Theorem grporcan 29809

Description: Right cancellation law for groups. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)

Hypothesis

Ref	Expression
grprcan.1	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
grporcan	⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))

Proof of Theorem grporcan

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grprcan.1	. . . . . . . 8 ⊢ 𝑋 = ran 𝐺
2		eqid 2732	. . . . . . . 8 ⊢ (GId‘𝐺) = (GId‘𝐺)
3	1, 2	grpoidinv2 29806	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐶 ∈ 𝑋) → ((((GId‘𝐺)𝐺𝐶) = 𝐶 ∧ (𝐶𝐺(GId‘𝐺)) = 𝐶) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺))))
4		simpr 485	. . . . . . . . 9 ⊢ (((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺)) → (𝐶𝐺𝑦) = (GId‘𝐺))
5	4	reximi 3084	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺)) → ∃𝑦 ∈ 𝑋 (𝐶𝐺𝑦) = (GId‘𝐺))
6	5	adantl 482	. . . . . . 7 ⊢ (((((GId‘𝐺)𝐺𝐶) = 𝐶 ∧ (𝐶𝐺(GId‘𝐺)) = 𝐶) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺))) → ∃𝑦 ∈ 𝑋 (𝐶𝐺𝑦) = (GId‘𝐺))
7	3, 6	syl 17	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐶 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 (𝐶𝐺𝑦) = (GId‘𝐺))
8	7	ad2ant2rl 747	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ∃𝑦 ∈ 𝑋 (𝐶𝐺𝑦) = (GId‘𝐺))
9		oveq1 7418	. . . . . . . . . . . 12 ⊢ ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) → ((𝐴𝐺𝐶)𝐺𝑦) = ((𝐵𝐺𝐶)𝐺𝑦))
10	9	ad2antll 727	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → ((𝐴𝐺𝐶)𝐺𝑦) = ((𝐵𝐺𝐶)𝐺𝑦))
11	1	grpoass 29794	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
12	11	3anassrs 1360	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
13	12	adantlrl 718	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
14	13	adantrr 715	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
15	1	grpoass 29794	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
16	15	3exp2 1354	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ GrpOp → (𝐵 ∈ 𝑋 → (𝐶 ∈ 𝑋 → (𝑦 ∈ 𝑋 → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦))))))
17	16	imp42 427	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
18	17	adantllr 717	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
19	18	adantrr 715	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
20	10, 14, 19	3eqtr3d 2780	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(𝐶𝐺𝑦)))
21	20	adantrrl 722	. . . . . . . . 9 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(𝐶𝐺𝑦)))
22		oveq2 7419	. . . . . . . . . . 11 ⊢ ((𝐶𝐺𝑦) = (GId‘𝐺) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐴𝐺(GId‘𝐺)))
23	22	ad2antrl 726	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐴𝐺(GId‘𝐺)))
24	23	adantl 482	. . . . . . . . 9 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐴𝐺(GId‘𝐺)))
25		oveq2 7419	. . . . . . . . . . 11 ⊢ ((𝐶𝐺𝑦) = (GId‘𝐺) → (𝐵𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(GId‘𝐺)))
26	25	ad2antrl 726	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → (𝐵𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(GId‘𝐺)))
27	26	adantl 482	. . . . . . . . 9 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐵𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(GId‘𝐺)))
28	21, 24, 27	3eqtr3d 2780	. . . . . . . 8 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(GId‘𝐺)) = (𝐵𝐺(GId‘𝐺)))
29	1, 2	grporid 29808	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
30	29	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
31	1, 2	grporid 29808	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺(GId‘𝐺)) = 𝐵)
32	31	ad2ant2r 745	. . . . . . . . 9 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐺(GId‘𝐺)) = 𝐵)
33	32	adantr 481	. . . . . . . 8 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐵𝐺(GId‘𝐺)) = 𝐵)
34	28, 30, 33	3eqtr3d 2780	. . . . . . 7 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → 𝐴 = 𝐵)
35	34	exp45 439	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝑦 ∈ 𝑋 → ((𝐶𝐺𝑦) = (GId‘𝐺) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) → 𝐴 = 𝐵))))
36	35	rexlimdv 3153	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (∃𝑦 ∈ 𝑋 (𝐶𝐺𝑦) = (GId‘𝐺) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) → 𝐴 = 𝐵)))
37	8, 36	mpd 15	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) → 𝐴 = 𝐵))
38		oveq1 7418	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴𝐺𝐶) = (𝐵𝐺𝐶))
39	37, 38	impbid1 224	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))
40	39	exp43 437	. 2 ⊢ (𝐺 ∈ GrpOp → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐶 ∈ 𝑋 → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵)))))
41	40	3imp2 1349	1 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∃wrex 3070  ran crn 5677  ‘cfv 6543  (class class class)co 7411  GrpOpcgr 29780  GIdcgi 29781

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-riota 7367  df-ov 7414  df-grpo 29784  df-gid 29785

This theorem is referenced by:  grpoinveu  29810  grpoid  29811  nvrcan  29915  ghomdiv  36846  rngorcan  36871  rngorz  36877