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Metamath Proof Explorer

Theorem 3wlkdlem3

Description: Lemma 3 for 3wlkd . (Contributed by Alexander van der Vekens, 10-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 ) ) )
Assertion 3wlkdlem3 
|- ( ph -> ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) )

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 1 fveq1i 
 |-  ( P ` 0 ) = ( <" A B C D "> ` 0 )
5 s4fv0 
 |-  ( A e. V -> ( <" A B C D "> ` 0 ) = A )
6 4 5 eqtrid 
 |-  ( A e. V -> ( P ` 0 ) = A )
7 1 fveq1i 
 |-  ( P ` 1 ) = ( <" A B C D "> ` 1 )
8 s4fv1 
 |-  ( B e. V -> ( <" A B C D "> ` 1 ) = B )
9 7 8 eqtrid 
 |-  ( B e. V -> ( P ` 1 ) = B )
10 6 9 anim12i 
 |-  ( ( A e. V /\ B e. V ) -> ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) )
11 1 fveq1i 
 |-  ( P ` 2 ) = ( <" A B C D "> ` 2 )
12 s4fv2 
 |-  ( C e. V -> ( <" A B C D "> ` 2 ) = C )
13 11 12 eqtrid 
 |-  ( C e. V -> ( P ` 2 ) = C )
14 1 fveq1i 
 |-  ( P ` 3 ) = ( <" A B C D "> ` 3 )
15 s4fv3 
 |-  ( D e. V -> ( <" A B C D "> ` 3 ) = D )
16 14 15 eqtrid 
 |-  ( D e. V -> ( P ` 3 ) = D )
17 13 16 anim12i 
 |-  ( ( C e. V /\ D e. V ) -> ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) )
18 10 17 anim12i 
 |-  ( ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) )
19 3 18 syl 
 |-  ( ph -> ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) )