Theorem xrsmul

Description: The multiplication operation of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015)

	Ref	Expression
Assertion	xrsmul	$$⊢{\cdot }_{𝑒}={\cdot }_{{ℝ}_{𝑠}^{*}}$$

Proof

Step	Hyp	Ref	Expression
1		xmulf	$$⊢{\cdot }_{𝑒}:{ℝ}^{*}\times {ℝ}^{*}⟶{ℝ}^{*}$$
2		xrex	$$⊢{ℝ}^{*}\in \mathrm{V}$$
3	2 2	xpex	$$⊢{ℝ}^{*}\times {ℝ}^{*}\in \mathrm{V}$$
4		fex2	$$⊢\left({\cdot }_{𝑒}:{ℝ}^{*}\times {ℝ}^{*}⟶{ℝ}^{*}\wedge {ℝ}^{*}\times {ℝ}^{*}\in \mathrm{V}\wedge {ℝ}^{*}\in \mathrm{V}\right)\to {\cdot }_{𝑒}\in \mathrm{V}$$
5	1 3 2 4	mp3an	$$⊢{\cdot }_{𝑒}\in \mathrm{V}$$
6		df-xrs	$$⊢{ℝ}_{𝑠}^{*}=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},{ℝ}^{*}⟩,⟨{+}_{\mathrm{ndx}},{+}_{𝑒}⟩,⟨{\cdot }_{\mathrm{ndx}},{\cdot }_{𝑒}⟩\right\}\cup \left\{⟨\mathrm{TopSet}\left(\mathrm{ndx}\right),\mathrm{ordTop}\left(\le \right)⟩,⟨{\le }_{\mathrm{ndx}},\le ⟩,⟨\mathrm{dist}\left(\mathrm{ndx}\right),\left(x\in {ℝ}^{*},y\in {ℝ}^{*}⟼if\left(x\le y,y{+}_{𝑒}\left(-x\right),x{+}_{𝑒}\left(-y\right)\right)\right)⟩\right\}$$
7	6	odrngmulr	$$⊢{\cdot }_{𝑒}\in \mathrm{V}\to {\cdot }_{𝑒}={\cdot }_{{ℝ}_{𝑠}^{*}}$$
8	5 7	ax-mp	$$⊢{\cdot }_{𝑒}={\cdot }_{{ℝ}_{𝑠}^{*}}$$