grprinvd - Metamath Proof Explorer

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Theorem grprinvd 19238

Description: The right inverse of a group element. Deduction associated with grprinv 19233. (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
grprinvd	⊢ (𝜑 → (𝑋 + (𝑁‘𝑋)) = 0 )

**Proof of Theorem grprinvd**
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	grprinv 19233	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = 0 )
8	1, 2, 7	syl2anc 596	1 ⊢ (𝜑 → (𝑋 + (𝑁‘𝑋)) = 0 )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 ‘cfv 6561 (class class class)co 7446 Basecbs 17419 +_gcplusg 17460 0_gc0g 17642 Grpcgrp 19176 inv_gcminusg 19177

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5172 df-mpt 5191 df-id 5551 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-riota 7403 df-ov 7449 df-0g 17644 df-mgm 18848 df-sgrp 18940 df-mnd 18956 df-grp 19179 df-minusg 19180

This theorem is used by: conjnmz 19498 rngmneg1 20422

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