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Theorem grplinvd 18975

Description: The left inverse of a group element. Deduction associated with grplinv 18970. (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 18970	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) + 𝑋) = 0 )
8	1, 2, 7	syl2anc 584	1 ⊢ (𝜑 → ((𝑁‘𝑋) + 𝑋) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6530 (class class class)co 7403 Basecbs 17226 +_gcplusg 17269 0_gc0g 17451 Grpcgrp 18914 inv_gcminusg 18915

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-fv 6538 df-riota 7360 df-ov 7406 df-0g 17453 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-grp 18917 df-minusg 18918

This theorem is referenced by: xpsinv 19041 rngmneg2 20126 rloccring 33211 qsdrngilem 33455 ply1dg1rt 33538 grpcominv1 42478

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