Theorem xleadd2a

Description: Commuted form of xleadd1a . (Contributed by Mario Carneiro, 20-Aug-2015)

	Ref	Expression
Assertion	xleadd2a	$$⊢\left(\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\wedge C\in {ℝ}^{*}\right)\wedge A\le B\right)\to C{+}_{𝑒}A\le C{+}_{𝑒}B$$

Proof

Step	Hyp	Ref	Expression
1		xleadd1a	$$⊢\left(\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\wedge C\in {ℝ}^{*}\right)\wedge A\le B\right)\to A{+}_{𝑒}C\le B{+}_{𝑒}C$$
2		xaddcom	$$⊢\left(A\in {ℝ}^{*}\wedge C\in {ℝ}^{*}\right)\to A{+}_{𝑒}C=C{+}_{𝑒}A$$
3	2	3adant2	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\wedge C\in {ℝ}^{*}\right)\to A{+}_{𝑒}C=C{+}_{𝑒}A$$
4	3	adantr	$$⊢\left(\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\wedge C\in {ℝ}^{*}\right)\wedge A\le B\right)\to A{+}_{𝑒}C=C{+}_{𝑒}A$$
5		xaddcom	$$⊢\left(B\in {ℝ}^{*}\wedge C\in {ℝ}^{*}\right)\to B{+}_{𝑒}C=C{+}_{𝑒}B$$
6	5	3adant1	$$⊢\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\wedge C\in {ℝ}^{*}\right)\to B{+}_{𝑒}C=C{+}_{𝑒}B$$
7	6	adantr	$$⊢\left(\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\wedge C\in {ℝ}^{*}\right)\wedge A\le B\right)\to B{+}_{𝑒}C=C{+}_{𝑒}B$$
8	1 4 7	3brtr3d	$$⊢\left(\left(A\in {ℝ}^{*}\wedge B\in {ℝ}^{*}\wedge C\in {ℝ}^{*}\right)\wedge A\le B\right)\to C{+}_{𝑒}A\le C{+}_{𝑒}B$$