Theorem grpasscan1 19092

Description: An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by AV, 30-Aug-2021.)

Hypotheses

Ref	Expression
grplcan.b	⊢ 𝐵 = (Base‘𝐺)
grplcan.p	⊢ + = (+g‘𝐺)
grpasscan1.n	⊢ 𝑁 = (invg‘𝐺)

Assertion

Ref	Expression
grpasscan1	⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((𝑁‘𝑋) + 𝑌)) = 𝑌)

Proof of Theorem grpasscan1

Step	Hyp	Ref	Expression
1		grplcan.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
2		grplcan.p	. . . . 5 ⊢ + = (+g‘𝐺)
3		eqid 2765	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
4		grpasscan1.n	. . . . 5 ⊢ 𝑁 = (invg‘𝐺)
5	1, 2, 3, 4	grprinv 19081	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = (0g‘𝐺))
6	5	3adant3 1150	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = (0g‘𝐺))
7	6	oveq1d 7431	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = ((0g‘𝐺) + 𝑌))
8	1, 4	grpinvcl 19078	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵)
9	1, 2	grpass 19033	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ (𝑁‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌)))
10	9	3exp2 1373	. . . . 5 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → ((𝑁‘𝑋) ∈ 𝐵 → (𝑌 ∈ 𝐵 → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌))))))
11	10	imp 412	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑁‘𝑋) ∈ 𝐵 → (𝑌 ∈ 𝐵 → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌)))))
12	8, 11	mpd 16	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑌 ∈ 𝐵 → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌))))
13	12	3impia 1135	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌)))
14	1, 2, 3	grplid 19058	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺) + 𝑌) = 𝑌)
15	14	3adant2 1149	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺) + 𝑌) = 𝑌)
16	7, 13, 15	3eqtr3d 2808	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((𝑁‘𝑋) + 𝑌)) = 𝑌)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ‘cfv 6540  (class class class)co 7416  Basecbs 17287  +_gcplusg 17328  0_gc0g 17510  Grpcgrp 19024  inv_gcminusg 19025

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-riota 7373  df-ov 7419  df-0g 17512  df-mgm 18716  df-sgrp 18799  df-mnd 18815  df-grp 19027  df-minusg 19028

This theorem is used by:  mulgaddcomlem  19187  ghmqusnsglem1  19374  ghmquskerlem1  19377  grplsmid  33753  nsgqusf1olem3  33764  qsdrnglem2  33818