Theorem grplcan 18272

Description: Left cancellation law for groups. (Contributed by NM, 25-Aug-2011.)

Hypotheses

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

Assertion

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

Proof of Theorem grplcan

Step	Hyp	Ref	Expression
1		oveq2 7172	. . . . . 6 ⊢ ((𝑍 + 𝑋) = (𝑍 + 𝑌) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)))
2	1	adantl 485	. . . . 5 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)))
3		grplcan.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐺)
4		grplcan.p	. . . . . . . . . . 11 ⊢ + = (+g‘𝐺)
5		eqid 2738	. . . . . . . . . . 11 ⊢ (0g‘𝐺) = (0g‘𝐺)
6		eqid 2738	. . . . . . . . . . 11 ⊢ (invg‘𝐺) = (invg‘𝐺)
7	3, 4, 5, 6	grplinv 18263	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → (((invg‘𝐺)‘𝑍) + 𝑍) = (0g‘𝐺))
8	7	adantlr 715	. . . . . . . . 9 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → (((invg‘𝐺)‘𝑍) + 𝑍) = (0g‘𝐺))
9	8	oveq1d 7179	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑋) = ((0g‘𝐺) + 𝑋))
10	3, 6	grpinvcl 18262	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
11	10	adantrl 716	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
12		simprr 773	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
13		simprl 771	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
14	11, 12, 13	3jca 1129	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵))
15	3, 4	grpass 18221	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘𝑍) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)))
16	14, 15	syldan 594	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)))
17	16	anassrs 471	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)))
18	3, 4, 5	grplid 18244	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((0g‘𝐺) + 𝑋) = 𝑋)
19	18	adantr 484	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → ((0g‘𝐺) + 𝑋) = 𝑋)
20	9, 17, 19	3eqtr3d 2781	. . . . . . 7 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
21	20	adantrl 716	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
22	21	adantr 484	. . . . 5 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)) = 𝑋)
23	7	adantrl 716	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) + 𝑍) = (0g‘𝐺))
24	23	oveq1d 7179	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑌) = ((0g‘𝐺) + 𝑌))
25	10	adantrl 716	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((invg‘𝐺)‘𝑍) ∈ 𝐵)
26		simprr 773	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
27		simprl 771	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
28	25, 26, 27	3jca 1129	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
29	3, 4	grpass 18221	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (((invg‘𝐺)‘𝑍) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑌) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)))
30	28, 29	syldan 594	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑌) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)))
31	3, 4, 5	grplid 18244	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((0g‘𝐺) + 𝑌) = 𝑌)
32	31	adantrr 717	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((0g‘𝐺) + 𝑌) = 𝑌)
33	24, 30, 32	3eqtr3d 2781	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
34	33	adantlr 715	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
35	34	adantr 484	. . . . 5 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
36	2, 22, 35	3eqtr3d 2781	. . . 4 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → 𝑋 = 𝑌)
37	36	exp53 451	. . 3 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → (𝑌 ∈ 𝐵 → (𝑍 ∈ 𝐵 → ((𝑍 + 𝑋) = (𝑍 + 𝑌) → 𝑋 = 𝑌)))))
38	37	3imp2 1350	. 2 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) → 𝑋 = 𝑌))
39		oveq2 7172	. 2 ⊢ (𝑋 = 𝑌 → (𝑍 + 𝑋) = (𝑍 + 𝑌))
40	38, 39	impbid1 228	1 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2113  ‘cfv 6333  (class class class)co 7164  Basecbs 16579  +_gcplusg 16661  0_gc0g 16809  Grpcgrp 18212  inv_gcminusg 18213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-fv 6341  df-riota 7121  df-ov 7167  df-0g 16811  df-mgm 17961  df-sgrp 18010  df-mnd 18021  df-grp 18215  df-minusg 18216

This theorem is referenced by:  grpidrcan  18275  grpinvinv  18277  grplmulf1o  18284  grplactcnv  18313  conjghm  18500  conjnmzb  18504  sylow3lem2  18864  gex2abl  19083  ringcom  19444  ringlz  19452  lmodlcan  19762  lmodfopne  19784  isnumbasgrplem2  40485  rnglz  44960  grptcmon  45814