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 ) ) )