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Theorem grprinvd 19013
Description: The right inverse of a group element. Deduction associated with grprinv 19008. (Contributed by SN, 29-Jan-2025.)
Hypotheses
Ref Expression
grplinvd.b 𝐵 = (Base‘𝐺)
grplinvd.p + = (+g𝐺)
grplinvd.u 0 = (0g𝐺)
grplinvd.n 𝑁 = (invg𝐺)
grplinvd.g (𝜑𝐺 ∈ Grp)
grplinvd.1 (𝜑𝑋𝐵)
Assertion
Ref Expression
grprinvd (𝜑 → (𝑋 + (𝑁𝑋)) = 0 )

Proof of Theorem grprinvd
StepHypRef Expression
1 grplinvd.g . 2 (𝜑𝐺 ∈ Grp)
2 grplinvd.1 . 2 (𝜑𝑋𝐵)
3 grplinvd.b . . 3 𝐵 = (Base‘𝐺)
4 grplinvd.p . . 3 + = (+g𝐺)
5 grplinvd.u . . 3 0 = (0g𝐺)
6 grplinvd.n . . 3 𝑁 = (invg𝐺)
73, 4, 5, 6grprinv 19008 . 2 ((𝐺 ∈ Grp ∧ 𝑋𝐵) → (𝑋 + (𝑁𝑋)) = 0 )
81, 2, 7syl2anc 584 1 (𝜑 → (𝑋 + (𝑁𝑋)) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Grpcgrp 18951  invgcminusg 18952
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 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955
This theorem is referenced by:  conjnmz  19270  rngmneg1  20164
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