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Theorem grpridd 5823
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 5641 . . . . . 6 (𝑦 = 𝑛 → (𝑦 + 𝑥) = (𝑛 + 𝑥))
32eqeq1d 2096 . . . . 5 (𝑦 = 𝑛 → ((𝑦 + 𝑥) = 𝑂 ↔ (𝑛 + 𝑥) = 𝑂))
43cbvrexv 2591 . . . 4 (∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∃𝑛𝐵 (𝑛 + 𝑥) = 𝑂)
51, 4sylib 120 . . 3 ((𝜑𝑥𝐵) → ∃𝑛𝐵 (𝑛 + 𝑥) = 𝑂)
6 grprinvlem.a . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
76caovassg 5785 . . . . . . 7 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
87adantlr 461 . . . . . 6 (((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
9 simprl 498 . . . . . 6 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → 𝑥𝐵)
10 simprrl 506 . . . . . 6 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → 𝑛𝐵)
118, 9, 10, 9caovassd 5786 . . . . 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 507 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑛 + 𝑥) = 𝑂)
1612, 13, 14, 6, 1, 9, 10, 15grprinvd 5822 . . . . . 6 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑥 + 𝑛) = 𝑂)
1716oveq1d 5649 . . . . 5 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → ((𝑥 + 𝑛) + 𝑥) = (𝑂 + 𝑥))
1815oveq2d 5650 . . . . 5 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑥 + (𝑛 + 𝑥)) = (𝑥 + 𝑂))
1911, 17, 183eqtr3d 2128 . . . 4 ((𝜑 ∧ (𝑥𝐵 ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂))) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
2019anassrs 392 . . 3 (((𝜑𝑥𝐵) ∧ (𝑛𝐵 ∧ (𝑛 + 𝑥) = 𝑂)) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
215, 20rexlimddv 2493 . 2 ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = (𝑥 + 𝑂))
2221, 14eqtr3d 2122 1 ((𝜑𝑥𝐵) → (𝑥 + 𝑂) = 𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 924   = wceq 1289  wcel 1438  wrex 2360  (class class class)co 5634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-iota 4967  df-fv 5010  df-ov 5637
This theorem is referenced by: (None)
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