Metamath Proof Explorer

Theorem 2wlkdlem10

Description: Lemma 10 for 3wlkd . (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 ) )
2wlkd.e 
|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) )
Assertion 2wlkdlem10 
|- ( ph -> A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )

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 2wlkd.e 
 |-  ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) )
6 1 2 3 4 5 2wlkdlem9 
 |-  ( ph -> ( { A , B } C_ ( I ` ( F ` 0 ) ) /\ { B , C } C_ ( I ` ( F ` 1 ) ) ) )
7 1 2 3 2wlkdlem3 
 |-  ( ph -> ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) )
8 preq12 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) -> { ( P ` 0 ) , ( P ` 1 ) } = { A , B } )
9 8 3adant3 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> { ( P ` 0 ) , ( P ` 1 ) } = { A , B } )
10 9 sseq1d 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) <-> { A , B } C_ ( I ` ( F ` 0 ) ) ) )
11 preq12 
 |-  ( ( ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> { ( P ` 1 ) , ( P ` 2 ) } = { B , C } )
12 11 3adant1 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> { ( P ` 1 ) , ( P ` 2 ) } = { B , C } )
13 12 sseq1d 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) <-> { B , C } C_ ( I ` ( F ` 1 ) ) ) )
14 10 13 anbi12d 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( ( { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) /\ { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) ) <-> ( { A , B } C_ ( I ` ( F ` 0 ) ) /\ { B , C } C_ ( I ` ( F ` 1 ) ) ) ) )
15 7 14 syl 
 |-  ( ph -> ( ( { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) /\ { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) ) <-> ( { A , B } C_ ( I ` ( F ` 0 ) ) /\ { B , C } C_ ( I ` ( F ` 1 ) ) ) ) )
16 6 15 mpbird 
 |-  ( ph -> ( { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) /\ { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) ) )
17 1 2 2wlkdlem2 
 |-  ( 0 ..^ ( # ` F ) ) = { 0 , 1 }
18 17 raleqi 
 |-  ( A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) <-> A. k e. { 0 , 1 } { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )
19 c0ex 
 |-  0 e. _V
20 1ex 
 |-  1 e. _V
21 fveq2 
 |-  ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
22 fv0p1e1 
 |-  ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` 1 ) )
23 21 22 preq12d 
 |-  ( k = 0 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 0 ) , ( P ` 1 ) } )
24 2fveq3 
 |-  ( k = 0 -> ( I ` ( F ` k ) ) = ( I ` ( F ` 0 ) ) )
25 23 24 sseq12d 
 |-  ( k = 0 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) <-> { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) ) )
26 fveq2 
 |-  ( k = 1 -> ( P ` k ) = ( P ` 1 ) )
27 oveq1 
 |-  ( k = 1 -> ( k + 1 ) = ( 1 + 1 ) )
28 1p1e2 
 |-  ( 1 + 1 ) = 2
29 27 28 eqtrdi 
 |-  ( k = 1 -> ( k + 1 ) = 2 )
30 29 fveq2d 
 |-  ( k = 1 -> ( P ` ( k + 1 ) ) = ( P ` 2 ) )
31 26 30 preq12d 
 |-  ( k = 1 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 1 ) , ( P ` 2 ) } )
32 2fveq3 
 |-  ( k = 1 -> ( I ` ( F ` k ) ) = ( I ` ( F ` 1 ) ) )
33 31 32 sseq12d 
 |-  ( k = 1 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) <-> { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) ) )
34 19 20 25 33 ralpr 
 |-  ( A. k e. { 0 , 1 } { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) <-> ( { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) /\ { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) ) )
35 18 34 bitri 
 |-  ( A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) <-> ( { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) /\ { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) ) )
36 16 35 sylibr 
 |-  ( ph -> A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )