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Metamath Proof Explorer

Theorem lenpfxcctswrd

Description: The length of the concatenation of the prefix of a word and the rest of the word is the length of the word. (Contributed by AV, 21-Oct-2018) (Revised by AV, 9-May-2020)

		Ref	Expression
	Assertion	lenpfxcctswrd	$⊢ (W \in Word V \land M \in (0 \dots \|W\|)) \to \|(W prefix M) ++ (W substr ⟨M, \|W\|⟩)\| = \|W\|$

Proof

Step	Hyp	Ref	Expression
1		pfxcctswrd	$⊢ (W \in Word V \land M \in (0 \dots \|W\|)) \to (W prefix M) ++ (W substr ⟨M, \|W\|⟩) = W$
2	1	fveq2d	$⊢ (W \in Word V \land M \in (0 \dots \|W\|)) \to \|(W prefix M) ++ (W substr ⟨M, \|W\|⟩)\| = \|W\|$