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Theorem grpinvval 17508

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 6698	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋))
2	1	eqeq1d 2653	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 ))
3	2	riotabidv 6653	. 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 17507	. 2 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
9		riotaex 6655	. 2 ⊢ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) ∈ V
10	3, 8, 9	fvmpt 6321	1 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 ℩crio 6650 (class class class)co 6690 Basecbs 15904 +_gcplusg 15988 0_gc0g 16147 inv_gcminusg 17470

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-minusg 17473

This theorem is referenced by: grplinv 17515 isgrpinv 17519 xrsinvgval 29805 ringinvval 29920

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