Theorem grpolcan 30819

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 7416	. . . . . 6 ⊢ ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)))
2	1	adantl 486	. . . . 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 2769	. . . . . . . . . . 11 ⊢ (GId‘𝐺) = (GId‘𝐺)
5		eqid 2769	. . . . . . . . . . 11 ⊢ (inv‘𝐺) = (inv‘𝐺)
6	3, 4, 5	grpolinv 30815	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐶 ∈ 𝑋) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
7	6	adantlr 727	. . . . . . . . 9 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
8	7	oveq1d 7423	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = ((GId‘𝐺)𝐺𝐴))
9	3, 5	grpoinvcl 30813	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐶 ∈ 𝑋) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
10	9	adantrl 728	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
11		simprr 784	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐶 ∈ 𝑋)
12		simprl 782	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
13	10, 11, 12	3jca 1144	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (((inv‘𝐺)‘𝐶) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋))
14	3	grpoass 30792	. . . . . . . . . 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 602	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)))
16	15	anassrs 472	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐴) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)))
17	3, 4	grpolid 30805	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
18	17	adantr 485	. . . . . . . 8 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋) → ((GId‘𝐺)𝐺𝐴) = 𝐴)
19	8, 16, 18	3eqtr3d 2812	. . . . . . 7 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
20	19	adantrl 728	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
21	20	adantr 485	. . . . 5 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐴)) = 𝐴)
22	6	adantrl 728	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (((inv‘𝐺)‘𝐶)𝐺𝐶) = (GId‘𝐺))
23	22	oveq1d 7423	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐵) = ((GId‘𝐺)𝐺𝐵))
24	9	adantrl 728	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
25		simprr 784	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐶 ∈ 𝑋)
26		simprl 782	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
27	24, 25, 26	3jca 1144	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (((inv‘𝐺)‘𝐶) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
28	3	grpoass 30792	. . . . . . . . 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 602	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((((inv‘𝐺)‘𝐶)𝐺𝐶)𝐺𝐵) = (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)))
30	3, 4	grpolid 30805	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
31	30	adantrr 729	. . . . . . . 8 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((GId‘𝐺)𝐺𝐵) = 𝐵)
32	23, 29, 31	3eqtr3d 2812	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
33	32	adantlr 727	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
34	33	adantr 485	. . . . 5 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → (((inv‘𝐺)‘𝐶)𝐺(𝐶𝐺𝐵)) = 𝐵)
35	2, 21, 34	3eqtr3d 2812	. . . 4 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝐶𝐺𝐴) = (𝐶𝐺𝐵)) → 𝐴 = 𝐵)
36	35	exp53 452	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐶 ∈ 𝑋 → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → 𝐴 = 𝐵)))))
37	36	3imp2 1366	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐶𝐺𝐴) = (𝐶𝐺𝐵) → 𝐴 = 𝐵))
38		oveq2 7416	. 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 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ran crn 5660  ‘cfv 6533  (class class class)co 7408  GrpOpcgr 30778  GIdcgi 30779  invcgn 30780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-ov 7411  df-grpo 30782  df-gid 30783  df-ginv 30784

This theorem is referenced by:  grpo2inv  30820  vclcan  30860  rngolcan  38452  rngolz  38456