Theorem grpasscan2d 42502

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 18941	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)
8	1, 2, 3, 7	syl3anc 1373	1 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  Grpcgrp 18872  inv_gcminusg 18873

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-riota 7347  df-ov 7393  df-0g 17411  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-grp 18875  df-minusg 18876

This theorem is referenced by:  grpcominv1  42503