Theorem grpidinv2 18856

Description: A group's properties using the explicit identity element. (Contributed by NM, 5-Feb-2010.) (Revised by AV, 1-Sep-2021.)

Hypotheses

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

Assertion

Ref	Expression
grpidinv2	⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → ((( 0 + 𝐴) = 𝐴 ∧ (𝐴 + 0 ) = 𝐴) ∧ ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))

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

Proof of Theorem grpidinv2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grplrinv.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		grplrinv.p	. . 3 ⊢ + = (+g‘𝐺)
3		grplrinv.i	. . 3 ⊢ 0 = (0g‘𝐺)
4	1, 2, 3	grplid 18827	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → ( 0 + 𝐴) = 𝐴)
5	1, 2, 3	grprid 18828	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → (𝐴 + 0 ) = 𝐴)
6	1, 2, 3	grplrinv 18855	. . 3 ⊢ (𝐺 ∈ Grp → ∀𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ))
7		oveq2 7401	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (𝑦 + 𝑧) = (𝑦 + 𝐴))
8	7	eqeq1d 2733	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝑦 + 𝑧) = 0 ↔ (𝑦 + 𝐴) = 0 ))
9		oveq1 7400	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (𝑧 + 𝑦) = (𝐴 + 𝑦))
10	9	eqeq1d 2733	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝑧 + 𝑦) = 0 ↔ (𝐴 + 𝑦) = 0 ))
11	8, 10	anbi12d 631	. . . . 5 ⊢ (𝑧 = 𝐴 → (((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) ↔ ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
12	11	rexbidv 3177	. . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) ↔ ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
13	12	rspcv 3605	. . 3 ⊢ (𝐴 ∈ 𝐵 → (∀𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) → ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
14	6, 13	mpan9 507	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 ))
15	4, 5, 14	jca31 515	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → ((( 0 + 𝐴) = 𝐴 ∧ (𝐴 + 0 ) = 𝐴) ∧ ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069  ‘cfv 6532  (class class class)co 7393  Basecbs 17126  +_gcplusg 17179  0_gc0g 17367  Grpcgrp 18794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540  df-riota 7349  df-ov 7396  df-0g 17369  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-grp 18797  df-minusg 18798

This theorem is referenced by:  grpidinv  18857