Theorem grprcan 19046

Description: Right cancellation law for groups. (Contributed by NM, 24-Aug-2011.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
grprcan.b	⊢ 𝐵 = (Base‘𝐺)
grprcan.p	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
grprcan	⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) ↔ 𝑋 = 𝑌))

Proof of Theorem grprcan

Dummy variables 𝑣 𝑢 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grprcan.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
2		grprcan.p	. . . . 5 ⊢ + = (+g‘𝐺)
3		eqid 2762	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
4	1, 2, 3	grpinvex 19016	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑍) = (0g‘𝐺))
5	4	3ad2antr3 1208	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑍) = (0g‘𝐺))
6		simprr 784	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + 𝑍) = (𝑌 + 𝑍))
7	6	oveq1d 7427	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑋 + 𝑍) + 𝑦) = ((𝑌 + 𝑍) + 𝑦))
8		simpll 778	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝐺 ∈ Grp)
9	1, 2	grpass 19015	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
10	8, 9	sylan 591	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
11		simplr1 1233	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋 ∈ 𝐵)
12		simplr3 1235	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑍 ∈ 𝐵)
13		simprll 790	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑦 ∈ 𝐵)
14	10, 11, 12, 13	caovassd 7611	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑋 + 𝑍) + 𝑦) = (𝑋 + (𝑍 + 𝑦)))
15		simplr2 1234	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑌 ∈ 𝐵)
16	10, 15, 12, 13	caovassd 7611	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → ((𝑌 + 𝑍) + 𝑦) = (𝑌 + (𝑍 + 𝑦)))
17	7, 14, 16	3eqtr3d 2805	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (𝑍 + 𝑦)) = (𝑌 + (𝑍 + 𝑦)))
18	1, 2	grpcl 19014	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
19	8, 18	syl3an1 1180	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
20	1, 3	grpidcl 19038	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
21	8, 20	syl 18	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (0g‘𝐺) ∈ 𝐵)
22	1, 2, 3	grplid 19040	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝐵) → ((0g‘𝐺) + 𝑢) = 𝑢)
23	8, 22	sylan 591	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢 ∈ 𝐵) → ((0g‘𝐺) + 𝑢) = 𝑢)
24	1, 2, 3	grpinvex 19016	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝐵) → ∃𝑣 ∈ 𝐵 (𝑣 + 𝑢) = (0g‘𝐺))
25	8, 24	sylan 591	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑢 ∈ 𝐵) → ∃𝑣 ∈ 𝐵 (𝑣 + 𝑢) = (0g‘𝐺))
26		simpr 489	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ 𝐵)
27	13	adantr 485	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍 ∈ 𝐵) → 𝑦 ∈ 𝐵)
28		simprlr 791	. . . . . . . . . 10 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑦 + 𝑍) = (0g‘𝐺))
29	28	adantr 485	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍 ∈ 𝐵) → (𝑦 + 𝑍) = (0g‘𝐺))
30	19, 21, 23, 10, 25, 26, 27, 29	grpinva 18738	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) ∧ 𝑍 ∈ 𝐵) → (𝑍 + 𝑦) = (0g‘𝐺))
31	12, 30	mpdan 699	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑍 + 𝑦) = (0g‘𝐺))
32	31	oveq2d 7428	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (𝑍 + 𝑦)) = (𝑋 + (0g‘𝐺)))
33	31	oveq2d 7428	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑌 + (𝑍 + 𝑦)) = (𝑌 + (0g‘𝐺)))
34	17, 32, 33	3eqtr3d 2805	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (0g‘𝐺)) = (𝑌 + (0g‘𝐺)))
35	1, 2, 3, 8, 11	grpridd 19043	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (0g‘𝐺)) = 𝑋)
36	1, 2, 3, 8, 15	grpridd 19043	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑌 + (0g‘𝐺)) = 𝑌)
37	34, 35, 36	3eqtr3d 2805	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋 = 𝑌)
38	37	expr 461	. . 3 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺))) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
39	5, 38	rexlimddv 3171	. 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
40		oveq1 7419	. 2 ⊢ (𝑋 = 𝑌 → (𝑋 + 𝑍) = (𝑌 + 𝑍))
41	39, 40	impbid1 228	1 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) ↔ 𝑋 = 𝑌))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∃wrex 3088  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  +_gcplusg 17316  0_gc0g 17498  Grpcgrp 19006

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-riota 7369  df-ov 7415  df-0g 17500  df-mgm 18704  df-sgrp 18783  df-mnd 18799  df-grp 19009

This theorem is used by:  grpinveu  19047  grpid  19048  grpidlcan  19077  grpraddf1o  19086  grpinvssd  19089  grpsubrcan  19093  grpsubadd  19100  sylow1lem4  19677  ogrpaddlt  20214  rngrz  20250  ringcom  20370  rhmunitinv  20619  lmodcom  21040  r1pid2  26330  cntrval2  33500  ply1dg1rt  33879  grpcominv1  43310  isnumbasgrplem2  43859  grptcepi  50400