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Theorem grpinvval 17777

Description: The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)

**Hypotheses**
Ref	Expression
grpinvval.b	⊢ 𝐵 = (Base‘𝐺)
grpinvval.p	⊢ + = (+_g‘𝐺)
grpinvval.o	⊢ 0 = (0_g‘𝐺)
grpinvval.n	⊢ 𝑁 = (inv_g‘𝐺)

**Assertion**
Ref	Expression
grpinvval	⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))

Distinct variable groups: 𝑦,𝐵 𝑦,𝐺 𝑦,𝑋 Allowed substitution hints: + (𝑦) 𝑁(𝑦) 0 (𝑦)

Proof of Theorem grpinvval Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6886	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋))
2	1	eqeq1d 2801	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 ))
3	2	riotabidv 6841	. 2 ⊢ (𝑥 = 𝑋 → (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))
4		grpinvval.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
5		grpinvval.p	. . 3 ⊢ + = (+_g‘𝐺)
6		grpinvval.o	. . 3 ⊢ 0 = (0_g‘𝐺)
7		grpinvval.n	. . 3 ⊢ 𝑁 = (inv_g‘𝐺)
8	4, 5, 6, 7	grpinvfval 17776	. 2 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
9		riotaex 6843	. 2 ⊢ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) ∈ V
10	3, 8, 9	fvmpt 6507	1 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ‘cfv 6101 ℩crio 6838 (class class class)co 6878 Basecbs 16184 +_gcplusg 16267 0_gc0g 16415 inv_gcminusg 17739

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-minusg 17742

This theorem is referenced by: grplinv 17784 isgrpinv 17788 xrsinvgval 30193 ringinvval 30308

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