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Theorem grpidinv2 12860
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 12838 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝐵) → ( 0 + 𝐴) = 𝐴)
51, 2, 3grprid 12839 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝐵) → (𝐴 + 0 ) = 𝐴)
61, 2, 3grplrinv 12859 . . 3 (𝐺 ∈ Grp → ∀𝑧𝐵𝑦𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ))
7 oveq2 5880 . . . . . . 7 (𝑧 = 𝐴 → (𝑦 + 𝑧) = (𝑦 + 𝐴))
87eqeq1d 2186 . . . . . 6 (𝑧 = 𝐴 → ((𝑦 + 𝑧) = 0 ↔ (𝑦 + 𝐴) = 0 ))
9 oveq1 5879 . . . . . . 7 (𝑧 = 𝐴 → (𝑧 + 𝑦) = (𝐴 + 𝑦))
109eqeq1d 2186 . . . . . 6 (𝑧 = 𝐴 → ((𝑧 + 𝑦) = 0 ↔ (𝐴 + 𝑦) = 0 ))
118, 10anbi12d 473 . . . . 5 (𝑧 = 𝐴 → (((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) ↔ ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
1211rexbidv 2478 . . . 4 (𝑧 = 𝐴 → (∃𝑦𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) ↔ ∃𝑦𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
1312rspcv 2837 . . 3 (𝐴𝐵 → (∀𝑧𝐵𝑦𝐵 ((𝑦 + 𝑧) = 0 ∧ (𝑧 + 𝑦) = 0 ) → ∃𝑦𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
146, 13mpan9 281 . 2 ((𝐺 ∈ Grp ∧ 𝐴𝐵) → ∃𝑦𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 ))
154, 5, 14jca31 309 1 ((𝐺 ∈ Grp ∧ 𝐴𝐵) → ((( 0 + 𝐴) = 𝐴 ∧ (𝐴 + 0 ) = 𝐴) ∧ ∃𝑦𝐵 ((𝑦 + 𝐴) = 0 ∧ (𝐴 + 𝑦) = 0 )))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wral 2455  wrex 2456  cfv 5215  (class class class)co 5872  Basecbs 12454  +gcplusg 12528  0gc0g 12693  Grpcgrp 12809
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-cnex 7899  ax-resscn 7900  ax-1re 7902  ax-addrcl 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-inn 8916  df-2 8974  df-ndx 12457  df-slot 12458  df-base 12460  df-plusg 12541  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813
This theorem is referenced by:  grpidinv  12861
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