MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpridd Structured version   Visualization version   GIF version

Theorem grpridd 18535
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 7365 . . . . . 6 (𝑦 = 𝑛 → (𝑦 + 𝑥) = (𝑛 + 𝑥))
32eqeq1d 2735 . . . . 5 (𝑦 = 𝑛 → ((𝑦 + 𝑥) = 𝑂 ↔ (𝑛 + 𝑥) = 𝑂))
43cbvrexvw 3225 . . . 4 (∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∃𝑛𝐵 (𝑛 + 𝑥) = 𝑂)
51, 4sylib 217 . . 3 ((𝜑𝑥𝐵) → ∃𝑛𝐵 (𝑛 + 𝑥) = 𝑂)
6 grprinvlem.a . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
76caovassg 7553 . . . . . . 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 7554 . . . . 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 18534 . . . . . 6 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑥 + 𝑛) = 𝑂)
1716oveq1d 7373 . . . . 5 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → ((𝑥 + 𝑛) + 𝑥) = (𝑂 + 𝑥))
1815oveq2d 7374 . . . . 5 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑥 + (𝑛 + 𝑥)) = (𝑥 + 𝑂))
1911, 17, 183eqtr3d 2781 . . . 4 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
2019anassrs 469 . . 3 (((𝜑𝑥𝐵) ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂)) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
215, 20rexlimddv 3155 . 2 ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
2221, 14eqtr3d 2775 1 ((𝜑𝑥𝐵) → (𝑥 + 𝑂) = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wrex 3070  (class class class)co 7358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361
This theorem is referenced by:  isgrpde  18776
  Copyright terms: Public domain W3C validator