Metamath Proof Explorer

Theorem 3wlkdlem10

Description: Lemma 10 for 3wlkd . (Contributed by Alexander van der Vekens, 12-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 ) )
3wlkd.e 
|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )
Assertion 3wlkdlem10 
|- ( ph -> A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )

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 3wlkd.e 
 |-  ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )
6 1 2 3 4 5 3wlkdlem9 
 |-  ( ph -> ( { A , B } C_ ( I ` ( F ` 0 ) ) /\ { B , C } C_ ( I ` ( F ` 1 ) ) /\ { C , D } C_ ( I ` ( F ` 2 ) ) ) )
7 1 2 3 3wlkdlem3 
 |-  ( ph -> ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) )
8 preq12 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) -> { ( P ` 0 ) , ( P ` 1 ) } = { A , B } )
9 8 adantr 
 |-  ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> { ( P ` 0 ) , ( P ` 1 ) } = { A , B } )
10 9 sseq1d 
 |-  ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) <-> { A , B } C_ ( I ` ( F ` 0 ) ) ) )
11 simplr 
 |-  ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( P ` 1 ) = B )
12 simprl 
 |-  ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( P ` 2 ) = C )
13 11 12 preq12d 
 |-  ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> { ( P ` 1 ) , ( P ` 2 ) } = { B , C } )
14 13 sseq1d 
 |-  ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) <-> { B , C } C_ ( I ` ( F ` 1 ) ) ) )
15 preq12 
 |-  ( ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) -> { ( P ` 2 ) , ( P ` 3 ) } = { C , D } )
16 15 adantl 
 |-  ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> { ( P ` 2 ) , ( P ` 3 ) } = { C , D } )
17 16 sseq1d 
 |-  ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( { ( P ` 2 ) , ( P ` 3 ) } C_ ( I ` ( F ` 2 ) ) <-> { C , D } C_ ( I ` ( F ` 2 ) ) ) )
18 10 14 17 3anbi123d 
 |-  ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( ( { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) /\ { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) /\ { ( P ` 2 ) , ( P ` 3 ) } C_ ( I ` ( F ` 2 ) ) ) <-> ( { A , B } C_ ( I ` ( F ` 0 ) ) /\ { B , C } C_ ( I ` ( F ` 1 ) ) /\ { C , D } C_ ( I ` ( F ` 2 ) ) ) ) )
19 7 18 syl 
 |-  ( ph -> ( ( { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) /\ { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) /\ { ( P ` 2 ) , ( P ` 3 ) } C_ ( I ` ( F ` 2 ) ) ) <-> ( { A , B } C_ ( I ` ( F ` 0 ) ) /\ { B , C } C_ ( I ` ( F ` 1 ) ) /\ { C , D } C_ ( I ` ( F ` 2 ) ) ) ) )
20 6 19 mpbird 
 |-  ( ph -> ( { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) /\ { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) /\ { ( P ` 2 ) , ( P ` 3 ) } C_ ( I ` ( F ` 2 ) ) ) )
21 1 2 3wlkdlem2 
 |-  ( 0 ..^ ( # ` F ) ) = { 0 , 1 , 2 }
22 21 raleqi 
 |-  ( A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) <-> A. k e. { 0 , 1 , 2 } { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )
23 c0ex 
 |-  0 e. _V
24 1ex 
 |-  1 e. _V
25 2ex 
 |-  2 e. _V
26 fveq2 
 |-  ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
27 fv0p1e1 
 |-  ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` 1 ) )
28 26 27 preq12d 
 |-  ( k = 0 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 0 ) , ( P ` 1 ) } )
29 2fveq3 
 |-  ( k = 0 -> ( I ` ( F ` k ) ) = ( I ` ( F ` 0 ) ) )
30 28 29 sseq12d 
 |-  ( k = 0 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) <-> { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) ) )
31 fveq2 
 |-  ( k = 1 -> ( P ` k ) = ( P ` 1 ) )
32 oveq1 
 |-  ( k = 1 -> ( k + 1 ) = ( 1 + 1 ) )
33 1p1e2 
 |-  ( 1 + 1 ) = 2
34 32 33 eqtrdi 
 |-  ( k = 1 -> ( k + 1 ) = 2 )
35 34 fveq2d 
 |-  ( k = 1 -> ( P ` ( k + 1 ) ) = ( P ` 2 ) )
36 31 35 preq12d 
 |-  ( k = 1 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 1 ) , ( P ` 2 ) } )
37 2fveq3 
 |-  ( k = 1 -> ( I ` ( F ` k ) ) = ( I ` ( F ` 1 ) ) )
38 36 37 sseq12d 
 |-  ( k = 1 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) <-> { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) ) )
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 preq12d 
 |-  ( k = 2 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 2 ) , ( P ` 3 ) } )
45 2fveq3 
 |-  ( k = 2 -> ( I ` ( F ` k ) ) = ( I ` ( F ` 2 ) ) )
46 44 45 sseq12d 
 |-  ( k = 2 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) <-> { ( P ` 2 ) , ( P ` 3 ) } C_ ( I ` ( F ` 2 ) ) ) )
47 23 24 25 30 38 46 raltp 
 |-  ( A. k e. { 0 , 1 , 2 } { ( 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 ) ) /\ { ( P ` 2 ) , ( P ` 3 ) } C_ ( I ` ( F ` 2 ) ) ) )
48 22 47 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 ) ) /\ { ( P ` 2 ) , ( P ` 3 ) } C_ ( I ` ( F ` 2 ) ) ) )
49 20 48 sylibr 
 |-  ( ph -> A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )