Metamath Proof Explorer

Theorem lmat22e12

Description: Entry of a 2x2 literal matrix. (Contributed by Thierry Arnoux, 12-Sep-2020)

Ref Expression
Hypotheses lmat22.m 
|- M = ( litMat ` <" <" A B "> <" C D "> "> )
lmat22.a 
|- ( ph -> A e. V )
lmat22.b 
|- ( ph -> B e. V )
lmat22.c 
|- ( ph -> C e. V )
lmat22.d 
|- ( ph -> D e. V )
Assertion lmat22e12 
|- ( ph -> ( 1 M 2 ) = B )

Proof

Step Hyp Ref Expression
1 lmat22.m 
 |-  M = ( litMat ` <" <" A B "> <" C D "> "> )
2 lmat22.a 
 |-  ( ph -> A e. V )
3 lmat22.b 
 |-  ( ph -> B e. V )
4 lmat22.c 
 |-  ( ph -> C e. V )
5 lmat22.d 
 |-  ( ph -> D e. V )
6 2nn 
 |-  2 e. NN
7 6 a1i 
 |-  ( ph -> 2 e. NN )
8 2 3 s2cld 
 |-  ( ph -> <" A B "> e. Word V )
9 4 5 s2cld 
 |-  ( ph -> <" C D "> e. Word V )
10 8 9 s2cld 
 |-  ( ph -> <" <" A B "> <" C D "> "> e. Word Word V )
11 s2len 
 |-  ( # ` <" <" A B "> <" C D "> "> ) = 2
12 11 a1i 
 |-  ( ph -> ( # ` <" <" A B "> <" C D "> "> ) = 2 )
13 1 2 3 4 5 lmat22lem 
 |-  ( ( ph /\ i e. ( 0 ..^ 2 ) ) -> ( # ` ( <" <" A B "> <" C D "> "> ` i ) ) = 2 )
14 0nn0 
 |-  0 e. NN0
15 1nn0 
 |-  1 e. NN0
16 1le2 
 |-  1 <_ 2
17 6 nnrei 
 |-  2 e. RR
18 17 leidi 
 |-  2 <_ 2
19 0p1e1 
 |-  ( 0 + 1 ) = 1
20 1p1e2 
 |-  ( 1 + 1 ) = 2
21 s2cli 
 |-  <" A B "> e. Word _V
22 s2fv0 
 |-  ( <" A B "> e. Word _V -> ( <" <" A B "> <" C D "> "> ` 0 ) = <" A B "> )
23 21 22 ax-mp 
 |-  ( <" <" A B "> <" C D "> "> ` 0 ) = <" A B ">
24 s2fv1 
 |-  ( B e. V -> ( <" A B "> ` 1 ) = B )
25 3 24 syl 
 |-  ( ph -> ( <" A B "> ` 1 ) = B )
26 1 7 10 12 13 14 15 16 18 19 20 23 25 lmatfvlem 
 |-  ( ph -> ( 1 M 2 ) = B )