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Theorem grpidinv2 17682
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 17660 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝐵) → ( 0 + 𝐴) = 𝐴)
51, 2, 3grprid 17661 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝐵) → (𝐴 + 0 ) = 𝐴)
61, 2, 3grplrinv 17681 . . 3 (𝐺 ∈ Grp → ∀𝑧𝐵𝑦𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ))
7 oveq2 6804 . . . . . . 7 (𝑧 = 𝐴 → (𝑦 + 𝑧) = (𝑦 + 𝐴))
87eqeq1d 2773 . . . . . 6 (𝑧 = 𝐴 → ((𝑦 + 𝑧) = 0 ↔ (𝑦 + 𝐴) = 0 ))
9 oveq1 6803 . . . . . . 7 (𝑧 = 𝐴 → (𝑧 + 𝑦) = (𝐴 + 𝑦))
109eqeq1d 2773 . . . . . 6 (𝑧 = 𝐴 → ((𝑧 + 𝑦) = 0 ↔ (𝐴 + 𝑦) = 0 ))
118, 10anbi12d 616 . . . . 5 (𝑧 = 𝐴 → (((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) ↔ ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
1211rexbidv 3200 . . . 4 (𝑧 = 𝐴 → (∃𝑦𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) ↔ ∃𝑦𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
1312rspcv 3456 . . 3 (𝐴𝐵 → (∀𝑧𝐵𝑦𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) → ∃𝑦𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
146, 13mpan9 496 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝐵) → ∃𝑦𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 ))
154, 5, 14jca31 504 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 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  cfv 6030  (class class class)co 6796  Basecbs 16064  +gcplusg 16149  0gc0g 16308  Grpcgrp 17630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634
This theorem is referenced by:  grpidinv  17683
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