Metamath Proof Explorer

Theorem ccat2s1fstOLD

Description: Obsolete version of ccat2s1fst as of 28-Jan-2024. The first symbol of the concatenation of a word with two single symbols. (Contributed by Alexander van der Vekens, 22-Sep-2018) (New usage is discouraged.) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	ccat2s1fstOLD	\|- ( ( ( W e. Word V /\ 0 < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = ( W ` 0 ) )

Proof

Step	Hyp	Ref	Expression
1		0nn0	\|- 0 e. NN0
2		ccat2s1fvwOLD	\|- ( ( ( W e. Word V /\ 0 e. NN0 /\ 0 < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = ( W ` 0 ) )
3	1 2	mp3anl2	\|- ( ( ( W e. Word V /\ 0 < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = ( W ` 0 ) )

Step

Hyp

Ref

Expression

0nn0

 |-  0 e. NN0

ccat2s1fvwOLD

 |-  ( ( ( W e. Word V /\ 0 e. NN0 /\ 0 < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = ( W ` 0 ) )

1 2

mp3anl2

 |-  ( ( ( W e. Word V /\ 0 < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` 0 ) = ( W ` 0 ) )