Theorem usgra2wlkspthlem1 24311

Description: Lemma 1 for usgra2wlkspth 24313. (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 12518	. . 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 6291	. . . . . . . . . . . . . . . . 17 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = ( 0..^ 2 ) )
5		fzo0to2pr 11866	. . . . . . . . . . . . . . . . 17 |- ( 0..^ 2 ) = { 0 , 1 }
6	4, 5	syl6req 2525	. . . . . . . . . . . . . . . 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 5717	. . . . . . . . . . . 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 4106	. . . . . . . . . . . . . . . . . . . 20 |- ( ( P ` 0 ) = A -> { ( P ` 0 ) , ( P ` 1 ) } = { A , ( P ` 1 ) } )
13	12	eqeq2d 2481	. . . . . . . . . . . . . . . . . . 19 |- ( ( P ` 0 ) = A -> ( ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } <-> ( E ` ( F ` 0 ) ) = { A , ( P ` 1 ) } ) )
14		preq2 4107	. . . . . . . . . . . . . . . . . . . 20 |- ( ( P ` 2 ) = B -> { ( P ` 1 ) , ( P ` 2 ) } = { ( P ` 1 ) , B } )
15	14	eqeq2d 2481	. . . . . . . . . . . . . . . . . . 19 |- ( ( P ` 2 ) = B -> ( ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } <-> ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) )
16	13, 15	bi2anan9 871	. . . . . . . . . . . . . . . . . 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 1016	. . . . . . . . . . . . . . . . 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 5865	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( F ` 0 ) = ( F ` 1 ) -> ( E ` ( F ` 0 ) ) = ( E ` ( F ` 1 ) ) )
19	18	eqeq1d 2469	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2494	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( E ` ( F ` 1 ) ) = { A , ( P ` 1 ) } /\ ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , B } ) -> { A , ( P ` 1 ) } = { ( P ` 1 ) , B } )
23		fvex 5875	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( P ` 0 ) e. _V
24	23	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( P ` 0 ) = A -> ( P ` 0 ) e. _V )
25		eleq1 2539	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( A = ( P ` 0 ) -> ( A e. _V <-> ( P ` 0 ) e. _V ) )
26	25	eqcoms 2479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5875	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5875	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( P ` 2 ) e. _V
33	32	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( P ` 2 ) = B -> ( P ` 2 ) e. _V )
34		eleq1 2539	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( B = ( P ` 2 ) -> ( B e. _V <-> ( P ` 2 ) e. _V ) )
35	34	eqcoms 2479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4205	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2493	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2691	. . . . . . . . . . . . . . . . . . . . 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 1193	. . . . . . . . . . . . . . . . 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 5717	. . . . . . . . . . . . . . . 16 |- ( ( # ` F ) = 2 -> ( F : ( 0..^ ( # ` F ) ) --> dom E <-> F : ( 0..^ 2 ) --> dom E ) )
59		fveq2 5865	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 2 -> ( P ` ( # ` F ) ) = ( P ` 2 ) )
60	59	eqeq1d 2469	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` F ) = 2 -> ( ( P ` ( # ` F ) ) = B <-> ( P ` 2 ) = B ) )
61	60	3anbi2d 1304	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` F ) = 2 -> ( ( ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B /\ A =/= B ) <-> ( ( P ` 0 ) = A /\ ( P ` 2 ) = B /\ A =/= B ) ) )
62	4, 5	syl6eq 2524	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` F ) = 2 -> ( 0..^ ( # ` F ) ) = { 0 , 1 } )
63	62	raleqdv 3064	. . . . . . . . . . . . . . . . . . 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 9589	. . . . . . . . . . . . . . . . . . . 20 |- 0 e. _V
65		1ex 9590	. . . . . . . . . . . . . . . . . . . 20 |- 1 e. _V
66		fveq2 5865	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 0 -> ( F ` i ) = ( F ` 0 ) )
67	66	fveq2d 5869	. . . . . . . . . . . . . . . . . . . . 21 |- ( i = 0 -> ( E ` ( F ` i ) ) = ( E ` ( F ` 0 ) ) )
68		fveq2 5865	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 0 -> ( P ` i ) = ( P ` 0 ) )
69		oveq1 6290	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( i = 0 -> ( i + 1 ) = ( 0 + 1 ) )
70	69	fveq2d 5869	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( i = 0 -> ( P ` ( i + 1 ) ) = ( P ` ( 0 + 1 ) ) )
71		0p1e1 10646	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 0 + 1 ) = 1
72	71	fveq2i 5868	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( P ` ( 0 + 1 ) ) = ( P ` 1 )
73	70, 72	syl6eq 2524	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 0 -> ( P ` ( i + 1 ) ) = ( P ` 1 ) )
74	68, 73	preq12d 4114	. . . . . . . . . . . . . . . . . . . . 21 |- ( i = 0 -> { ( P ` i ) , ( P ` ( i + 1 ) ) } = { ( P ` 0 ) , ( P ` 1 ) } )
75	67, 74	eqeq12d 2489	. . . . . . . . . . . . . . . . . . . 20 |- ( i = 0 -> ( ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) )
76		fveq2 5865	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 1 -> ( F ` i ) = ( F ` 1 ) )
77	76	fveq2d 5869	. . . . . . . . . . . . . . . . . . . . 21 |- ( i = 1 -> ( E ` ( F ` i ) ) = ( E ` ( F ` 1 ) ) )
78		fveq2 5865	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 1 -> ( P ` i ) = ( P ` 1 ) )
79		oveq1 6290	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( i = 1 -> ( i + 1 ) = ( 1 + 1 ) )
80	79	fveq2d 5869	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( i = 1 -> ( P ` ( i + 1 ) ) = ( P ` ( 1 + 1 ) ) )
81		1p1e2 10648	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( 1 + 1 ) = 2
82	81	fveq2i 5868	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( P ` ( 1 + 1 ) ) = ( P ` 2 )
83	80, 82	syl6eq 2524	. . . . . . . . . . . . . . . . . . . . . 22 |- ( i = 1 -> ( P ` ( i + 1 ) ) = ( P ` 2 ) )
84	78, 83	preq12d 4114	. . . . . . . . . . . . . . . . . . . . 21 |- ( i = 1 -> { ( P ` i ) , ( P ` ( i + 1 ) ) } = { ( P ` 1 ) , ( P ` 2 ) } )
85	77, 84	eqeq12d 2489	. . . . . . . . . . . . . . . . . . . 20 |- ( i = 1 -> ( ( E ` ( F ` i ) ) = { ( P ` i ) , ( P ` ( i + 1 ) ) } <-> ( E ` ( F ` 1 ) ) = { ( P ` 1 ) , ( P ` 2 ) } ) )
86	64, 65, 75, 85	ralpr 4080	. . . . . . . . . . . . . . . . . . 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 10602	. . . . . . . . . . . 12 |- 0 =/= 1
98	96, 97	pm3.2i 455	. . . . . . . . . . 11 |- ( ( 0 e. _V /\ 1 e. _V ) /\ 0 =/= 1 )
99		eqid 2467	. . . . . . . . . . . 12 |- { 0 , 1 } = { 0 , 1 }
100	99	f12dfv 6166	. . . . . . . . . . 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 5592	. . . . . . . . 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 1190	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 973   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   _Vcvv 3113   {cpr 4029   `'ccnv 4998   dom cdm 4999   ran crn 5000   Fun wfun 5581   -->wf 5583   -1-1->wf1 5584   ` cfv 5587  (class class class)co 6283   0cc0 9491   1c1 9492   + caddc 9494   2c2 10584  ..^cfzo 11791   #chash 12372  Word cword 12499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8319  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-n0 10795  df-z 10864  df-uz 11082  df-fz 11672  df-fzo 11792  df-hash 12373  df-word 12507

This theorem is referenced by:  usgra2wlkspth  24313