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Theorem grpidinv2 17695
 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 + = (+g𝐺)
grplrinv.i 0 = (0g𝐺)
Assertion
Ref Expression
grpidinv2 ((𝐺 ∈ Grp ∧ 𝐴𝐵) → ((( 0 + 𝐴) = 𝐴 ∧ (𝐴 + 0 ) = 𝐴) ∧ ∃𝑦𝐵 ((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.
StepHypRef Expression
1 grplrinv.b . . 3 𝐵 = (Base‘𝐺)
2 grplrinv.p . . 3 + = (+g𝐺)
3 grplrinv.i . . 3 0 = (0g𝐺)
41, 2, 3grplid 17673 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝐵) → ( 0 + 𝐴) = 𝐴)
51, 2, 3grprid 17674 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝐵) → (𝐴 + 0 ) = 𝐴)
61, 2, 3grplrinv 17694 . . 3 (𝐺 ∈ Grp → ∀𝑧𝐵𝑦𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ))
7 oveq2 6822 . . . . . . 7 (𝑧 = 𝐴 → (𝑦 + 𝑧) = (𝑦 + 𝐴))
87eqeq1d 2762 . . . . . 6 (𝑧 = 𝐴 → ((𝑦 + 𝑧) = 0 ↔ (𝑦 + 𝐴) = 0 ))
9 oveq1 6821 . . . . . . 7 (𝑧 = 𝐴 → (𝑧 + 𝑦) = (𝐴 + 𝑦))
109eqeq1d 2762 . . . . . 6 (𝑧 = 𝐴 → ((𝑧 + 𝑦) = 0 ↔ (𝐴 + 𝑦) = 0 ))
118, 10anbi12d 749 . . . . 5 (𝑧 = 𝐴 → (((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) ↔ ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
1211rexbidv 3190 . . . 4 (𝑧 = 𝐴 → (∃𝑦𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) ↔ ∃𝑦𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
1312rspcv 3445 . . 3 (𝐴𝐵 → (∀𝑧𝐵𝑦𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) → ∃𝑦𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
146, 13mpan9 487 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝐵) → ∃𝑦𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 ))
154, 5, 14jca31 558 1 ((𝐺 ∈ Grp ∧ 𝐴𝐵) → ((( 0 + 𝐴) = 𝐴 ∧ (𝐴 + 0 ) = 𝐴) ∧ ∃𝑦𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051  ‘cfv 6049  (class class class)co 6814  Basecbs 16079  +gcplusg 16163  0gc0g 16322  Grpcgrp 17643 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 6775  df-ov 6817  df-0g 16324  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-grp 17646  df-minusg 17647 This theorem is referenced by:  grpidinv  17696
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