Metamath Proof Explorer

Theorem 2wlkdlem5

Description: Lemma 5 for 2wlkd . (Contributed by AV, 14-Feb-2021)

Ref Expression
Hypotheses 2wlkd.p 
|- P = <" A B C ">
2wlkd.f 
|- F = <" J K ">
2wlkd.s 
|- ( ph -> ( A e. V /\ B e. V /\ C e. V ) )
2wlkd.n 
|- ( ph -> ( A =/= B /\ B =/= C ) )
Assertion 2wlkdlem5 
|- ( ph -> A. k e. ( 0 ..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )

Proof

Step Hyp Ref Expression
1 2wlkd.p 
 |-  P = <" A B C ">
2 2wlkd.f 
 |-  F = <" J K ">
3 2wlkd.s 
 |-  ( ph -> ( A e. V /\ B e. V /\ C e. V ) )
4 2wlkd.n 
 |-  ( ph -> ( A =/= B /\ B =/= C ) )
5 1 2 3 2wlkdlem3 
 |-  ( ph -> ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) )
6 simp1 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( P ` 0 ) = A )
7 simp2 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( P ` 1 ) = B )
8 6 7 neeq12d 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( ( P ` 0 ) =/= ( P ` 1 ) <-> A =/= B ) )
9 simp3 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( P ` 2 ) = C )
10 7 9 neeq12d 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( ( P ` 1 ) =/= ( P ` 2 ) <-> B =/= C ) )
11 8 10 anbi12d 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) <-> ( A =/= B /\ B =/= C ) ) )
12 11 bicomd 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( ( A =/= B /\ B =/= C ) <-> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) )
13 5 12 syl 
 |-  ( ph -> ( ( A =/= B /\ B =/= C ) <-> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) ) )
14 4 13 mpbid 
 |-  ( ph -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) )
15 1 2 2wlkdlem2 
 |-  ( 0 ..^ ( # ` F ) ) = { 0 , 1 }
16 15 raleqi 
 |-  ( A. k e. ( 0 ..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> A. k e. { 0 , 1 } ( P ` k ) =/= ( P ` ( k + 1 ) ) )
17 c0ex 
 |-  0 e. _V
18 1ex 
 |-  1 e. _V
19 fveq2 
 |-  ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
20 fv0p1e1 
 |-  ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` 1 ) )
21 19 20 neeq12d 
 |-  ( k = 0 -> ( ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( P ` 0 ) =/= ( P ` 1 ) ) )
22 fveq2 
 |-  ( k = 1 -> ( P ` k ) = ( P ` 1 ) )
23 oveq1 
 |-  ( k = 1 -> ( k + 1 ) = ( 1 + 1 ) )
24 1p1e2 
 |-  ( 1 + 1 ) = 2
25 23 24 eqtrdi 
 |-  ( k = 1 -> ( k + 1 ) = 2 )
26 25 fveq2d 
 |-  ( k = 1 -> ( P ` ( k + 1 ) ) = ( P ` 2 ) )
27 22 26 neeq12d 
 |-  ( k = 1 -> ( ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( P ` 1 ) =/= ( P ` 2 ) ) )
28 17 18 21 27 ralpr 
 |-  ( A. k e. { 0 , 1 } ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) )
29 16 28 bitri 
 |-  ( A. k e. ( 0 ..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) ) )
30 14 29 sylibr 
 |-  ( ph -> A. k e. ( 0 ..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )