Theorem grpinveu 19004

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 18973	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )
5		eqtr3 2760	. . . . . . . . . . . 12 ⊢ (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → (𝑦 + 𝑋) = (𝑧 + 𝑋))
6	1, 2	grprcan 19003	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Grp ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑦 + 𝑋) = (𝑧 + 𝑋) ↔ 𝑦 = 𝑧))
7	5, 6	imbitrid 244	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Grp ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧))
8	7	3exp2 1353	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → (𝑦 ∈ 𝐵 → (𝑧 ∈ 𝐵 → (𝑋 ∈ 𝐵 → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧)))))
9	8	com24 95	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → (𝑧 ∈ 𝐵 → (𝑦 ∈ 𝐵 → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧)))))
10	9	imp41 425	. . . . . . . 8 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧))
11	10	an32s 652	. . . . . . 7 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → (((𝑦 + 𝑋) = 0 ∧ (𝑧 + 𝑋) = 0 ) → 𝑦 = 𝑧))
12	11	expd 415	. . . . . 6 ⊢ ((((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → ((𝑦 + 𝑋) = 0 → ((𝑧 + 𝑋) = 0 → 𝑦 = 𝑧)))
13	12	ralrimdva 3151	. . . . 5 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝑦 + 𝑋) = 0 → ∀𝑧 ∈ 𝐵 ((𝑧 + 𝑋) = 0 → 𝑦 = 𝑧)))
14	13	ancld 550	. . . 4 ⊢ (((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((𝑦 + 𝑋) = 0 → ((𝑦 + 𝑋) = 0 ∧ ∀𝑧 ∈ 𝐵 ((𝑧 + 𝑋) = 0 → 𝑦 = 𝑧))))
15	14	reximdva 3165	. . 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 15	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑋) = 0 ∧ ∀𝑧 ∈ 𝐵 ((𝑧 + 𝑋) = 0 → 𝑦 = 𝑧)))
17		oveq1 7437	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 + 𝑋) = (𝑧 + 𝑋))
18	17	eqeq1d 2736	. . 3 ⊢ (𝑦 = 𝑧 → ((𝑦 + 𝑋) = 0 ↔ (𝑧 + 𝑋) = 0 ))
19	18	reu8 3741	. 2 ⊢ (∃!𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ↔ ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑋) = 0 ∧ ∀𝑧 ∈ 𝐵 ((𝑧 + 𝑋) = 0 → 𝑦 = 𝑧)))
20	16, 19	sylibr 234	1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃!𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067  ∃!wreu 3375  ‘cfv 6562  (class class class)co 7430  Basecbs 17244  +_gcplusg 17297  0_gc0g 17485  Grpcgrp 18963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-riota 7387  df-ov 7433  df-0g 17487  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966

This theorem is referenced by:  grpinvf  19016  grplinv  19019  isgrpinv  19023