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Theorem grpinvval 18947

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 7368	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋))
2	1	eqeq1d 2739	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 ))
3	2	riotabidv 7319	. 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 18945	. 2 ⊢ 𝑁 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ))
9		riotaex 7321	. 2 ⊢ (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) ∈ V
10	3, 8, 9	fvmpt 6941	1 ⊢ (𝑋 ∈ 𝐵 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ‘cfv 6492 ℩crio 7316 (class class class)co 7360 Basecbs 17170 +_gcplusg 17211 0_gc0g 17393 inv_gcminusg 18901

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-riota 7317 df-ov 7363 df-minusg 18904

This theorem is referenced by: grplinv 18956 isgrpinv 18960 xrsinvgval 33083 ringinvval 33311 ressply1invg 33644 linvh 42549 primrootsunit1 42550

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