Theorem grpidinv2 18936

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 18906	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → ( 0 + 𝐴) = 𝐴)
5	1, 2, 3	grprid 18907	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵) → (𝐴 + 0 ) = 𝐴)
6	1, 2, 3	grplrinv 18935	. . 3 ⊢ (𝐺 ∈ Grp → ∀𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ))
7		oveq2 7398	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (𝑦 + 𝑧) = (𝑦 + 𝐴))
8	7	eqeq1d 2732	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝑦 + 𝑧) = 0 ↔ (𝑦 + 𝐴) = 0 ))
9		oveq1 7397	. . . . . . 7 ⊢ (𝑧 = 𝐴 → (𝑧 + 𝑦) = (𝐴 + 𝑦))
10	9	eqeq1d 2732	. . . . . 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 3158	. . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) ↔ ∃𝑦 ∈ 𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
13	12	rspcv 3587	. . 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 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  0_gc0g 17409  Grpcgrp 18872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-riota 7347  df-ov 7393  df-0g 17411  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-grp 18875  df-minusg 18876

This theorem is referenced by:  grpidinv  18937