Theorem grpoinveu 30880

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 30876	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
4		simpl 487	. . . . . 6 ⊢ (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
5	4	reximi 3103	. . . . 5 ⊢ (∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → ∃𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)
6	5	adantl 486	. . . 4 ⊢ ((((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)) → ∃𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)
7	3, 6	syl 18	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)
8		eqtr3 2785	. . . . . . . . . . . 12 ⊢ (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → (𝑦𝐺𝐴) = (𝑧𝐺𝐴))
9	1	grporcan 30879	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑦𝐺𝐴) = (𝑧𝐺𝐴) ↔ 𝑦 = 𝑧))
10	8, 9	imbitrid 247	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
11	10	3exp2 1373	. . . . . . . . . 10 ⊢ (𝐺 ∈ GrpOp → (𝑦 ∈ 𝑋 → (𝑧 ∈ 𝑋 → (𝐴 ∈ 𝑋 → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧)))))
12	11	com24 96	. . . . . . . . 9 ⊢ (𝐺 ∈ GrpOp → (𝐴 ∈ 𝑋 → (𝑧 ∈ 𝑋 → (𝑦 ∈ 𝑋 → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧)))))
13	12	imp41 430	. . . . . . . 8 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
14	13	an32s 664	. . . . . . 7 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
15	14	expd 420	. . . . . 6 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧)))
16	15	ralrimdva 3165	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧)))
17	16	ancld 559	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧))))
18	17	reximdva 3178	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (∃𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈 → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧))))
19	7, 18	mpd 16	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧)))
20		oveq1 7417	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦𝐺𝐴) = (𝑧𝐺𝐴))
21	20	eqeq1d 2765	. . 3 ⊢ (𝑦 = 𝑧 → ((𝑦𝐺𝐴) = 𝑈 ↔ (𝑧𝐺𝐴) = 𝑈))
22	21	reu8 3696	. 2 ⊢ (∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈 ↔ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧)))
23	19, 22	sylibr 237	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  ∃!wreu 3367  ran crn 5662  ‘cfv 6536  (class class class)co 7410  GrpOpcgr 30850  GIdcgi 30851

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-riota 7367  df-ov 7413  df-grpo 30854  df-gid 30855

This theorem is used by:  grpoinvcl  30885  grpoinv  30886