Theorem grpidinv2 18882

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 18852	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → ( 0 + 𝐴) = 𝐴)
5	1, 2, 3	grprid 18853	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → (𝐴 + 0 ) = 𝐴)
6	1, 2, 3	grplrinv 18881	. . 3 ⊢ (𝐺 ∈ Grp → ∀𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ))
7		oveq2 7417	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (𝑦 + 𝑧) = (𝑦 + 𝐴))
8	7	eqeq1d 2735	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝑦 + 𝑧) = 0 ↔ (𝑦 + 𝐴) = 0 ))
9		oveq1 7416	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (𝑧 + 𝑦) = (𝐴 + 𝑦))
10	9	eqeq1d 2735	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝑧 + 𝑦) = 0 ↔ (𝐴 + 𝑦) = 0 ))
11	8, 10	anbi12d 632	. . . . 5 ⊢ (𝑧 = 𝐴 → (((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) ↔ ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
12	11	rexbidv 3179	. . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) ↔ ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
13	12	rspcv 3609	. . 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 508	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 ))
15	4, 5, 14	jca31 516	1 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → ((( 0 + 𝐴) = 𝐴 ∧ (𝐴 + 0 ) = 𝐴) ∧ ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  +_gcplusg 17197  0_gc0g 17385  Grpcgrp 18819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-riota 7365  df-ov 7412  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823

This theorem is referenced by:  grpidinv  18883