Theorem grpolcan 28313

Description: Left cancellation law for groups. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
grplcan.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
grpolcan	⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))

Proof of Theorem grpolcan

Step	Hyp	Ref	Expression
1		oveq2 7143	. . . . . 6 ⊢ ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)))
2	1	adantl 485	. . . . 5 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)))
3		grplcan.1	. . . . . . . . . . 11 ⊢ 𝑋 = ran 𝐺
4		eqid 2798	. . . . . . . . . . 11 ⊢ (GId‘𝐺) = (GId‘𝐺)
5		eqid 2798	. . . . . . . . . . 11 ⊢ (inv‘𝐺) = (inv‘𝐺)
6	3, 4, 5	grpolinv 28309	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐶 ∈ 𝑋) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
7	6	adantlr 714	. . . . . . . . 9 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
8	7	oveq1d 7150	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = ((GId‘𝐺)𝐺𝐴))
9	3, 5	grpoinvcl 28307	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐶 ∈ 𝑋) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
10	9	adantrl 715	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
11		simprr 772	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐶 ∈ 𝑋)
12		simprl 770	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
13	10, 11, 12	3jca 1125	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (((inv‘𝐺)‘𝐶) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋))
14	3	grpoass 28286	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ (((inv‘𝐺)‘𝐶) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)))
15	13, 14	syldan 594	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)))
16	15	anassrs 471	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)))
17	3, 4	grpolid 28299	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
18	17	adantr 484	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
19	8, 16, 18	3eqtr3d 2841	. . . . . . 7 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
20	19	adantrl 715	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
21	20	adantr 484	. . . . 5 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
22	6	adantrl 715	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
23	22	oveq1d 7150	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐵) = ((GId‘𝐺)𝐺𝐵))
24	9	adantrl 715	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
25		simprr 772	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐶 ∈ 𝑋)
26		simprl 770	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
27	24, 25, 26	3jca 1125	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (((inv‘𝐺)‘𝐶) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
28	3	grpoass 28286	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (((inv‘𝐺)‘𝐶) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐵) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)))
29	27, 28	syldan 594	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐵) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)))
30	3, 4	grpolid 28299	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
31	30	adantrr 716	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
32	23, 29, 31	3eqtr3d 2841	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
33	32	adantlr 714	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
34	33	adantr 484	. . . . 5 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
35	2, 21, 34	3eqtr3d 2841	. . . 4 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → 𝐴 = 𝐵)
36	35	exp53 451	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐶 ∈ 𝑋 → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → 𝐴 = 𝐵)))))
37	36	3imp2 1346	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → 𝐴 = 𝐵))
38		oveq2 7143	. 2 ⊢ (𝐴 = 𝐵 → (𝐶𝐺𝐴) = (𝐶𝐺𝐵))
39	37, 38	impbid1 228	1 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ran crn 5520  ‘cfv 6324  (class class class)co 7135  GrpOpcgr 28272  GIdcgi 28273  invcgn 28274

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  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 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-grpo 28276  df-gid 28277  df-ginv 28278

This theorem is referenced by:  grpo2inv  28314  vclcan  28354  rngolcan  35356  rngolz  35360