Theorem isgrpde

Description: Deduce a group from its properties. In this version of isgrpd , we don't assume there is an expression for the inverse of x . (Contributed by Jeff Madsen, 1-Dec-2009) (Revised by NM, 6-Jan-2015)

	Ref	Expression
Hypotheses	isgrpd.b	$$⊢\phi \to B={\mathrm{Base}}_G$$
	isgrpd.p	$$⊢\phi \to \underset{\dot{}}{+}={+}_G$$
	isgrpd.c	$$⊢\left(\phi \wedge x\in B\wedge y\in B\right)\to x\underset{\dot{}}{+}y\in B$$
	isgrpd.a	$$⊢\left(\phi \wedge \left(x\in B\wedge y\in B\wedge z\in B\right)\right)\to \left(x\underset{\dot{}}{+}y\right)\underset{\dot{}}{+}z=x\underset{\dot{}}{+}\left(y\underset{\dot{}}{+}z\right)$$
	isgrpd.z	$$⊢\phi \to \underset{\dot{}}{0}\in B$$
	isgrpd.i	$$⊢\left(\phi \wedge x\in B\right)\to \underset{\dot{}}{0}\underset{\dot{}}{+}x=x$$
	isgrpde.n	$$⊢\left(\phi \wedge x\in B\right)\to \exists y\in By\underset{\dot{}}{+}x=\underset{\dot{}}{0}$$
Assertion	isgrpde	$$⊢\phi \to G\in \mathrm{Grp}$$

Proof

Step	Hyp	Ref	Expression
1		isgrpd.b	$$⊢\phi \to B={\mathrm{Base}}_G$$
2		isgrpd.p	$$⊢\phi \to \underset{\dot{}}{+}={+}_G$$
3		isgrpd.c	$$⊢\left(\phi \wedge x\in B\wedge y\in B\right)\to x\underset{\dot{}}{+}y\in B$$
4		isgrpd.a	$$⊢\left(\phi \wedge \left(x\in B\wedge y\in B\wedge z\in B\right)\right)\to \left(x\underset{\dot{}}{+}y\right)\underset{\dot{}}{+}z=x\underset{\dot{}}{+}\left(y\underset{\dot{}}{+}z\right)$$
5		isgrpd.z	$$⊢\phi \to \underset{\dot{}}{0}\in B$$
6		isgrpd.i	$$⊢\left(\phi \wedge x\in B\right)\to \underset{\dot{}}{0}\underset{\dot{}}{+}x=x$$
7		isgrpde.n	$$⊢\left(\phi \wedge x\in B\right)\to \exists y\in By\underset{\dot{}}{+}x=\underset{\dot{}}{0}$$
8	3 5 6 4 7	grprida	$$⊢\left(\phi \wedge x\in B\right)\to x\underset{\dot{}}{+}\underset{\dot{}}{0}=x$$
9	1 2 5 6 8	grpidd	$$⊢\phi \to \underset{\dot{}}{0}=0_G$$
10	1 2 3 4 5 6 8	ismndd	$$⊢\phi \to G\in \mathrm{Mnd}$$
11	1 2 9 10 7	isgrpd2e	$$⊢\phi \to G\in \mathrm{Grp}$$