Metamath Proof Explorer

Theorem 3wlkdlem5

Description: Lemma 5 for 3wlkd . (Contributed by Alexander van der Vekens, 11-Nov-2017) (Revised by AV, 7-Feb-2021)

Ref Expression
Hypotheses 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 3wlkdlem5 
|- ( ph -> A. k e. ( 0 ..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )

Proof

Step Hyp Ref Expression
1 3wlkd.p 
 |-  P = <" A B C D ">
2 3wlkd.f 
 |-  F = <" J K L ">
3 3wlkd.s 
 |-  ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )
4 3wlkd.n 
 |-  ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )
5 simpl 
 |-  ( ( A =/= B /\ A =/= C ) -> A =/= B )
6 simpl 
 |-  ( ( B =/= C /\ B =/= D ) -> B =/= C )
7 id 
 |-  ( C =/= D -> C =/= D )
8 5 6 7 3anim123i 
 |-  ( ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) -> ( A =/= B /\ B =/= C /\ C =/= D ) )
9 4 8 syl 
 |-  ( ph -> ( A =/= B /\ B =/= C /\ C =/= D ) )
10 1 2 3 3wlkdlem3 
 |-  ( ph -> ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) )
11 simpl 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) -> ( P ` 0 ) = A )
12 simpr 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) -> ( P ` 1 ) = B )
13 11 12 neeq12d 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) -> ( ( P ` 0 ) =/= ( P ` 1 ) <-> A =/= B ) )
14 13 adantr 
 |-  ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( ( P ` 0 ) =/= ( P ` 1 ) <-> A =/= B ) )
15 12 adantr 
 |-  ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( P ` 1 ) = B )
16 simpl 
 |-  ( ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) -> ( P ` 2 ) = C )
17 16 adantl 
 |-  ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( P ` 2 ) = C )
18 15 17 neeq12d 
 |-  ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( ( P ` 1 ) =/= ( P ` 2 ) <-> B =/= C ) )
19 simpr 
 |-  ( ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) -> ( P ` 3 ) = D )
20 16 19 neeq12d 
 |-  ( ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) -> ( ( P ` 2 ) =/= ( P ` 3 ) <-> C =/= D ) )
21 20 adantl 
 |-  ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( ( P ` 2 ) =/= ( P ` 3 ) <-> C =/= D ) )
22 14 18 21 3anbi123d 
 |-  ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) <-> ( A =/= B /\ B =/= C /\ C =/= D ) ) )
23 10 22 syl 
 |-  ( ph -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) <-> ( A =/= B /\ B =/= C /\ C =/= D ) ) )
24 9 23 mpbird 
 |-  ( ph -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) )
25 1 2 3wlkdlem2 
 |-  ( 0 ..^ ( # ` F ) ) = { 0 , 1 , 2 }
26 25 raleqi 
 |-  ( A. k e. ( 0 ..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> A. k e. { 0 , 1 , 2 } ( P ` k ) =/= ( P ` ( k + 1 ) ) )
27 c0ex 
 |-  0 e. _V
28 1ex 
 |-  1 e. _V
29 2ex 
 |-  2 e. _V
30 fveq2 
 |-  ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
31 fv0p1e1 
 |-  ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` 1 ) )
32 30 31 neeq12d 
 |-  ( k = 0 -> ( ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( P ` 0 ) =/= ( P ` 1 ) ) )
33 fveq2 
 |-  ( k = 1 -> ( P ` k ) = ( P ` 1 ) )
34 oveq1 
 |-  ( k = 1 -> ( k + 1 ) = ( 1 + 1 ) )
35 1p1e2 
 |-  ( 1 + 1 ) = 2
36 34 35 eqtrdi 
 |-  ( k = 1 -> ( k + 1 ) = 2 )
37 36 fveq2d 
 |-  ( k = 1 -> ( P ` ( k + 1 ) ) = ( P ` 2 ) )
38 33 37 neeq12d 
 |-  ( k = 1 -> ( ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( P ` 1 ) =/= ( P ` 2 ) ) )
39 fveq2 
 |-  ( k = 2 -> ( P ` k ) = ( P ` 2 ) )
40 oveq1 
 |-  ( k = 2 -> ( k + 1 ) = ( 2 + 1 ) )
41 2p1e3 
 |-  ( 2 + 1 ) = 3
42 40 41 eqtrdi 
 |-  ( k = 2 -> ( k + 1 ) = 3 )
43 42 fveq2d 
 |-  ( k = 2 -> ( P ` ( k + 1 ) ) = ( P ` 3 ) )
44 39 43 neeq12d 
 |-  ( k = 2 -> ( ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( P ` 2 ) =/= ( P ` 3 ) ) )
45 27 28 29 32 38 44 raltp 
 |-  ( A. k e. { 0 , 1 , 2 } ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) )
46 26 45 bitri 
 |-  ( A. k e. ( 0 ..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) )
47 24 46 sylibr 
 |-  ( ph -> A. k e. ( 0 ..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )