Theorem grpasscan2d 41802

Description: An associative cancellation law for groups. (Contributed by SN, 29-Jan-2025.)

Hypotheses

Ref	Expression
grpasscan2d.b	⊢ 𝐵 = (Base‘𝐺)
grpasscan2d.p	⊢ + = (+g‘𝐺)
grpasscan2d.n	⊢ 𝑁 = (invg‘𝐺)
grpasscan2d.g	⊢ (𝜑 → 𝐺 ∈ Grp)
grpasscan2d.1	⊢ (𝜑 → 𝑋 ∈ 𝐵)
grpasscan2d.2	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
grpasscan2d	⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)

Proof of Theorem grpasscan2d

Step	Hyp	Ref	Expression
1		grpasscan2d.g	. 2 ⊢ (𝜑 → 𝐺 ∈ Grp)
2		grpasscan2d.1	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		grpasscan2d.2	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
4		grpasscan2d.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
5		grpasscan2d.p	. . 3 ⊢ + = (+g‘𝐺)
6		grpasscan2d.n	. . 3 ⊢ 𝑁 = (invg‘𝐺)
7	4, 5, 6	grpasscan2 18961	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)
8	1, 2, 3, 7	syl3anc 1368	1 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ‘cfv 6542  (class class class)co 7415  Basecbs 17177  +_gcplusg 17230  Grpcgrp 18892  inv_gcminusg 18893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-riota 7371  df-ov 7418  df-0g 17420  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-grp 18895  df-minusg 18896

This theorem is referenced by:  grpcominv1  41803