Theorem grpoinveu 28296

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 28292	. . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
4		simpl 485	. . . . . 6 ⊢ (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
5	4	reximi 3243	. . . . 5 ⊢ (∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → ∃𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)
6	5	adantl 484	. . . 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 2843	. . . . . . . . . . . 12 ⊢ (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → (𝑦𝐺𝐴) = (𝑧𝐺𝐴))
9	1	grporcan 28295	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑦𝐺𝐴) = (𝑧𝐺𝐴) ↔ 𝑦 = 𝑧))
10	8, 9	syl5ib 246	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ GrpOp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
11	10	3exp2 1350	. . . . . . . . . 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 428	. . . . . . . 8 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
14	13	an32s 650	. . . . . . 7 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝑧𝐺𝐴) = 𝑈) → 𝑦 = 𝑧))
15	14	expd 418	. . . . . 6 ⊢ ((((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧)))
16	15	ralrimdva 3189	. . . . 5 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧)))
17	16	ancld 553	. . . 4 ⊢ (((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑦𝐺𝐴) = 𝑈 → ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧))))
18	17	reximdva 3274	. . 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 7163	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦𝐺𝐴) = (𝑧𝐺𝐴))
21	20	eqeq1d 2823	. . 3 ⊢ (𝑦 = 𝑧 → ((𝑦𝐺𝐴) = 𝑈 ↔ (𝑧𝐺𝐴) = 𝑈))
22	21	reu8 3724	. 2 ⊢ (∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈 ↔ ∃𝑦 ∈ 𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ ∀𝑧 ∈ 𝑋 ((𝑧𝐺𝐴) = 𝑈 → 𝑦 = 𝑧)))
23	19, 22	sylibr 236	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ∃!𝑦 ∈ 𝑋 (𝑦𝐺𝐴) = 𝑈)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139  ∃!wreu 3140  ran crn 5556  ‘cfv 6355  (class class class)co 7156  GrpOpcgr 28266  GIdcgi 28267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fo 6361  df-fv 6363  df-riota 7114  df-ov 7159  df-grpo 28270  df-gid 28271

This theorem is referenced by:  grpoinvcl  28301  grpoinv  28302