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Theorem grplrinv 17674
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 + = (+g𝐺)
grplrinv.i 0 = (0g𝐺)
Assertion
Ref Expression
grplrinv (𝐺 ∈ Grp → ∀𝑥𝐵𝑦𝐵 ((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
StepHypRef Expression
1 grplrinv.b . . . 4 𝐵 = (Base‘𝐺)
2 eqid 2760 . . . 4 (invg𝐺) = (invg𝐺)
31, 2grpinvcl 17668 . . 3 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ((invg𝐺)‘𝑥) ∈ 𝐵)
4 oveq1 6820 . . . . . 6 (𝑦 = ((invg𝐺)‘𝑥) → (𝑦 + 𝑥) = (((invg𝐺)‘𝑥) + 𝑥))
54eqeq1d 2762 . . . . 5 (𝑦 = ((invg𝐺)‘𝑥) → ((𝑦 + 𝑥) = 0 ↔ (((invg𝐺)‘𝑥) + 𝑥) = 0 ))
6 oveq2 6821 . . . . . 6 (𝑦 = ((invg𝐺)‘𝑥) → (𝑥 + 𝑦) = (𝑥 + ((invg𝐺)‘𝑥)))
76eqeq1d 2762 . . . . 5 (𝑦 = ((invg𝐺)‘𝑥) → ((𝑥 + 𝑦) = 0 ↔ (𝑥 + ((invg𝐺)‘𝑥)) = 0 ))
85, 7anbi12d 749 . . . 4 (𝑦 = ((invg𝐺)‘𝑥) → (((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ) ↔ ((((invg𝐺)‘𝑥) + 𝑥) = 0 ∧ (𝑥 + ((invg𝐺)‘𝑥)) = 0 )))
98adantl 473 . . 3 (((𝐺 ∈ Grp ∧ 𝑥𝐵) ∧ 𝑦 = ((invg𝐺)‘𝑥)) → (((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ) ↔ ((((invg𝐺)‘𝑥) + 𝑥) = 0 ∧ (𝑥 + ((invg𝐺)‘𝑥)) = 0 )))
10 grplrinv.p . . . . 5 + = (+g𝐺)
11 grplrinv.i . . . . 5 0 = (0g𝐺)
121, 10, 11, 2grplinv 17669 . . . 4 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (((invg𝐺)‘𝑥) + 𝑥) = 0 )
131, 10, 11, 2grprinv 17670 . . . 4 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝑥 + ((invg𝐺)‘𝑥)) = 0 )
1412, 13jca 555 . . 3 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ((((invg𝐺)‘𝑥) + 𝑥) = 0 ∧ (𝑥 + ((invg𝐺)‘𝑥)) = 0 ))
153, 9, 14rspcedvd 3456 . 2 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ∃𝑦𝐵 ((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ))
1615ralrimiva 3104 1 (𝐺 ∈ Grp → ∀𝑥𝐵𝑦𝐵 ((𝑦 + 𝑥) = 0 ∧ (𝑥 + 𝑦) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  cfv 6049  (class class class)co 6813  Basecbs 16059  +gcplusg 16143  0gc0g 16302  Grpcgrp 17623  invgcminusg 17624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-0g 16304  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-grp 17626  df-minusg 17627
This theorem is referenced by:  grpidinv2  17675
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