Theorem grpinveu 13113

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 13085	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )
5		eqtr3 2213	. . . . . . . . . . . 12 ⊢ (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → (𝑦 + 𝑋) = (𝑧 + 𝑋))
6	1, 2	grprcan 13112	. . . . . . . . . . . 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 1227	. . . . . . . . . 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 2574	. . . . 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 2596	. . 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 5926	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 + 𝑋) = (𝑧 + 𝑋))
18	17	eqeq1d 2202	. . 3 ⊢ (𝑦 = 𝑧 → ((𝑦 + 𝑋) = 0 ↔ (𝑧 + 𝑋) = 0 ))
19	18	reu8 2957	. 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 980   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ∃!wreu 2474  ‘cfv 5255  (class class class)co 5919  Basecbs 12621  +_gcplusg 12698  0_gc0g 12870  Grpcgrp 13075

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-riota 5874  df-ov 5922  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078

This theorem is referenced by:  grpinvf  13122  grplinv  13125  isgrpinv  13129