ) )
62	4, 5	syl6eq 2491	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = { 0 , 1 } )
63	62	raleqdv 2935	. . . . . . . . . . . . . . . . . . 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 9392	. . . . . . . . . . . . . . . . . . . 20 |- 0 e. _V
65		1ex 9393	. . . . . . . . . . . . . . . . . . . 20 |- 1 e. _V
66		fveq2 5703	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 0 -> ( F ` i ) = ( F ` 0 ) )
67	66	fveq2d 5707	. . . . . . . . . . . . . . . . . . . . 21 |- ( i = 0 -> ( E ` ( F ` i ) ) = ( E ` ( F ` 0 ) ) )
68		fveq2 5703	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 0 -> ( P ` i ) = ( P ` 0 ) )
69		oveq1 6110	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( i = 0 -> ( i + 1 ) = ( 0 + 1 ) )
70	69	fveq2d 5707	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( i = 0 -> ( P ` ( i + 1 ) ) = ( P ` ( 0 + 1 ) ) )
71		0p1e1 10445	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 0 + 1 ) = 1
72	71	fveq2i 5706	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( P ` ( 0 + 1 ) ) = ( P ` 1 )
73	70, 72	syl6eq 2491	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 0 -> ( P ` ( i + 1 ) ) = ( P ` 1 ) )
74	68, 73	preq12d 3974	. . . . . . . . . . . . . . . . . . . . 21 |- ( i = 0 -> { ( P ` i ) , ( P ` ( i + 1 ) ) } = { ( P ` 0 ) , ( P ` 1 ) } )
75	67, 74	eqeq12d 2457	. . . . . . . . . . . . . . . . . . . 20 |- ( i = 0 -> ( ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) )
76		fveq2 5703	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 1 -> ( F ` i ) = ( F ` 1 ) )
77	76	fveq2d 5707	. . . . . . . . . . . . . . . . . . . . 21 |- ( i = 1 -> ( E ` ( F ` i ) ) = ( E ` ( F ` 1 ) ) )
78		fveq2 5703	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 1 -> ( P ` i ) = ( P ` 1 ) )
79		oveq1 6110	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( i = 1 -> ( i + 1 ) = ( 1 + 1 ) )
80	79	fveq2d 5707	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( i = 1 -> ( P ` ( i + 1 ) ) = ( P ` ( 1 + 1 ) ) )
81		1p1e2 10447	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 1 + 1 ) = 2
82	81	fveq2i 5706	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( P ` ( 1 + 1 ) ) = ( P ` 2 )
83	80, 82	syl6eq 2491	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 1 -> ( P ` ( i + 1 ) ) = ( P ` 2 ) )
84	78, 83	preq12d 3974	. . . . . . . . . . . . . . . . . . . . 21 |- ( i = 1 -> { ( P ` i ) , ( P ` ( i + 1 ) ) } = { ( P ` 1 ) , ( P ` 2 ) } )
85	77, 84	eqeq12d 2457	. . . . . . . . . . . . . . . . . . . 20 |- ( i = 1 -> ( ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) )
86	64, 65, 75, 85	ralpr 3941	. . . . . . . . . . . . . . . . . . 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 710	. . . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . 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 532	. . . . . . . . . 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 455	. . . . . . . . . . . 12 |- ( 0 e. _V /\ 1 e. _V )
97		0ne1 10401	. . . . . . . . . . . 12 |- 0 =/= 1
98	96, 97	pm3.2i 455	. . . . . . . . . . 11 |- ( ( 0 e. _V /\ 1 e. _V ) /\ 0 =/= 1 )
99		eqid 2443	. . . . . . . . . . . 12 |- { 0 , 1 } = { 0 , 1 }
100	99	f12dfv 30158	. . . . . . . . . . 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 5435	. . . . . . . . 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 463	. . . . . . 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 1181	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 965   = wceq 1369   e. wcel 1756   =/= wne 2618   A.wral 2727   _Vcvv 2984   {cpr 3891   `'ccnv 4851   dom cdm 4852   ran crn 4853   Fun wfun 5424   -->wf 5426   -1-1->wf1 5427   ` cfv 5430  (class class class)co 6103   0cc0 9294   1c1 9295   + caddc 9297   2c2 10383  ..^cfzo 11560   #chash 12115  Word cword 12233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450  df-fzo 11561  df-hash 12116  df-word 12241

This theorem is referenced by:  usgra2wlkspth  30310