This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ccat2s1fvwALT

Description: Alternate proof of ccat2s1fvw using words of length 2, see df-s2 . A symbol of the concatenation of a word with two single symbols corresponding to the symbol of the word. (Contributed by AV, 22-Sep-2018) (Proof shortened by Mario Carneiro/AV, 21-Oct-2018) (Revised by AV, 28-Jan-2024) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Assertion ccat2s1fvwALT 
|- ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( W ` I ) )

Proof

Step Hyp Ref Expression
1 ccatw2s1ccatws2 
 |-  ( W e. Word V -> ( ( W ++ <" X "> ) ++ <" Y "> ) = ( W ++ <" X Y "> ) )
2 1 fveq1d 
 |-  ( W e. Word V -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( ( W ++ <" X Y "> ) ` I ) )
3 2 3ad2ant1 
 |-  ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( ( W ++ <" X Y "> ) ` I ) )
4 simp1 
 |-  ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> W e. Word V )
5 s2cli 
 |-  <" X Y "> e. Word _V
6 5 a1i 
 |-  ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> <" X Y "> e. Word _V )
7 simp2 
 |-  ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> I e. NN0 )
8 lencl 
 |-  ( W e. Word V -> ( # ` W ) e. NN0 )
9 8 nn0zd 
 |-  ( W e. Word V -> ( # ` W ) e. ZZ )
10 9 3ad2ant1 
 |-  ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> ( # ` W ) e. ZZ )
11 simp3 
 |-  ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> I < ( # ` W ) )
12 elfzo0z 
 |-  ( I e. ( 0 ..^ ( # ` W ) ) <-> ( I e. NN0 /\ ( # ` W ) e. ZZ /\ I < ( # ` W ) ) )
13 7 10 11 12 syl3anbrc 
 |-  ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> I e. ( 0 ..^ ( # ` W ) ) )
14 ccatval1 
 |-  ( ( W e. Word V /\ <" X Y "> e. Word _V /\ I e. ( 0 ..^ ( # ` W ) ) ) -> ( ( W ++ <" X Y "> ) ` I ) = ( W ` I ) )
15 4 6 13 14 syl3anc 
 |-  ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> ( ( W ++ <" X Y "> ) ` I ) = ( W ` I ) )
16 3 15 eqtrd 
 |-  ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( W ` I ) )