Theorem grprcan 19013

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 2740	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
4	1, 2, 3	grpinvex 18983	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑍) = (0g‘𝐺))
5	4	3ad2antr3 1190	. . 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 7463	. . . . . . 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 18982	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
10	8, 9	sylan 579	. . . . . . . 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 1215	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋 ∈ 𝐵)
12		simplr3 1217	. . . . . . . 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 7649	. . . . . . 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 1216	. . . . . . . 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 7649	. . . . . . 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 2788	. . . . . 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 18981	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
19	8, 18	syl3an1 1163	. . . . . . . . 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 19005	. . . . . . . . . 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 19007	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝐵) → ((0g‘𝐺) + 𝑢) = 𝑢)
23	8, 22	sylan 579	. . . . . . . . 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 18983	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝐵) → ∃𝑣 ∈ 𝐵 (𝑣 + 𝑢) = (0g‘𝐺))
25	8, 24	sylan 579	. . . . . . . . 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 484	. . . . . . . . 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 480	. . . . . . . . 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 480	. . . . . . . . 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 18712	. . . . . . . 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 7464	. . . . . 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 7464	. . . . . 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 2788	. . . . 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 19010	. . . . 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 19010	. . . . 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 2788	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋 = 𝑌)
38	37	expr 456	. . 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 3167	. 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
40		oveq1 7455	. 2 ⊢ (𝑋 = 𝑌 → (𝑋 + 𝑍) = (𝑌 + 𝑍))
41	39, 40	impbid1 225	1 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∃wrex 3076  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  0_gc0g 17499  Grpcgrp 18973

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-pr 5447

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-ss 3993  df-nul 4353  df-if 4549  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-iota 6525  df-fun 6575  df-fv 6581  df-riota 7404  df-ov 7451  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976

This theorem is referenced by:  grpinveu  19014  grpid  19015  grpidlcan  19044  grpraddf1o  19054  grpinvssd  19057  grpsubrcan  19061  grpsubadd  19068  sylow1lem4  19643  rngrz  20193  ringcom  20303  rhmunitinv  20537  lmodcom  20928  r1pid2  26221  ogrpaddlt  33067  ply1dg1rt  33569  r1pid2OLD  33594  grpcominv1  42463  isnumbasgrplem2  43061  grptcepi  48764