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Theorem grpridd 17885
 Description: Deduce right identity from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
grprinvlem.c ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
grprinvlem.o (𝜑𝑂𝐵)
grprinvlem.i ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = 𝑥)
grprinvlem.a ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
grprinvlem.n ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂)
Assertion
Ref Expression
grpridd ((𝜑𝑥𝐵) → (𝑥 + 𝑂) = 𝑥)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝑂,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥, + ,𝑦,𝑧

Proof of Theorem grpridd
Dummy variables 𝑢 𝑛 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grprinvlem.n . . . 4 ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂)
2 oveq1 7156 . . . . . 6 (𝑦 = 𝑛 → (𝑦 + 𝑥) = (𝑛 + 𝑥))
32eqeq1d 2826 . . . . 5 (𝑦 = 𝑛 → ((𝑦 + 𝑥) = 𝑂 ↔ (𝑛 + 𝑥) = 𝑂))
43cbvrexvw 3435 . . . 4 (∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∃𝑛𝐵 (𝑛 + 𝑥) = 𝑂)
51, 4sylib 221 . . 3 ((𝜑𝑥𝐵) → ∃𝑛𝐵 (𝑛 + 𝑥) = 𝑂)
6 grprinvlem.a . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
76caovassg 7340 . . . . . . 7 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
87adantlr 714 . . . . . 6 (((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
9 simprl 770 . . . . . 6 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → 𝑥𝐵)
10 simprrl 780 . . . . . 6 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → 𝑛𝐵)
118, 9, 10, 9caovassd 7341 . . . . 5 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → ((𝑥 + 𝑛) + 𝑥) = (𝑥 + (𝑛 + 𝑥)))
12 grprinvlem.c . . . . . . 7 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
13 grprinvlem.o . . . . . . 7 (𝜑𝑂𝐵)
14 grprinvlem.i . . . . . . 7 ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = 𝑥)
15 simprrr 781 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑛 + 𝑥) = 𝑂)
1612, 13, 14, 6, 1, 9, 10, 15grprinvd 17884 . . . . . 6 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑥 + 𝑛) = 𝑂)
1716oveq1d 7164 . . . . 5 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → ((𝑥 + 𝑛) + 𝑥) = (𝑂 + 𝑥))
1815oveq2d 7165 . . . . 5 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑥 + (𝑛 + 𝑥)) = (𝑥 + 𝑂))
1911, 17, 183eqtr3d 2867 . . . 4 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
2019anassrs 471 . . 3 (((𝜑𝑥𝐵) ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂)) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
215, 20rexlimddv 3283 . 2 ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
2221, 14eqtr3d 2861 1 ((𝜑𝑥𝐵) → (𝑥 + 𝑂) = 𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∃wrex 3134  (class class class)co 7149 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-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ral 3138  df-rex 3139  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7152 This theorem is referenced by:  isgrpde  18124
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