Theorem gsumval1

Description: Value of the group sum operation when every element being summed is an identity of G . (Contributed by Mario Carneiro, 7-Dec-2014)

	Ref	Expression
Hypotheses	gsumval1.b	$$⊢B={\mathrm{Base}}_G$$
	gsumval1.z	$$⊢\underset{\dot{}}{0}=0_G$$
	gsumval1.p	$$⊢\underset{\dot{}}{+}={+}_G$$
	gsumval1.o	$$⊢O=\left\{x\in B|\forall y\in B\left(x\underset{\dot{}}{+}y=y\wedge y\underset{\dot{}}{+}x=y\right)\right\}$$
	gsumval1.g	$$⊢\phi \to G\in V$$
	gsumval1.a	$$⊢\phi \to A\in W$$
	gsumval1.f	$$⊢\phi \to F:A⟶O$$
Assertion	gsumval1	$$⊢\phi \to {\sum }_{G}F=\underset{\dot{}}{0}$$

Proof

Step	Hyp	Ref	Expression
1		gsumval1.b	$$⊢B={\mathrm{Base}}_G$$
2		gsumval1.z	$$⊢\underset{\dot{}}{0}=0_G$$
3		gsumval1.p	$$⊢\underset{\dot{}}{+}={+}_G$$
4		gsumval1.o	$$⊢O=\left\{x\in B|\forall y\in B\left(x\underset{\dot{}}{+}y=y\wedge y\underset{\dot{}}{+}x=y\right)\right\}$$
5		gsumval1.g	$$⊢\phi \to G\in V$$
6		gsumval1.a	$$⊢\phi \to A\in W$$
7		gsumval1.f	$$⊢\phi \to F:A⟶O$$
8		eqidd	$$⊢\phi \to F^{-1}\left[\mathrm{V}\setminus O\right]=F^{-1}\left[\mathrm{V}\setminus O\right]$$
9	4	ssrab3	$$⊢O\subseteq B$$
10		fss	$$⊢\left(F:A⟶O\wedge O\subseteq B\right)\to F:A⟶B$$
11	7 9 10	sylancl	$$⊢\phi \to F:A⟶B$$
12	1 2 3 4 8 5 6 11	gsumval	$$⊢\phi \to {\sum }_{G}F=if\left(\mathrm{ran}F\subseteq O,\underset{\dot{}}{0},if\left(A\in \mathrm{ran}\dots ,\left(\iota z|\exists m\exists n\in {\mathbb{Z}}_{\ge m}\left(A=\left(m\dots n\right)\wedge z=seqm\left(\underset{\dot{}}{+},F\right)\left(n\right)\right)\right),\left(\iota z|\exists f\left(f:\left(1\dots \left|F^{-1}\left[\mathrm{V}\setminus O\right]\right|\right)\underset{1-1\; onto}{⟶}{F}^{-1}\left[\mathrm{V}\setminus O\right]\wedge z=seq1\left(\underset{\dot{}}{+},\left(F\circ f\right)\right)\left(\left|F^{-1}\left[\mathrm{V}\setminus O\right]\right|\right)\right)\right)\right)\right)$$
13		frn	$$⊢F:A⟶O\to \mathrm{ran}F\subseteq O$$
14		iftrue	$$⊢\mathrm{ran}F\subseteq O\to if\left(\mathrm{ran}F\subseteq O,\underset{\dot{}}{0},if\left(A\in \mathrm{ran}\dots ,\left(\iota z|\exists m\exists n\in {\mathbb{Z}}_{\ge m}\left(A=\left(m\dots n\right)\wedge z=seqm\left(\underset{\dot{}}{+},F\right)\left(n\right)\right)\right),\left(\iota z|\exists f\left(f:\left(1\dots \left|F^{-1}\left[\mathrm{V}\setminus O\right]\right|\right)\underset{1-1\; onto}{⟶}{F}^{-1}\left[\mathrm{V}\setminus O\right]\wedge z=seq1\left(\underset{\dot{}}{+},\left(F\circ f\right)\right)\left(\left|F^{-1}\left[\mathrm{V}\setminus O\right]\right|\right)\right)\right)\right)\right)=\underset{\dot{}}{0}$$
15	7 13 14	3syl	$$⊢\phi \to if\left(\mathrm{ran}F\subseteq O,\underset{\dot{}}{0},if\left(A\in \mathrm{ran}\dots ,\left(\iota z|\exists m\exists n\in {\mathbb{Z}}_{\ge m}\left(A=\left(m\dots n\right)\wedge z=seqm\left(\underset{\dot{}}{+},F\right)\left(n\right)\right)\right),\left(\iota z|\exists f\left(f:\left(1\dots \left|F^{-1}\left[\mathrm{V}\setminus O\right]\right|\right)\underset{1-1\; onto}{⟶}{F}^{-1}\left[\mathrm{V}\setminus O\right]\wedge z=seq1\left(\underset{\dot{}}{+},\left(F\circ f\right)\right)\left(\left|F^{-1}\left[\mathrm{V}\setminus O\right]\right|\right)\right)\right)\right)\right)=\underset{\dot{}}{0}$$
16	12 15	eqtrd	$$⊢\phi \to {\sum }_{G}F=\underset{\dot{}}{0}$$