Theorem grplcan 18923

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 7363	. . . . . 6 ⊢ ((𝑍 + 𝑋) = (𝑍 + 𝑌) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)))
2	1	adantl 481	. . . . 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 2733	. . . . . . . . . . 11 ⊢ (0g‘𝐺) = (0g‘𝐺)
6		eqid 2733	. . . . . . . . . . 11 ⊢ (invg‘𝐺) = (invg‘𝐺)
7	3, 4, 5, 6	grplinv 18912	. . . . . . . . . 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 7370	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑋) = ((0g‘𝐺) + 𝑋))
10	3, 6	grpinvcl 18910	. . . . . . . . . . . 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 772	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
13		simprl 770	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
14	11, 12, 13	3jca 1128	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵))
15	3, 4	grpass 18865	. . . . . . . . . 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 591	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)))
17	16	anassrs 467	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑋) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑋)))
18	3, 4, 5	grplid 18890	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((0g‘𝐺) + 𝑋) = 𝑋)
19	18	adantr 480	. . . . . . . 8 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑍 ∈ 𝐵) → ((0g‘𝐺) + 𝑋) = 𝑋)
20	9, 17, 19	3eqtr3d 2776	. . . . . . 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 480	. . . . 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 7370	. . . . . . . 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 772	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
27		simprl 770	. . . . . . . . . 10 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
28	25, 26, 27	3jca 1128	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((invg‘𝐺)‘𝑍) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵))
29	3, 4	grpass 18865	. . . . . . . . 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 591	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((((invg‘𝐺)‘𝑍) + 𝑍) + 𝑌) = (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)))
31	3, 4, 5	grplid 18890	. . . . . . . . 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 2776	. . . . . . 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 480	. . . . 5 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → (((invg‘𝐺)‘𝑍) + (𝑍 + 𝑌)) = 𝑌)
36	2, 22, 35	3eqtr3d 2776	. . . 4 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ (𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ (𝑍 + 𝑋) = (𝑍 + 𝑌)) → 𝑋 = 𝑌)
37	36	exp53 447	. . 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 7363	. 2 ⊢ (𝑋 = 𝑌 → (𝑍 + 𝑋) = (𝑍 + 𝑌))
40	38, 39	impbid1 225	1 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑍 + 𝑋) = (𝑍 + 𝑌) ↔ 𝑋 = 𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ‘cfv 6489  (class class class)co 7355  Basecbs 17130  +_gcplusg 17171  0_gc0g 17353  Grpcgrp 18856  inv_gcminusg 18857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-riota 7312  df-ov 7358  df-0g 17355  df-mgm 18558  df-sgrp 18637  df-mnd 18653  df-grp 18859  df-minusg 18860

This theorem is referenced by:  grpidrcan  18926  grpinvinv  18928  grplmulf1o  18936  grplactcnv  18966  conjghm  19171  conjnmzb  19175  sylow3lem2  19550  gex2abl  19773  rnglz  20093  ringcom  20208  lmodlcan  20820  lmodfopne  20843  r1peuqusdeg1  35698  isnumbasgrplem2  43211  grptcmon  49708