Theorem grpoinveu 30598

Description: The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpoinveu.1	⊢ 𝑋 = ran 𝐺
grpoinveu.2	⊢ 𝑈 = (GId‘𝐺)

Assertion

Ref	Expression
grpoinveu	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐺   𝑦,𝑈   𝑦,𝑋

Proof of Theorem grpoinveu

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpoinveu.1	. . . . 5 ⊢ 𝑋 = ran 𝐺
2		grpoinveu.2	. . . . 5 ⊢ 𝑈 = (GId‘𝐺)
3	1, 2	grpoidinv2 30594	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
4		simpl 482	. . . . . 6 ⊢ (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
5	4	reximi 3075	. . . . 5 ⊢ (∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → ∃𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)
6	5	adantl 481	. . . 4 ⊢ ((((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → ∃𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)
7	3, 6	syl 17	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)
8		eqtr3 2759	. . . . . . . . . . . 12 ⊢ (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → (𝑦𝐺𝐴) = (𝑧𝐺𝐴))
9	1	grporcan 30597	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑦𝐺𝐴) = (𝑧𝐺𝐴) ↔ 𝑦 = 𝑧))
10	8, 9	imbitrid 244	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
11	10	3exp2 1356	. . . . . . . . . 10 ⊢ (𝐺 ∈ GrpOp → (𝑦 ∈ 𝑋 → (𝑧 ∈ 𝑋 → (𝐴 ∈ 𝑋 → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧)))))
12	11	com24 95	. . . . . . . . 9 ⊢ (𝐺 ∈ GrpOp → (𝐴 ∈ 𝑋 → (𝑧 ∈ 𝑋 → (𝑦 ∈ 𝑋 → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧)))))
13	12	imp41 425	. . . . . . . 8 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
14	13	an32s 653	. . . . . . 7 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
15	14	expd 415	. . . . . 6 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧)))
16	15	ralrimdva 3137	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧)))
17	16	ancld 550	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧))))
18	17	reximdva 3150	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (∃𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈 → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧))))
19	7, 18	mpd 15	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧)))
20		oveq1 7367	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦𝐺𝐴) = (𝑧𝐺𝐴))
21	20	eqeq1d 2739	. . 3 ⊢ (𝑦 = 𝑧 → ((𝑦𝐺𝐴) = 𝑈 ↔ (𝑧𝐺𝐴) = 𝑈))
22	21	reu8 3692	. 2 ⊢ (∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈 ↔ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧)))
23	19, 22	sylibr 234	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061  ∃!wreu 3349  ran crn 5626  ‘cfv 6493  (class class class)co 7360  GrpOpcgr 30568  GIdcgi 30569

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 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-riota 7317  df-ov 7363  df-grpo 30572  df-gid 30573

This theorem is referenced by:  grpoinvcl  30603  grpoinv  30604