Theorem grprcan 18938

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 2737	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
4	1, 2, 3	grpinvex 18908	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑍) = (0g‘𝐺))
5	4	3ad2antr3 1192	. . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑍) = (0g‘𝐺))
6		simprr 773	. . . . . . . 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 7373	. . . . . . 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 767	. . . . . . . . 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 18907	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
10	8, 9	sylan 581	. . . . . . . 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 1217	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑋 ∈ 𝐵)
12		simplr3 1219	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ ((𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑍) = (0g‘𝐺)) ∧ (𝑋 + 𝑍) = (𝑌 + 𝑍))) → 𝑍 ∈ 𝐵)
13		simprll 779	. . . . . . . 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 7557	. . . . . . 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 1218	. . . . . . . 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 7557	. . . . . . 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 18906	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
19	8, 18	syl3an1 1164	. . . . . . . . 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 18930	. . . . . . . . . 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 18932	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝐵) → ((0g‘𝐺) + 𝑢) = 𝑢)
23	8, 22	sylan 581	. . . . . . . . 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 18908	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝐵) → ∃𝑣 ∈ 𝐵 (𝑣 + 𝑢) = (0g‘𝐺))
25	8, 24	sylan 581	. . . . . . . . 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 780	. . . . . . . . . 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 18631	. . . . . . . 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 688	. . . . . . 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 7374	. . . . . 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 7374	. . . . . 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 18935	. . . . 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 18935	. . . . 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 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 3145	. 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑍) = (𝑌 + 𝑍) → 𝑋 = 𝑌))
40		oveq1 7365	. 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 1542   ∈ wcel 2114  ∃wrex 3062  ‘cfv 6490  (class class class)co 7358  Basecbs 17168  +_gcplusg 17209  0_gc0g 17391  Grpcgrp 18898

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-riota 7315  df-ov 7361  df-0g 17393  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-grp 18901

This theorem is referenced by:  grpinveu  18939  grpid  18940  grpidlcan  18969  grpraddf1o  18979  grpinvssd  18982  grpsubrcan  18986  grpsubadd  18993  sylow1lem4  19565  ogrpaddlt  20102  rngrz  20136  ringcom  20250  rhmunitinv  20477  lmodcom  20892  r1pid2  26139  cntrval2  33252  ply1dg1rt  33660  r1pid2OLD  33689  grpcominv1  42964  isnumbasgrplem2  43547  grptcepi  50066