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Theorem grpidinv 18155
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 2824 . . 3 (0g𝐺) = (0g𝐺)
31, 2grpidcl 18127 . 2 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝐵)
4 oveq1 7152 . . . . . . 7 (𝑢 = (0g𝐺) → (𝑢 + 𝑥) = ((0g𝐺) + 𝑥))
54eqeq1d 2826 . . . . . 6 (𝑢 = (0g𝐺) → ((𝑢 + 𝑥) = 𝑥 ↔ ((0g𝐺) + 𝑥) = 𝑥))
6 oveq2 7153 . . . . . . 7 (𝑢 = (0g𝐺) → (𝑥 + 𝑢) = (𝑥 + (0g𝐺)))
76eqeq1d 2826 . . . . . 6 (𝑢 = (0g𝐺) → ((𝑥 + 𝑢) = 𝑥 ↔ (𝑥 + (0g𝐺)) = 𝑥))
85, 7anbi12d 633 . . . . 5 (𝑢 = (0g𝐺) → (((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥) ↔ (((0g𝐺) + 𝑥) = 𝑥 ∧ (𝑥 + (0g𝐺)) = 𝑥)))
9 eqeq2 2836 . . . . . . 7 (𝑢 = (0g𝐺) → ((𝑦 + 𝑥) = 𝑢 ↔ (𝑦 + 𝑥) = (0g𝐺)))
10 eqeq2 2836 . . . . . . 7 (𝑢 = (0g𝐺) → ((𝑥 + 𝑦) = 𝑢 ↔ (𝑥 + 𝑦) = (0g𝐺)))
119, 10anbi12d 633 . . . . . 6 (𝑢 = (0g𝐺) → (((𝑦 + 𝑥) = 𝑢 ∧ (𝑥 + 𝑦) = 𝑢) ↔ ((𝑦 + 𝑥) = (0g𝐺) ∧ (𝑥 + 𝑦) = (0g𝐺))))
1211rexbidv 3290 . . . . 5 (𝑢 = (0g𝐺) → (∃𝑦𝐵 ((𝑦 + 𝑥) = 𝑢 ∧ (𝑥 + 𝑦) = 𝑢) ↔ ∃𝑦𝐵 ((𝑦 + 𝑥) = (0g𝐺) ∧ (𝑥 + 𝑦) = (0g𝐺))))
138, 12anbi12d 633 . . . 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 3192 . . 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 485 . 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 18154 . . 3 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ((((0g𝐺) + 𝑥) = 𝑥 ∧ (𝑥 + (0g𝐺)) = 𝑥) ∧ ∃𝑦𝐵 ((𝑦 + 𝑥) = (0g𝐺) ∧ (𝑥 + 𝑦) = (0g𝐺))))
1817ralrimiva 3177 . 2 (𝐺 ∈ Grp → ∀𝑥𝐵 ((((0g𝐺) + 𝑥) = 𝑥 ∧ (𝑥 + (0g𝐺)) = 𝑥) ∧ ∃𝑦𝐵 ((𝑦 + 𝑥) = (0g𝐺) ∧ (𝑥 + 𝑦) = (0g𝐺))))
193, 15, 18rspcedvd 3612 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 209  wa 399   = wceq 1538  wcel 2115  wral 3133  wrex 3134  cfv 6343  (class class class)co 7145  Basecbs 16479  +gcplusg 16561  0gc0g 16709  Grpcgrp 18099
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-riota 7103  df-ov 7148  df-0g 16711  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-grp 18102  df-minusg 18103
This theorem is referenced by: (None)
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