Theorem grprcan 18129

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 2798	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
4	1, 2, 3	grpinvex 18105	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑍) = (0g‘𝐺))
5	4	3ad2antr3 1187	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑍) = (0g‘𝐺))
6		simprr 772	. . . . . . . 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 7150	. . . . . . 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 766	. . . . . . . . 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 18104	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
10	8, 9	sylan 583	. . . . . . . 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 1212	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋 ∈ 𝐵)
12		simplr3 1214	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑍 ∈ 𝐵)
13		simprll 778	. . . . . . . 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 7327	. . . . . . 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 1213	. . . . . . . 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 7327	. . . . . . 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 2841	. . . . . 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 18103	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
19	8, 18	syl3an1 1160	. . . . . . . . 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 18123	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
21	8, 20	syl 17	. . . . . . . . 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 18125	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝐵) → ((0g‘𝐺) + 𝑢) = 𝑢)
23	8, 22	sylan 583	. . . . . . . . 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 18105	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝐵) → ∃𝑣 ∈ 𝐵 (𝑣 + 𝑢) = (0g‘𝐺))
25	8, 24	sylan 583	. . . . . . . . 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 488	. . . . . . . . 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 484	. . . . . . . . 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 779	. . . . . . . . . 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 484	. . . . . . . . 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	grprinvd 17876	. . . . . . . 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 686	. . . . . . 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 7151	. . . . . 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 7151	. . . . . 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 2841	. . . . 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	grprid 18126	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + (0g‘𝐺)) = 𝑋)
36	8, 11, 35	syl2anc 587	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑋 + (0g‘𝐺)) = 𝑋)
37	1, 2, 3	grprid 18126	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑌 + (0g‘𝐺)) = 𝑌)
38	8, 15, 37	syl2anc 587	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → (𝑌 + (0g‘𝐺)) = 𝑌)
39	34, 36, 38	3eqtr3d 2841	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋 = 𝑌)
40	39	expr 460	. . 3 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺))) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
41	5, 40	rexlimddv 3250	. 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
42		oveq1 7142	. 2 ⊢ (𝑋 = 𝑌 → (𝑋 + 𝑍) = (𝑌 + 𝑍))
43	41, 42	impbid1 228	1 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  +_gcplusg 16557  0_gc0g 16705  Grpcgrp 18095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-riota 7093  df-ov 7138  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098

This theorem is referenced by:  grpinveu  18130  grpid  18131  grpidlcan  18157  grpinvssd  18168  grpsubrcan  18172  grpsubadd  18179  sylow1lem4  18718  ringcom  19325  ringrz  19334  lmodcom  19673  ogrpaddlt  30768  rhmunitinv  30946  isnumbasgrplem2  40048