Theorem grplrinv 13132

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 2193	. . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	grpinvcl 13123	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
4		oveq1 5926	. . . . . 6 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → (𝑦 + 𝑥) = (((invg‘𝐺)‘𝑥) + 𝑥))
5	4	eqeq1d 2202	. . . . 5 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → ((𝑦 + 𝑥) = 0 ↔ (((invg‘𝐺)‘𝑥) + 𝑥) = 0 ))
6		oveq2 5927	. . . . . 6 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → (𝑥 + 𝑦) = (𝑥 + ((invg‘𝐺)‘𝑥)))
7	6	eqeq1d 2202	. . . . 5 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → ((𝑥 + 𝑦) = 0 ↔ (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 ))
8	5, 7	anbi12d 473	. . . 4 ⊢ (𝑦 = ((invg‘𝐺)‘𝑥) → (((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ) ↔ ((((invg‘𝐺)‘𝑥) + 𝑥) = 0 ∧ (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 )))
9	8	adantl 277	. . 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 13125	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (((invg‘𝐺)‘𝑥) + 𝑥) = 0 )
13	1, 10, 11, 2	grprinv 13126	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 )
14	12, 13	jca 306	. . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((((invg‘𝐺)‘𝑥) + 𝑥) = 0 ∧ (𝑥 + ((invg‘𝐺)‘𝑥)) = 0 ))
15	3, 9, 14	rspcedvd 2871	. 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ))
16	15	ralrimiva 2567	1 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ‘cfv 5255  (class class class)co 5919  Basecbs 12621  +_gcplusg 12698  0_gc0g 12870  Grpcgrp 13075  inv_gcminusg 13076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079

This theorem is referenced by:  grpidinv2  13133