Theorem grprcan 18854

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 2732	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
4	1, 2, 3	grpinvex 18825	. . . 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 771	. . . . . . . 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 7420	. . . . . . 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 765	. . . . . . . . 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 18824	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
10	8, 9	sylan 580	. . . . . . . 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 777	. . . . . . . 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 7602	. . . . . . 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 7602	. . . . . . 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 2780	. . . . . 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 18823	. . . . . . . . . 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 18846	. . . . . . . . . 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 18848	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝐵) → ((0g‘𝐺) + 𝑢) = 𝑢)
23	8, 22	sylan 580	. . . . . . . . 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 18825	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝐵) → ∃𝑣 ∈ 𝐵 (𝑣 + 𝑢) = (0g‘𝐺))
25	8, 24	sylan 580	. . . . . . . . 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 485	. . . . . . . . 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 481	. . . . . . . . 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 778	. . . . . . . . . 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 481	. . . . . . . . 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 18589	. . . . . . . 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 685	. . . . . . 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 7421	. . . . . 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 7421	. . . . . 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 2780	. . . . 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 18851	. . . . 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 18851	. . . . 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 2780	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋 = 𝑌)
38	37	expr 457	. . 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 3161	. 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
40		oveq1 7412	. 2 ⊢ (𝑋 = 𝑌 → (𝑋 + 𝑍) = (𝑌 + 𝑍))
41	39, 40	impbid1 224	1 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∃wrex 3070  ‘cfv 6540  (class class class)co 7405  Basecbs 17140  +_gcplusg 17193  0_gc0g 17381  Grpcgrp 18815

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-riota 7361  df-ov 7408  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818

This theorem is referenced by:  grpinveu  18855  grpid  18856  grpidlcan  18885  grpinvssd  18896  grpsubrcan  18900  grpsubadd  18907  sylow1lem4  19463  ringcom  20090  ringrz  20101  rhmunitinv  20282  lmodcom  20510  ogrpaddlt  32222  grpcominv1  41079  isnumbasgrplem2  41831  rngrz  46651  grptcepi  47670