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

Theorem grprida 18701
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
grpinva.c ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
grpinva.o (𝜑𝑂𝐵)
grpinva.i ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = 𝑥)
grpinva.a ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
grpinva.r ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂)
Assertion
Ref Expression
grprida ((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 grprida
Dummy variables 𝑢 𝑛 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpinva.r . . . 4 ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂)
2 oveq1 7438 . . . . . 6 (𝑦 = 𝑛 → (𝑦 + 𝑥) = (𝑛 + 𝑥))
32eqeq1d 2737 . . . . 5 (𝑦 = 𝑛 → ((𝑦 + 𝑥) = 𝑂 ↔ (𝑛 + 𝑥) = 𝑂))
43cbvrexvw 3236 . . . 4 (∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∃𝑛𝐵 (𝑛 + 𝑥) = 𝑂)
51, 4sylib 218 . . 3 ((𝜑𝑥𝐵) → ∃𝑛𝐵 (𝑛 + 𝑥) = 𝑂)
6 grpinva.a . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
76caovassg 7631 . . . . . . 7 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
87adantlr 715 . . . . . 6 (((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
9 simprl 771 . . . . . 6 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → 𝑥𝐵)
10 simprrl 781 . . . . . 6 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → 𝑛𝐵)
118, 9, 10, 9caovassd 7632 . . . . 5 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → ((𝑥 + 𝑛) + 𝑥) = (𝑥 + (𝑛 + 𝑥)))
12 grpinva.c . . . . . . 7 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
13 grpinva.o . . . . . . 7 (𝜑𝑂𝐵)
14 grpinva.i . . . . . . 7 ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = 𝑥)
15 simprrr 782 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑛 + 𝑥) = 𝑂)
1612, 13, 14, 6, 1, 9, 10, 15grpinva 18700 . . . . . 6 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑥 + 𝑛) = 𝑂)
1716oveq1d 7446 . . . . 5 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → ((𝑥 + 𝑛) + 𝑥) = (𝑂 + 𝑥))
1815oveq2d 7447 . . . . 5 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑥 + (𝑛 + 𝑥)) = (𝑥 + 𝑂))
1911, 17, 183eqtr3d 2783 . . . 4 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
2019anassrs 467 . . 3 (((𝜑𝑥𝐵) ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂)) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
215, 20rexlimddv 3159 . 2 ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
2221, 14eqtr3d 2777 1 ((𝜑𝑥𝐵) → (𝑥 + 𝑂) = 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434
This theorem is referenced by:  isgrpde  18988
  Copyright terms: Public domain W3C validator