Theorem sq11d

Description: The square function is one-to-one for nonnegative reals. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	resqcld.1	$$⊢\phi \to A\in ℝ$$
	lt2sqd.2	$$⊢\phi \to B\in ℝ$$
	lt2sqd.3	$$⊢\phi \to 0\le A$$
	lt2sqd.4	$$⊢\phi \to 0\le B$$
	sq11d.5	$$⊢\phi \to A^2=B^2$$
Assertion	sq11d	$$⊢\phi \to A=B$$

Proof

Step	Hyp	Ref	Expression
1		resqcld.1	$$⊢\phi \to A\in ℝ$$
2		lt2sqd.2	$$⊢\phi \to B\in ℝ$$
3		lt2sqd.3	$$⊢\phi \to 0\le A$$
4		lt2sqd.4	$$⊢\phi \to 0\le B$$
5		sq11d.5	$$⊢\phi \to A^2=B^2$$
6		sq11	$$⊢\left(\left(A\in ℝ\wedge 0\le A\right)\wedge \left(B\in ℝ\wedge 0\le B\right)\right)\to \left(A^2=B^2\leftrightarrow A=B\right)$$
7	1 3 2 4 6	syl22anc	$$⊢\phi \to \left(A^2=B^2\leftrightarrow A=B\right)$$
8	5 7	mpbid	$$⊢\phi \to A=B$$