Theorem 3pthdlem1 27024

Description: Lemma 1 for 3pthd 27034. (Contributed by AV, 9-Feb-2021.)

Hypotheses

Ref	Expression
3wlkd.p	|- P = <" A B C D ">
3wlkd.f	|- F = <" J K L ">
3wlkd.s	|- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )
3wlkd.n	|- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )

Assertion

Ref	Expression
3pthdlem1	|- ( ph -> A. k e. ( 0..^ ( # ` P ) ) A. j e. ( 1..^ ( # ` F ) ) ( k =/= j -> ( P ` k ) =/= ( P ` j ) ) )

Distinct variable groups:   A, k   B, k   C, k   D, k   k, J   k, K   k, L   k, V   k, F   P, k   j, F, k   P, j

Allowed substitution hints:   ph( j, k)   A( j)   B( j)   C( j)   D( j)   J( j)   K( j)   L( j)   V( j)

Proof of Theorem 3pthdlem1

Step	Hyp	Ref	Expression
1		3wlkd.p	. . . . 5 |- P = <" A B C D ">
2		3wlkd.f	. . . . 5 |- F = <" J K L ">
3		3wlkd.s	. . . . 5 |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )
4	1, 2, 3	3wlkdlem3 27021	. . . 4 |- ( ph -> ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) )
5		3wlkd.n	. . . 4 |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )
6		simpr1l 1118	. . . . . . . 8 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> A =/= B )
7		simpl 473	. . . . . . . . . . 11 |- ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) -> ( P ` 0 ) = A )
8	7	adantr 481	. . . . . . . . . 10 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( P ` 0 ) = A )
9		simpr 477	. . . . . . . . . . 11 |- ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) -> ( P ` 1 ) = B )
10	9	adantr 481	. . . . . . . . . 10 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( P ` 1 ) = B )
11	8, 10	neeq12d 2855	. . . . . . . . 9 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( ( P ` 0 ) =/= ( P ` 1 ) <-> A =/= B ) )
12	11	adantr 481	. . . . . . . 8 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( ( P ` 0 ) =/= ( P ` 1 ) <-> A =/= B ) )
13	6, 12	mpbird 247	. . . . . . 7 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( P ` 0 ) =/= ( P ` 1 ) )
14	13	a1d 25	. . . . . 6 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( 0 =/= 1 -> ( P ` 0 ) =/= ( P ` 1 ) ) )
15		simpr1r 1119	. . . . . . . 8 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> A =/= C )
16		simpl 473	. . . . . . . . . . 11 |- ( ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) -> ( P ` 2 ) = C )
17	16	adantl 482	. . . . . . . . . 10 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( P ` 2 ) = C )
18	8, 17	neeq12d 2855	. . . . . . . . 9 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( ( P ` 0 ) =/= ( P ` 2 ) <-> A =/= C ) )
19	18	adantr 481	. . . . . . . 8 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( ( P ` 0 ) =/= ( P ` 2 ) <-> A =/= C ) )
20	15, 19	mpbird 247	. . . . . . 7 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( P ` 0 ) =/= ( P ` 2 ) )
21	20	a1d 25	. . . . . 6 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( 0 =/= 2 -> ( P ` 0 ) =/= ( P ` 2 ) ) )
22	14, 21	jca 554	. . . . 5 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( ( 0 =/= 1 -> ( P ` 0 ) =/= ( P ` 1 ) ) /\ ( 0 =/= 2 -> ( P ` 0 ) =/= ( P ` 2 ) ) ) )
23		eqid 2622	. . . . . . . 8 |- 1 = 1
24	23	2a1i 12	. . . . . . 7 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( ( P ` 1 ) = ( P ` 1 ) -> 1 = 1 ) )
25	24	necon3d 2815	. . . . . 6 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( 1 =/= 1 -> ( P ` 1 ) =/= ( P ` 1 ) ) )
26		simpr2l 1120	. . . . . . . 8 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> B =/= C )
27	10, 17	neeq12d 2855	. . . . . . . . 9 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( ( P ` 1 ) =/= ( P ` 2 ) <-> B =/= C ) )
28	27	adantr 481	. . . . . . . 8 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( ( P ` 1 ) =/= ( P ` 2 ) <-> B =/= C ) )
29	26, 28	mpbird 247	. . . . . . 7 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( P ` 1 ) =/= ( P ` 2 ) )
30	29	a1d 25	. . . . . 6 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( 1 =/= 2 -> ( P ` 1 ) =/= ( P ` 2 ) ) )
31	25, 30	jca 554	. . . . 5 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( ( 1 =/= 1 -> ( P ` 1 ) =/= ( P ` 1 ) ) /\ ( 1 =/= 2 -> ( P ` 1 ) =/= ( P ` 2 ) ) ) )
32	29	necomd 2849	. . . . . . 7 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( P ` 2 ) =/= ( P ` 1 ) )
33	32	a1d 25	. . . . . 6 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( 2 =/= 1 -> ( P ` 2 ) =/= ( P ` 1 ) ) )
34		eqid 2622	. . . . . . . 8 |- 2 = 2
35	34	2a1i 12	. . . . . . 7 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( ( P ` 2 ) = ( P ` 2 ) -> 2 = 2 ) )
36	35	necon3d 2815	. . . . . 6 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( 2 =/= 2 -> ( P ` 2 ) =/= ( P ` 2 ) ) )
37		simpr2r 1121	. . . . . . . . . 10 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> B =/= D )
38		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) -> ( P ` 3 ) = D )
39	38	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( P ` 3 ) = D )
40	10, 39	neeq12d 2855	. . . . . . . . . . 11 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( ( P ` 1 ) =/= ( P ` 3 ) <-> B =/= D ) )
41	40	adantr 481	. . . . . . . . . 10 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( ( P ` 1 ) =/= ( P ` 3 ) <-> B =/= D ) )
42	37, 41	mpbird 247	. . . . . . . . 9 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( P ` 1 ) =/= ( P ` 3 ) )
43	42	necomd 2849	. . . . . . . 8 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( P ` 3 ) =/= ( P ` 1 ) )
44	43	a1d 25	. . . . . . 7 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( 3 =/= 1 -> ( P ` 3 ) =/= ( P ` 1 ) ) )
45		simp3 1063	. . . . . . . . . . 11 |- ( ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) -> C =/= D )
46	45	necomd 2849	. . . . . . . . . 10 |- ( ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) -> D =/= C )
47	46	adantl 482	. . . . . . . . 9 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> D =/= C )
48		simpl 473	. . . . . . . . . . . . 13 |- ( ( ( P ` 3 ) = D /\ ( P ` 2 ) = C ) -> ( P ` 3 ) = D )
49		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( P ` 3 ) = D /\ ( P ` 2 ) = C ) -> ( P ` 2 ) = C )
50	48, 49	neeq12d 2855	. . . . . . . . . . . 12 |- ( ( ( P ` 3 ) = D /\ ( P ` 2 ) = C ) -> ( ( P ` 3 ) =/= ( P ` 2 ) <-> D =/= C ) )
51	50	ancoms 469	. . . . . . . . . . 11 |- ( ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) -> ( ( P ` 3 ) =/= ( P ` 2 ) <-> D =/= C ) )
52	51	adantl 482	. . . . . . . . . 10 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( ( P ` 3 ) =/= ( P ` 2 ) <-> D =/= C ) )
53	52	adantr 481	. . . . . . . . 9 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( ( P ` 3 ) =/= ( P ` 2 ) <-> D =/= C ) )
54	47, 53	mpbird 247	. . . . . . . 8 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( P ` 3 ) =/= ( P ` 2 ) )
55	54	a1d 25	. . . . . . 7 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( 3 =/= 2 -> ( P ` 3 ) =/= ( P ` 2 ) ) )
56	44, 55	jca 554	. . . . . 6 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( ( 3 =/= 1 -> ( P ` 3 ) =/= ( P ` 1 ) ) /\ ( 3 =/= 2 -> ( P ` 3 ) =/= ( P ` 2 ) ) ) )
57	33, 36, 56	jca31 557	. . . . 5 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( ( ( 2 =/= 1 -> ( P ` 2 ) =/= ( P ` 1 ) ) /\ ( 2 =/= 2 -> ( P ` 2 ) =/= ( P ` 2 ) ) ) /\ ( ( 3 =/= 1 -> ( P ` 3 ) =/= ( P ` 1 ) ) /\ ( 3 =/= 2 -> ( P ` 3 ) =/= ( P ` 2 ) ) ) ) )
58	22, 31, 57	jca31 557	. . . 4 |- ( ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) /\ ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) ) -> ( ( ( ( 0 =/= 1 -> ( P ` 0 ) =/= ( P ` 1 ) ) /\ ( 0 =/= 2 -> ( P ` 0 ) =/= ( P ` 2 ) ) ) /\ ( ( 1 =/= 1 -> ( P ` 1 ) =/= ( P ` 1 ) ) /\ ( 1 =/= 2 -> ( P ` 1 ) =/= ( P ` 2 ) ) ) ) /\ ( ( ( 2 =/= 1 -> ( P ` 2 ) =/= ( P ` 1 ) ) /\ ( 2 =/= 2 -> ( P ` 2 ) =/= ( P ` 2 ) ) ) /\ ( ( 3 =/= 1 -> ( P ` 3 ) =/= ( P ` 1 ) ) /\ ( 3 =/= 2 -> ( P ` 3 ) =/= ( P ` 2 ) ) ) ) ) )
59	4, 5, 58	syl2anc 693	. . 3 |- ( ph -> ( ( ( ( 0 =/= 1 -> ( P ` 0 ) =/= ( P ` 1 ) ) /\ ( 0 =/= 2 -> ( P ` 0 ) =/= ( P ` 2 ) ) ) /\ ( ( 1 =/= 1 -> ( P ` 1 ) =/= ( P ` 1 ) ) /\ ( 1 =/= 2 -> ( P ` 1 ) =/= ( P ` 2 ) ) ) ) /\ ( ( ( 2 =/= 1 -> ( P ` 2 ) =/= ( P ` 1 ) ) /\ ( 2 =/= 2 -> ( P ` 2 ) =/= ( P ` 2 ) ) ) /\ ( ( 3 =/= 1 -> ( P ` 3 ) =/= ( P ` 1 ) ) /\ ( 3 =/= 2 -> ( P ` 3 ) =/= ( P ` 2 ) ) ) ) ) )
60	1	fveq2i 6194	. . . . . . . 8 |- ( # ` P ) = ( # ` <" A B C D "> )
61		s4len 13644	. . . . . . . 8 |- ( # ` <" A B C D "> ) = 4
62	60, 61	eqtri 2644	. . . . . . 7 |- ( # ` P ) = 4
63	62	oveq2i 6661	. . . . . 6 |- ( 0..^ ( # ` P ) ) = ( 0..^ 4 )
64		fzo0to42pr 12555	. . . . . 6 |- ( 0..^ 4 ) = ( { 0 , 1 } u. { 2 , 3 } )
65	63, 64	eqtri 2644	. . . . 5 |- ( 0..^ ( # ` P ) ) = ( { 0 , 1 } u. { 2 , 3 } )
66	65	raleqi 3142	. . . 4 |- ( A. k e. ( 0..^ ( # ` P ) ) ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) <-> A. k e. ( { 0 , 1 } u. { 2 , 3 } ) ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) )
67		ralunb 3794	. . . 4 |- ( A. k e. ( { 0 , 1 } u. { 2 , 3 } ) ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) <-> ( A. k e. { 0 , 1 } ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) /\ A. k e. { 2 , 3 } ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) ) )
68		c0ex 10034	. . . . . 6 |- 0 e. _V
69		1ex 10035	. . . . . 6 |- 1 e. _V
70		neeq1 2856	. . . . . . . 8 |- ( k = 0 -> ( k =/= 1 <-> 0 =/= 1 ) )
71		fveq2 6191	. . . . . . . . 9 |- ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
72	71	neeq1d 2853	. . . . . . . 8 |- ( k = 0 -> ( ( P ` k ) =/= ( P ` 1 ) <-> ( P ` 0 ) =/= ( P ` 1 ) ) )
73	70, 72	imbi12d 334	. . . . . . 7 |- ( k = 0 -> ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) <-> ( 0 =/= 1 -> ( P ` 0 ) =/= ( P ` 1 ) ) ) )
74		neeq1 2856	. . . . . . . 8 |- ( k = 0 -> ( k =/= 2 <-> 0 =/= 2 ) )
75	71	neeq1d 2853	. . . . . . . 8 |- ( k = 0 -> ( ( P ` k ) =/= ( P ` 2 ) <-> ( P ` 0 ) =/= ( P ` 2 ) ) )
76	74, 75	imbi12d 334	. . . . . . 7 |- ( k = 0 -> ( ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) <-> ( 0 =/= 2 -> ( P ` 0 ) =/= ( P ` 2 ) ) ) )
77	73, 76	anbi12d 747	. . . . . 6 |- ( k = 0 -> ( ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) <-> ( ( 0 =/= 1 -> ( P ` 0 ) =/= ( P ` 1 ) ) /\ ( 0 =/= 2 -> ( P ` 0 ) =/= ( P ` 2 ) ) ) ) )
78		neeq1 2856	. . . . . . . 8 |- ( k = 1 -> ( k =/= 1 <-> 1 =/= 1 ) )
79		fveq2 6191	. . . . . . . . 9 |- ( k = 1 -> ( P ` k ) = ( P ` 1 ) )
80	79	neeq1d 2853	. . . . . . . 8 |- ( k = 1 -> ( ( P ` k ) =/= ( P ` 1 ) <-> ( P ` 1 ) =/= ( P ` 1 ) ) )
81	78, 80	imbi12d 334	. . . . . . 7 |- ( k = 1 -> ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) <-> ( 1 =/= 1 -> ( P ` 1 ) =/= ( P ` 1 ) ) ) )
82		neeq1 2856	. . . . . . . 8 |- ( k = 1 -> ( k =/= 2 <-> 1 =/= 2 ) )
83	79	neeq1d 2853	. . . . . . . 8 |- ( k = 1 -> ( ( P ` k ) =/= ( P ` 2 ) <-> ( P ` 1 ) =/= ( P ` 2 ) ) )
84	82, 83	imbi12d 334	. . . . . . 7 |- ( k = 1 -> ( ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) <-> ( 1 =/= 2 -> ( P ` 1 ) =/= ( P ` 2 ) ) ) )
85	81, 84	anbi12d 747	. . . . . 6 |- ( k = 1 -> ( ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) <-> ( ( 1 =/= 1 -> ( P ` 1 ) =/= ( P ` 1 ) ) /\ ( 1 =/= 2 -> ( P ` 1 ) =/= ( P ` 2 ) ) ) ) )
86	68, 69, 77, 85	ralpr 4238	. . . . 5 |- ( A. k e. { 0 , 1 } ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) <-> ( ( ( 0 =/= 1 -> ( P ` 0 ) =/= ( P ` 1 ) ) /\ ( 0 =/= 2 -> ( P ` 0 ) =/= ( P ` 2 ) ) ) /\ ( ( 1 =/= 1 -> ( P ` 1 ) =/= ( P ` 1 ) ) /\ ( 1 =/= 2 -> ( P ` 1 ) =/= ( P ` 2 ) ) ) ) )
87		2ex 11092	. . . . . 6 |- 2 e. _V
88		3ex 11096	. . . . . 6 |- 3 e. _V
89		neeq1 2856	. . . . . . . 8 |- ( k = 2 -> ( k =/= 1 <-> 2 =/= 1 ) )
90		fveq2 6191	. . . . . . . . 9 |- ( k = 2 -> ( P ` k ) = ( P ` 2 ) )
91	90	neeq1d 2853	. . . . . . . 8 |- ( k = 2 -> ( ( P ` k ) =/= ( P ` 1 ) <-> ( P ` 2 ) =/= ( P ` 1 ) ) )
92	89, 91	imbi12d 334	. . . . . . 7 |- ( k = 2 -> ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) <-> ( 2 =/= 1 -> ( P ` 2 ) =/= ( P ` 1 ) ) ) )
93		neeq1 2856	. . . . . . . 8 |- ( k = 2 -> ( k =/= 2 <-> 2 =/= 2 ) )
94	90	neeq1d 2853	. . . . . . . 8 |- ( k = 2 -> ( ( P ` k ) =/= ( P ` 2 ) <-> ( P ` 2 ) =/= ( P ` 2 ) ) )
95	93, 94	imbi12d 334	. . . . . . 7 |- ( k = 2 -> ( ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) <-> ( 2 =/= 2 -> ( P ` 2 ) =/= ( P ` 2 ) ) ) )
96	92, 95	anbi12d 747	. . . . . 6 |- ( k = 2 -> ( ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) <-> ( ( 2 =/= 1 -> ( P ` 2 ) =/= ( P ` 1 ) ) /\ ( 2 =/= 2 -> ( P ` 2 ) =/= ( P ` 2 ) ) ) ) )
97		neeq1 2856	. . . . . . . 8 |- ( k = 3 -> ( k =/= 1 <-> 3 =/= 1 ) )
98		fveq2 6191	. . . . . . . . 9 |- ( k = 3 -> ( P ` k ) = ( P ` 3 ) )
99	98	neeq1d 2853	. . . . . . . 8 |- ( k = 3 -> ( ( P ` k ) =/= ( P ` 1 ) <-> ( P ` 3 ) =/= ( P ` 1 ) ) )
100	97, 99	imbi12d 334	. . . . . . 7 |- ( k = 3 -> ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) <-> ( 3 =/= 1 -> ( P ` 3 ) =/= ( P ` 1 ) ) ) )
101		neeq1 2856	. . . . . . . 8 |- ( k = 3 -> ( k =/= 2 <-> 3 =/= 2 ) )
102	98	neeq1d 2853	. . . . . . . 8 |- ( k = 3 -> ( ( P ` k ) =/= ( P ` 2 ) <-> ( P ` 3 ) =/= ( P ` 2 ) ) )
103	101, 102	imbi12d 334	. . . . . . 7 |- ( k = 3 -> ( ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) <-> ( 3 =/= 2 -> ( P ` 3 ) =/= ( P ` 2 ) ) ) )
104	100, 103	anbi12d 747	. . . . . 6 |- ( k = 3 -> ( ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) <-> ( ( 3 =/= 1 -> ( P ` 3 ) =/= ( P ` 1 ) ) /\ ( 3 =/= 2 -> ( P ` 3 ) =/= ( P ` 2 ) ) ) ) )
105	87, 88, 96, 104	ralpr 4238	. . . . 5 |- ( A. k e. { 2 , 3 } ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) <-> ( ( ( 2 =/= 1 -> ( P ` 2 ) =/= ( P ` 1 ) ) /\ ( 2 =/= 2 -> ( P ` 2 ) =/= ( P ` 2 ) ) ) /\ ( ( 3 =/= 1 -> ( P ` 3 ) =/= ( P ` 1 ) ) /\ ( 3 =/= 2 -> ( P ` 3 ) =/= ( P ` 2 ) ) ) ) )
106	86, 105	anbi12i 733	. . . 4 |- ( ( A. k e. { 0 , 1 } ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) /\ A. k e. { 2 , 3 } ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) ) <-> ( ( ( ( 0 =/= 1 -> ( P ` 0 ) =/= ( P ` 1 ) ) /\ ( 0 =/= 2 -> ( P ` 0 ) =/= ( P ` 2 ) ) ) /\ ( ( 1 =/= 1 -> ( P ` 1 ) =/= ( P ` 1 ) ) /\ ( 1 =/= 2 -> ( P ` 1 ) =/= ( P ` 2 ) ) ) ) /\ ( ( ( 2 =/= 1 -> ( P ` 2 ) =/= ( P ` 1 ) ) /\ ( 2 =/= 2 -> ( P ` 2 ) =/= ( P ` 2 ) ) ) /\ ( ( 3 =/= 1 -> ( P ` 3 ) =/= ( P ` 1 ) ) /\ ( 3 =/= 2 -> ( P ` 3 ) =/= ( P ` 2 ) ) ) ) ) )
107	66, 67, 106	3bitri 286	. . 3 |- ( A. k e. ( 0..^ ( # ` P ) ) ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) <-> ( ( ( ( 0 =/= 1 -> ( P ` 0 ) =/= ( P ` 1 ) ) /\ ( 0 =/= 2 -> ( P ` 0 ) =/= ( P ` 2 ) ) ) /\ ( ( 1 =/= 1 -> ( P ` 1 ) =/= ( P ` 1 ) ) /\ ( 1 =/= 2 -> ( P ` 1 ) =/= ( P ` 2 ) ) ) ) /\ ( ( ( 2 =/= 1 -> ( P ` 2 ) =/= ( P ` 1 ) ) /\ ( 2 =/= 2 -> ( P ` 2 ) =/= ( P ` 2 ) ) ) /\ ( ( 3 =/= 1 -> ( P ` 3 ) =/= ( P ` 1 ) ) /\ ( 3 =/= 2 -> ( P ` 3 ) =/= ( P ` 2 ) ) ) ) ) )
108	59, 107	sylibr 224	. 2 |- ( ph -> A. k e. ( 0..^ ( # ` P ) ) ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) )
109	2	fveq2i 6194	. . . . . . . 8 |- ( # ` F ) = ( # ` <" J K L "> )
110		s3len 13639	. . . . . . . 8 |- ( # ` <" J K L "> ) = 3
111	109, 110	eqtri 2644	. . . . . . 7 |- ( # ` F ) = 3
112	111	oveq2i 6661	. . . . . 6 |- ( 1..^ ( # ` F ) ) = ( 1..^ 3 )
113		fzo13pr 12552	. . . . . 6 |- ( 1..^ 3 ) = { 1 , 2 }
114	112, 113	eqtri 2644	. . . . 5 |- ( 1..^ ( # ` F ) ) = { 1 , 2 }
115	114	raleqi 3142	. . . 4 |- ( A. j e. ( 1..^ ( # ` F ) ) ( k =/= j -> ( P ` k ) =/= ( P ` j ) ) <-> A. j e. { 1 , 2 } ( k =/= j -> ( P ` k ) =/= ( P ` j ) ) )
116		neeq2 2857	. . . . . 6 |- ( j = 1 -> ( k =/= j <-> k =/= 1 ) )
117		fveq2 6191	. . . . . . 7 |- ( j = 1 -> ( P ` j ) = ( P ` 1 ) )
118	117	neeq2d 2854	. . . . . 6 |- ( j = 1 -> ( ( P ` k ) =/= ( P ` j ) <-> ( P ` k ) =/= ( P ` 1 ) ) )
119	116, 118	imbi12d 334	. . . . 5 |- ( j = 1 -> ( ( k =/= j -> ( P ` k ) =/= ( P ` j ) ) <-> ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) ) )
120		neeq2 2857	. . . . . 6 |- ( j = 2 -> ( k =/= j <-> k =/= 2 ) )
121		fveq2 6191	. . . . . . 7 |- ( j = 2 -> ( P ` j ) = ( P ` 2 ) )
122	121	neeq2d 2854	. . . . . 6 |- ( j = 2 -> ( ( P ` k ) =/= ( P ` j ) <-> ( P ` k ) =/= ( P ` 2 ) ) )
123	120, 122	imbi12d 334	. . . . 5 |- ( j = 2 -> ( ( k =/= j -> ( P ` k ) =/= ( P ` j ) ) <-> ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) )
124	69, 87, 119, 123	ralpr 4238	. . . 4 |- ( A. j e. { 1 , 2 } ( k =/= j -> ( P ` k ) =/= ( P ` j ) ) <-> ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) )
125	115, 124	bitri 264	. . 3 |- ( A. j e. ( 1..^ ( # ` F ) ) ( k =/= j -> ( P ` k ) =/= ( P ` j ) ) <-> ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) )
126	125	ralbii 2980	. 2 |- ( A. k e. ( 0..^ ( # ` P ) ) A. j e. ( 1..^ ( # ` F ) ) ( k =/= j -> ( P ` k ) =/= ( P ` j ) ) <-> A. k e. ( 0..^ ( # ` P ) ) ( ( k =/= 1 -> ( P ` k ) =/= ( P ` 1 ) ) /\ ( k =/= 2 -> ( P ` k ) =/= ( P ` 2 ) ) ) )
127	108, 126	sylibr 224	1 |- ( ph -> A. k e. ( 0..^ ( # ` P ) ) A. j e. ( 1..^ ( # ` F ) ) ( k =/= j -> ( P ` k ) =/= ( P ` j ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   u. cun 3572   {cpr 4179   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   2c2 11070   3c3 11071   4c4 11072  ..^cfzo 12465   #chash 13117   <"cs3 13587   <"cs4 13588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-s4 13595

This theorem is referenced by:  3pthd  27034