Theorem grpasscan1 19064

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 2769	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
4		grpasscan1.n	. . . . 5 ⊢ 𝑁 = (invg‘𝐺)
5	1, 2, 3, 4	grprinv 19053	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = (0g‘𝐺))
6	5	3adant3 1148	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + (𝑁‘𝑋)) = (0g‘𝐺))
7	6	oveq1d 7423	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = ((0g‘𝐺) + 𝑌))
8	1, 4	grpinvcl 19050	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) ∈ 𝐵)
9	1, 2	grpass 19005	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ (𝑁‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌)))
10	9	3exp2 1371	. . . . 5 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → ((𝑁‘𝑋) ∈ 𝐵 → (𝑌 ∈ 𝐵 → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌))))))
11	10	imp 411	. . . 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 1133	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + (𝑁‘𝑋)) + 𝑌) = (𝑋 + ((𝑁‘𝑋) + 𝑌)))
14	1, 2, 3	grplid 19030	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺) + 𝑌) = 𝑌)
15	14	3adant2 1147	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺) + 𝑌) = 𝑌)
16	7, 13, 15	3eqtr3d 2812	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((𝑁‘𝑋) + 𝑌)) = 𝑌)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ‘cfv 6533  (class class class)co 7408  Basecbs 17265  +_gcplusg 17306  0_gc0g 17488  Grpcgrp 18996  inv_gcminusg 18997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-riota 7365  df-ov 7411  df-0g 17490  df-mgm 18694  df-sgrp 18773  df-mnd 18789  df-grp 18999  df-minusg 19000

This theorem is referenced by:  mulgaddcomlem  19159  ghmqusnsglem1  19346  ghmquskerlem1  19349  grplsmid  33653  nsgqusf1olem3  33664  qsdrnglem2  33719