Theorem grpinveu 13611

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 13583	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )
5		eqtr3 2249	. . . . . . . . . . . 12 ⊢ (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → (𝑦 + 𝑋) = (𝑧 + 𝑋))
6	1, 2	grprcan 13610	. . . . . . . . . . . 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 1249	. . . . . . . . . 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 568	. . . . . . 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 2610	. . . . 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 2632	. . 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 6020	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 + 𝑋) = (𝑧 + 𝑋))
18	17	eqeq1d 2238	. . 3 ⊢ (𝑦 = 𝑧 → ((𝑦 + 𝑋) = 0 ↔ (𝑧 + 𝑋) = 0 ))
19	18	reu8 3000	. 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 1002   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  ∃!wreu 2510  ‘cfv 5324  (class class class)co 6013  Basecbs 13072  +_gcplusg 13150  0_gc0g 13329  Grpcgrp 13573

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-riota 5966  df-ov 6016  df-inn 9134  df-2 9192  df-ndx 13075  df-slot 13076  df-base 13078  df-plusg 13163  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-grp 13576

This theorem is referenced by:  grpinvf  13620  grplinv  13623  isgrpinv  13627