Theorem grpidinv2 18549

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 18524	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → ( 0 + 𝐴) = 𝐴)
5	1, 2, 3	grprid 18525	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → (𝐴 + 0 ) = 𝐴)
6	1, 2, 3	grplrinv 18548	. . 3 ⊢ (𝐺 ∈ Grp → ∀𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ))
7		oveq2 7263	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (𝑦 + 𝑧) = (𝑦 + 𝐴))
8	7	eqeq1d 2740	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝑦 + 𝑧) = 0 ↔ (𝑦 + 𝐴) = 0 ))
9		oveq1 7262	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (𝑧 + 𝑦) = (𝐴 + 𝑦))
10	9	eqeq1d 2740	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝑧 + 𝑦) = 0 ↔ (𝐴 + 𝑦) = 0 ))
11	8, 10	anbi12d 630	. . . . 5 ⊢ (𝑧 = 𝐴 → (((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) ↔ ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
12	11	rexbidv 3225	. . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) ↔ ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
13	12	rspcv 3547	. . 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 506	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 ))
15	4, 5, 14	jca31 514	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → ((( 0 + 𝐴) = 𝐴 ∧ (𝐴 + 0 ) = 𝐴) ∧ ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  0_gc0g 17067  Grpcgrp 18492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-riota 7212  df-ov 7258  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-minusg 18496

This theorem is referenced by:  grpidinv  18550