Theorem grpinveu 13793

Description: The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 24-Aug-2011.)

Hypotheses

Ref	Expression
grpinveu.b	⊢ 𝐵 = (Base‘𝐺)
grpinveu.p	⊢ + = (+g‘𝐺)
grpinveu.o	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
grpinveu	⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )

Distinct variable groups:   𝑦,𝐵   𝑦,𝐺   𝑦, +   𝑦, 0   𝑦,𝑋

Proof of Theorem grpinveu

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinveu.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2		grpinveu.p	. . . 4 ⊢ + = (+g‘𝐺)
3		grpinveu.o	. . . 4 ⊢ 0 = (0g‘𝐺)
4	1, 2, 3	grpinvex 13765	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )
5		eqtr3 2254	. . . . . . . . . . . 12 ⊢ (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → (𝑦 + 𝑋) = (𝑧 + 𝑋))
6	1, 2	grprcan 13792	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑦 + 𝑋) = (𝑧 + 𝑋) ↔ 𝑦 = 𝑧))
7	5, 6	imbitrid 154	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧))
8	7	3exp2 1252	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → (𝑦 ∈ 𝐵 → (𝑧 ∈ 𝐵 → (𝑋 ∈ 𝐵 → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧)))))
9	8	com24 87	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → (𝑧 ∈ 𝐵 → (𝑦 ∈ 𝐵 → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧)))))
10	9	imp41 353	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧))
11	10	an32s 570	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧))
12	11	expd 258	. . . . . 6 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → ((𝑦 + 𝑋) = 0 → ((𝑧 + 𝑋) = 0 → 𝑦 = 𝑧)))
13	12	ralrimdva 2624	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝑦 + 𝑋) = 0 → ∀𝑧 ∈ 𝐵 ((𝑧 + 𝑋) = 0 → 𝑦 = 𝑧)))
14	13	ancld 325	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝑦 + 𝑋) = 0 → ((𝑦 + 𝑋) = 0 ∧ ∀𝑧 ∈ 𝐵 ((𝑧 + 𝑋) = 0 → 𝑦 = 𝑧))))
15	14	reximdva 2646	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 → ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑋) = 0 ∧ ∀𝑧 ∈ 𝐵 ((𝑧 + 𝑋) = 0 → 𝑦 = 𝑧))))
16	4, 15	mpd 13	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑋) = 0 ∧ ∀𝑧 ∈ 𝐵 ((𝑧 + 𝑋) = 0 → 𝑦 = 𝑧)))
17		oveq1 6065	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 + 𝑋) = (𝑧 + 𝑋))
18	17	eqeq1d 2243	. . 3 ⊢ (𝑦 = 𝑧 → ((𝑦 + 𝑋) = 0 ↔ (𝑧 + 𝑋) = 0 ))
19	18	reu8 3016	. 2 ⊢ (∃!𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ↔ ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑋) = 0 ∧ ∀𝑧 ∈ 𝐵 ((𝑧 + 𝑋) = 0 → 𝑦 = 𝑧)))
20	16, 19	sylibr 134	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  ∃!wreu 2524  ‘cfv 5357  (class class class)co 6058  Basecbs 13296  +_gcplusg 13374  0_gc0g 13553  Grpcgrp 13755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-inn 9255  df-2 9313  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758

This theorem is referenced by:  grpinvf  13802  grplinv  13805  isgrpinv  13809