Metamath Proof Explorer

Theorem 2wlkdlem4

Description: Lemma 4 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 ) )
Assertion 2wlkdlem4 
|- ( ph -> A. k e. ( 0 ... ( # ` F ) ) ( P ` k ) e. V )

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 1 2 3 2wlkdlem3 
 |-  ( ph -> ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) )
5 simp1 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( P ` 0 ) = A )
6 5 eleq1d 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( ( P ` 0 ) e. V <-> A e. V ) )
7 simp2 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( P ` 1 ) = B )
8 7 eleq1d 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( ( P ` 1 ) e. V <-> B e. V ) )
9 simp3 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( P ` 2 ) = C )
10 9 eleq1d 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( ( P ` 2 ) e. V <-> C e. V ) )
11 6 8 10 3anbi123d 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) <-> ( A e. V /\ B e. V /\ C e. V ) ) )
12 11 bicomd 
 |-  ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B /\ ( P ` 2 ) = C ) -> ( ( A e. V /\ B e. V /\ C e. V ) <-> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) ) )
13 4 12 syl 
 |-  ( ph -> ( ( A e. V /\ B e. V /\ C e. V ) <-> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) ) )
14 3 13 mpbid 
 |-  ( ph -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) )
15 2 fveq2i 
 |-  ( # ` F ) = ( # ` <" J K "> )
16 s2len 
 |-  ( # ` <" J K "> ) = 2
17 15 16 eqtri 
 |-  ( # ` F ) = 2
18 17 oveq2i 
 |-  ( 0 ... ( # ` F ) ) = ( 0 ... 2 )
19 fz0tp 
 |-  ( 0 ... 2 ) = { 0 , 1 , 2 }
20 18 19 eqtri 
 |-  ( 0 ... ( # ` F ) ) = { 0 , 1 , 2 }
21 20 raleqi 
 |-  ( A. k e. ( 0 ... ( # ` F ) ) ( P ` k ) e. V <-> A. k e. { 0 , 1 , 2 } ( P ` k ) e. V )
22 c0ex 
 |-  0 e. _V
23 1ex 
 |-  1 e. _V
24 2ex 
 |-  2 e. _V
25 fveq2 
 |-  ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
26 25 eleq1d 
 |-  ( k = 0 -> ( ( P ` k ) e. V <-> ( P ` 0 ) e. V ) )
27 fveq2 
 |-  ( k = 1 -> ( P ` k ) = ( P ` 1 ) )
28 27 eleq1d 
 |-  ( k = 1 -> ( ( P ` k ) e. V <-> ( P ` 1 ) e. V ) )
29 fveq2 
 |-  ( k = 2 -> ( P ` k ) = ( P ` 2 ) )
30 29 eleq1d 
 |-  ( k = 2 -> ( ( P ` k ) e. V <-> ( P ` 2 ) e. V ) )
31 22 23 24 26 28 30 raltp 
 |-  ( A. k e. { 0 , 1 , 2 } ( P ` k ) e. V <-> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) )
32 21 31 bitri 
 |-  ( A. k e. ( 0 ... ( # ` F ) ) ( P ` k ) e. V <-> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V /\ ( P ` 2 ) e. V ) )
33 14 32 sylibr 
 |-  ( ph -> A. k e. ( 0 ... ( # ` F ) ) ( P ` k ) e. V )