Theorem grpasscan2d 42462

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 19042	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)
8	1, 2, 3, 7	syl3anc 1371	1 ⊢ (𝜑 → ((𝑋 + (𝑁‘𝑌)) + 𝑌) = 𝑋)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  Grpcgrp 18973  inv_gcminusg 18974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-riota 7404  df-ov 7451  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977

This theorem is referenced by:  grpcominv1  42463