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Theorem grpidinv 18885
Description: A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (Revised by AV, 1-Sep-2021.)
Hypotheses
Ref Expression
grpidinv.b 𝐵 = (Base‘𝐺)
grpidinv.p + = (+g𝐺)
Assertion
Ref Expression
grpidinv (𝐺 ∈ Grp → ∃𝑢𝐵𝑥𝐵 (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∃𝑦𝐵 ((𝑦 + 𝑥) = 𝑢 ∧ (𝑥 + 𝑦) = 𝑢)))
Distinct variable groups:   𝑢,𝐺,𝑥,𝑦   𝑢,𝐵,𝑦   𝑢, + ,𝑦
Allowed substitution hints:   𝐵(𝑥)   + (𝑥)

Proof of Theorem grpidinv
StepHypRef Expression
1 grpidinv.b . . 3 𝐵 = (Base‘𝐺)
2 eqid 2732 . . 3 (0g𝐺) = (0g𝐺)
31, 2grpidcl 18852 . 2 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝐵)
4 oveq1 7418 . . . . . . 7 (𝑢 = (0g𝐺) → (𝑢 + 𝑥) = ((0g𝐺) + 𝑥))
54eqeq1d 2734 . . . . . 6 (𝑢 = (0g𝐺) → ((𝑢 + 𝑥) = 𝑥 ↔ ((0g𝐺) + 𝑥) = 𝑥))
6 oveq2 7419 . . . . . . 7 (𝑢 = (0g𝐺) → (𝑥 + 𝑢) = (𝑥 + (0g𝐺)))
76eqeq1d 2734 . . . . . 6 (𝑢 = (0g𝐺) → ((𝑥 + 𝑢) = 𝑥 ↔ (𝑥 + (0g𝐺)) = 𝑥))
85, 7anbi12d 631 . . . . 5 (𝑢 = (0g𝐺) → (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ↔ (((0g𝐺) + 𝑥) = 𝑥 ∧ (𝑥 + (0g𝐺)) = 𝑥)))
9 eqeq2 2744 . . . . . . 7 (𝑢 = (0g𝐺) → ((𝑦 + 𝑥) = 𝑢 ↔ (𝑦 + 𝑥) = (0g𝐺)))
10 eqeq2 2744 . . . . . . 7 (𝑢 = (0g𝐺) → ((𝑥 + 𝑦) = 𝑢 ↔ (𝑥 + 𝑦) = (0g𝐺)))
119, 10anbi12d 631 . . . . . 6 (𝑢 = (0g𝐺) → (((𝑦 + 𝑥) = 𝑢 ∧ (𝑥 + 𝑦) = 𝑢) ↔ ((𝑦 + 𝑥) = (0g𝐺) ∧ (𝑥 + 𝑦) = (0g𝐺))))
1211rexbidv 3178 . . . . 5 (𝑢 = (0g𝐺) → (∃𝑦𝐵 ((𝑦 + 𝑥) = 𝑢 ∧ (𝑥 + 𝑦) = 𝑢) ↔ ∃𝑦𝐵 ((𝑦 + 𝑥) = (0g𝐺) ∧ (𝑥 + 𝑦) = (0g𝐺))))
138, 12anbi12d 631 . . . 4 (𝑢 = (0g𝐺) → ((((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∃𝑦𝐵 ((𝑦 + 𝑥) = 𝑢 ∧ (𝑥 + 𝑦) = 𝑢)) ↔ ((((0g𝐺) + 𝑥) = 𝑥 ∧ (𝑥 + (0g𝐺)) = 𝑥) ∧ ∃𝑦𝐵 ((𝑦 + 𝑥) = (0g𝐺) ∧ (𝑥 + 𝑦) = (0g𝐺)))))
1413ralbidv 3177 . . 3 (𝑢 = (0g𝐺) → (∀𝑥𝐵 (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∃𝑦𝐵 ((𝑦 + 𝑥) = 𝑢 ∧ (𝑥 + 𝑦) = 𝑢)) ↔ ∀𝑥𝐵 ((((0g𝐺) + 𝑥) = 𝑥 ∧ (𝑥 + (0g𝐺)) = 𝑥) ∧ ∃𝑦𝐵 ((𝑦 + 𝑥) = (0g𝐺) ∧ (𝑥 + 𝑦) = (0g𝐺)))))
1514adantl 482 . 2 ((𝐺 ∈ Grp ∧ 𝑢 = (0g𝐺)) → (∀𝑥𝐵 (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∃𝑦𝐵 ((𝑦 + 𝑥) = 𝑢 ∧ (𝑥 + 𝑦) = 𝑢)) ↔ ∀𝑥𝐵 ((((0g𝐺) + 𝑥) = 𝑥 ∧ (𝑥 + (0g𝐺)) = 𝑥) ∧ ∃𝑦𝐵 ((𝑦 + 𝑥) = (0g𝐺) ∧ (𝑥 + 𝑦) = (0g𝐺)))))
16 grpidinv.p . . . 4 + = (+g𝐺)
171, 16, 2grpidinv2 18884 . . 3 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ((((0g𝐺) + 𝑥) = 𝑥 ∧ (𝑥 + (0g𝐺)) = 𝑥) ∧ ∃𝑦𝐵 ((𝑦 + 𝑥) = (0g𝐺) ∧ (𝑥 + 𝑦) = (0g𝐺))))
1817ralrimiva 3146 . 2 (𝐺 ∈ Grp → ∀𝑥𝐵 ((((0g𝐺) + 𝑥) = 𝑥 ∧ (𝑥 + (0g𝐺)) = 𝑥) ∧ ∃𝑦𝐵 ((𝑦 + 𝑥) = (0g𝐺) ∧ (𝑥 + 𝑦) = (0g𝐺))))
193, 15, 18rspcedvd 3614 1 (𝐺 ∈ Grp → ∃𝑢𝐵𝑥𝐵 (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ∧ ∃𝑦𝐵 ((𝑦 + 𝑥) = 𝑢 ∧ (𝑥 + 𝑦) = 𝑢)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3061  wrex 3070  cfv 6543  (class class class)co 7411  Basecbs 17146  +gcplusg 17199  0gc0g 17387  Grpcgrp 18821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-riota 7367  df-ov 7414  df-0g 17389  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-grp 18824  df-minusg 18825
This theorem is referenced by: (None)
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