grplinvd - Metamath Proof Explorer

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Theorem grplinvd 19105

Description: The left inverse of a group element. Deduction associated with grplinv 19100. (Contributed by SN, 29-Jan-2025.)

**Hypotheses**
Ref	Expression
grplinvd.b	⊢ 𝐵 = (Base‘𝐺)
grplinvd.p	⊢ + = (+_g‘𝐺)
grplinvd.u	⊢ 0 = (0_g‘𝐺)
grplinvd.n	⊢ 𝑁 = (inv_g‘𝐺)
grplinvd.g	⊢ (𝜑 → 𝐺 ∈ Grp)
grplinvd.1	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
grplinvd	⊢ (𝜑 → ((𝑁‘𝑋) + 𝑋) = 0 )

**Proof of Theorem grplinvd**
Step	Hyp	Ref	Expression
1		grplinvd.g	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
2		grplinvd.1	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		grplinvd.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
4		grplinvd.p	. . 3 ⊢ + = (+_g‘𝐺)
5		grplinvd.u	. . 3 ⊢ 0 = (0_g‘𝐺)
6		grplinvd.n	. . 3 ⊢ 𝑁 = (inv_g‘𝐺)
7	3, 4, 5, 6	grplinv 19100	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 )
8	1, 2, 7	syl2anc 596	1 ⊢ (𝜑 → ((𝑁‘𝑋) + 𝑋) = 0 )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2146 ‘cfv 6540 (class class class)co 7419 Basecbs 17291 +_gcplusg 17332 0_gc0g 17514 Grpcgrp 19044 inv_gcminusg 19045

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-riota 7376 df-ov 7422 df-0g 17516 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-grp 19047 df-minusg 19048

This theorem is used by: xpsinv 19170 rngmneg2 20290 rloccring 33655 qsdrngilem 33840 ply1dg1rt 33934 grpcominv1 43340

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