Theorem grplrinv 19015

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 19006	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
4		oveq1 7439	. . . . . 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 7440	. . . . . 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 481	. . 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 19008	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (((invg‘𝐺)‘𝑥) + 𝑥) = 0 )
13	1, 10, 11, 2	grprinv 19009	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 )
14	12, 13	jca 511	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((((invg‘𝐺)‘𝑥) + 𝑥) = 0 ∧ (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 ))
15	3, 9, 14	rspcedvd 3623	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ))
16	15	ralrimiva 3145	1 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060  ∃wrex 3069  ‘cfv 6560  (class class class)co 7432  Basecbs 17248  +_gcplusg 17298  0_gc0g 17485  Grpcgrp 18952  inv_gcminusg 18953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-riota 7389  df-ov 7435  df-0g 17487  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955  df-minusg 18956

This theorem is referenced by:  grpidinv2  19016