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Theorem grprinvd 19068

Description: The right inverse of a group element. Deduction associated with grprinv 19063. (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 19063	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = 0 )
8	1, 2, 7	syl2anc 595	1 ⊢ (𝜑 → (𝑋 + (𝑁‘𝑋)) = 0 )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ∈ wcel 2142 ‘cfv 6536 (class class class)co 7412 Basecbs 17275 +_gcplusg 17316 0_gc0g 17498 Grpcgrp 19006 inv_gcminusg 19007

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5555 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-riota 7369 df-ov 7415 df-0g 17500 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-grp 19009 df-minusg 19010

This theorem is used by: conjnmz 19328 rngmneg1 20251

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