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Theorem grpinvnzcl 18940

Description: The inverse of a nonzero group element is a nonzero group element. (Contributed by Stefan O'Rear, 27-Feb-2015.)

**Hypotheses**
Ref	Expression
grpinvnzcl.b	⊢ 𝐵 = (Base‘𝐺)
grpinvnzcl.z	⊢ 0 = (0_g‘𝐺)
grpinvnzcl.n	⊢ 𝑁 = (inv_g‘𝐺)

**Assertion**
Ref	Expression
grpinvnzcl	⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ∖ { 0 })) → (𝑁‘𝑋) ∈ (𝐵 ∖ { 0 }))

**Proof of Theorem grpinvnzcl**
Step	Hyp	Ref	Expression
1		eldifi 4121	. . 3 ⊢ (𝑋 ∈ (𝐵 ∖ { 0 }) → 𝑋 ∈ 𝐵)
2		grpinvnzcl.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3		grpinvnzcl.n	. . . 4 ⊢ 𝑁 = (inv_g‘𝐺)
4	2, 3	grpinvcl 18917	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵)
5	1, 4	sylan2 592	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ∖ { 0 })) → (𝑁‘𝑋) ∈ 𝐵)
6		eldifsn 4785	. . 3 ⊢ (𝑋 ∈ (𝐵 ∖ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ))
7		grpinvnzcl.z	. . . . 5 ⊢ 0 = (0_g‘𝐺)
8	2, 7, 3	grpinvnz 18939	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝑁‘𝑋) ≠ 0 )
9	8	3expb 1117	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )) → (𝑁‘𝑋) ≠ 0 )
10	6, 9	sylan2b 593	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ∖ { 0 })) → (𝑁‘𝑋) ≠ 0 )
11		eldifsn 4785	. 2 ⊢ ((𝑁‘𝑋) ∈ (𝐵 ∖ { 0 }) ↔ ((𝑁‘𝑋) ∈ 𝐵 ∧ (𝑁‘𝑋) ≠ 0 ))
12	5, 10, 11	sylanbrc 582	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵 ∖ { 0 })) → (𝑁‘𝑋) ∈ (𝐵 ∖ { 0 }))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∖ cdif 3940 {csn 4623 ‘cfv 6537 Basecbs 17153 0_gc0g 17394 Grpcgrp 18863 inv_gcminusg 18864

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-riota 7361 df-ov 7408 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867

This theorem is referenced by: islindf4 21733 baerlem5amN 41100 baerlem5bmN 41101 baerlem5abmN 41102 lindslinindsimp1 47418 lindslinindsimp2lem5 47423

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