Theorem grporcan 30589

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 2736	. . . . . . . 8 ⊢ (GId‘𝐺) = (GId‘𝐺)
3	1, 2	grpoidinv2 30586	. . . . . . 7 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐶 ∈ 𝑋) → ((((GId‘𝐺)𝐺𝐶) = 𝐶 ∧ (𝐶𝐺(GId‘𝐺)) = 𝐶) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺))))
4		simpr 484	. . . . . . . . 9 ⊢ (((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺)) → (𝐶𝐺𝑦) = (GId‘𝐺))
5	4	reximi 3075	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐶) = (GId‘𝐺) ∧ (𝐶𝐺𝑦) = (GId‘𝐺)) → ∃𝑦 ∈ 𝑋 (𝐶𝐺𝑦) = (GId‘𝐺))
6	5	adantl 481	. . . . . . 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 750	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ∃𝑦 ∈ 𝑋 (𝐶𝐺𝑦) = (GId‘𝐺))
9		oveq1 7374	. . . . . . . . . . . 12 ⊢ ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) → ((𝐴𝐺𝐶)𝐺𝑦) = ((𝐵𝐺𝐶)𝐺𝑦))
10	9	ad2antll 730	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → ((𝐴𝐺𝐶)𝐺𝑦) = ((𝐵𝐺𝐶)𝐺𝑦))
11	1	grpoass 30574	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
12	11	3anassrs 1362	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
13	12	adantlrl 721	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
14	13	adantrr 718	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → ((𝐴𝐺𝐶)𝐺𝑦) = (𝐴𝐺(𝐶𝐺𝑦)))
15	1	grpoass 30574	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
16	15	3exp2 1356	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ GrpOp → (𝐵 ∈ 𝑋 → (𝐶 ∈ 𝑋 → (𝑦 ∈ 𝑋 → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦))))))
17	16	imp42 426	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ GrpOp ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
18	17	adantllr 720	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ 𝑦 ∈ 𝑋) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
19	18	adantrr 718	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → ((𝐵𝐺𝐶)𝐺𝑦) = (𝐵𝐺(𝐶𝐺𝑦)))
20	10, 14, 19	3eqtr3d 2779	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(𝐶𝐺𝑦)))
21	20	adantrrl 725	. . . . . . . . 9 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(𝐶𝐺𝑦)))
22		oveq2 7375	. . . . . . . . . . 11 ⊢ ((𝐶𝐺𝑦) = (GId‘𝐺) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐴𝐺(GId‘𝐺)))
23	22	ad2antrl 729	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐴𝐺(GId‘𝐺)))
24	23	adantl 481	. . . . . . . . 9 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(𝐶𝐺𝑦)) = (𝐴𝐺(GId‘𝐺)))
25		oveq2 7375	. . . . . . . . . . 11 ⊢ ((𝐶𝐺𝑦) = (GId‘𝐺) → (𝐵𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(GId‘𝐺)))
26	25	ad2antrl 729	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶))) → (𝐵𝐺(𝐶𝐺𝑦)) = (𝐵𝐺(GId‘𝐺)))
27	26	adantl 481	. . . . . . . . 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 2779	. . . . . . . 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 30588	. . . . . . . . 9 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
30	29	ad2antrr 727	. . . . . . . 8 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → (𝐴𝐺(GId‘𝐺)) = 𝐴)
31	1, 2	grporid 30588	. . . . . . . . . 10 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝐵𝐺(GId‘𝐺)) = 𝐵)
32	31	ad2ant2r 748	. . . . . . . . 9 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐺(GId‘𝐺)) = 𝐵)
33	32	adantr 480	. . . . . . . 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 2779	. . . . . . 7 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) ∧ (𝑦 ∈ 𝑋 ∧ ((𝐶𝐺𝑦) = (GId‘𝐺) ∧ (𝐴𝐺𝐶) = (𝐵𝐺𝐶)))) → 𝐴 = 𝐵)
35	34	exp45 438	. . . . . 6 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝑦 ∈ 𝑋 → ((𝐶𝐺𝑦) = (GId‘𝐺) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) → 𝐴 = 𝐵))))
36	35	rexlimdv 3136	. . . . 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 7374	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴𝐺𝐶) = (𝐵𝐺𝐶))
39	37, 38	impbid1 225	. . 3 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))
40	39	exp43 436	. 2 ⊢ (𝐺 ∈ GrpOp → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐶 ∈ 𝑋 → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵)))))
41	40	3imp2 1351	1 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐶) = (𝐵𝐺𝐶) ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  ran crn 5632  ‘cfv 6498  (class class class)co 7367  GrpOpcgr 30560  GIdcgi 30561

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fo 6504  df-fv 6506  df-riota 7324  df-ov 7370  df-grpo 30564  df-gid 30565

This theorem is referenced by:  grpoinveu  30590  grpoid  30591  nvrcan  30695  ghomdiv  38213  rngorcan  38238  rngorz  38244