Theorem grplrinv 18678

Description: In a group, every member has a left and right inverse. (Contributed 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
grplrinv	⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ))

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

Allowed substitution hints:   𝐵(𝑥)   + (𝑥)   0 (𝑥)

Proof of Theorem grplrinv

Step	Hyp	Ref	Expression
1		grplrinv.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2		eqid 2736	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	grpinvcl 18672	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
4		oveq1 7314	. . . . . 6 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → (𝑦 + 𝑥) = (((invg‘𝐺)‘𝑥) + 𝑥))
5	4	eqeq1d 2738	. . . . 5 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → ((𝑦 + 𝑥) = 0 ↔ (((invg‘𝐺)‘𝑥) + 𝑥) = 0 ))
6		oveq2 7315	. . . . . 6 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → (𝑥 + 𝑦) = (𝑥 + ((invg‘𝐺)‘𝑥)))
7	6	eqeq1d 2738	. . . . 5 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → ((𝑥 + 𝑦) = 0 ↔ (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 ))
8	5, 7	anbi12d 632	. . . 4 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → (((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ) ↔ ((((invg‘𝐺)‘𝑥) + 𝑥) = 0 ∧ (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 )))
9	8	adantl 483	. . 3 ⊢ (((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 = ((invg‘𝐺)‘𝑥)) → (((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ) ↔ ((((invg‘𝐺)‘𝑥) + 𝑥) = 0 ∧ (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 )))
10		grplrinv.p	. . . . 5 ⊢ + = (+g‘𝐺)
11		grplrinv.i	. . . . 5 ⊢ 0 = (0g‘𝐺)
12	1, 10, 11, 2	grplinv 18673	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (((invg‘𝐺)‘𝑥) + 𝑥) = 0 )
13	1, 10, 11, 2	grprinv 18674	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 )
14	12, 13	jca 513	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((((invg‘𝐺)‘𝑥) + 𝑥) = 0 ∧ (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 ))
15	3, 9, 14	rspcedvd 3568	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ))
16	15	ralrimiva 3140	1 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ∀wral 3062  ∃wrex 3071  ‘cfv 6458  (class class class)co 7307  Basecbs 16957  +_gcplusg 17007  0_gc0g 17195  Grpcgrp 18622  inv_gcminusg 18623

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-fv 6466  df-riota 7264  df-ov 7310  df-0g 17197  df-mgm 18371  df-sgrp 18420  df-mnd 18431  df-grp 18625  df-minusg 18626

This theorem is referenced by:  grpidinv2  18679