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Theorem grprinvd 5748
Description: Deduce right inverse 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)
grprinvd.x ((𝜑𝜓) → 𝑋𝐵)
grprinvd.n ((𝜑𝜓) → 𝑁𝐵)
grprinvd.e ((𝜑𝜓) → (𝑁 + 𝑋) = 𝑂)
Assertion
Ref Expression
grprinvd ((𝜑𝜓) → (𝑋 + 𝑁) = 𝑂)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝑂,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑦,𝑁,𝑧   𝑥, + ,𝑦,𝑧   𝑦,𝑋,𝑧   𝜓,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑧)   𝑁(𝑥)   𝑋(𝑥)

Proof of Theorem grprinvd
Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grprinvlem.c . 2 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
2 grprinvlem.o . 2 (𝜑𝑂𝐵)
3 grprinvlem.i . 2 ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = 𝑥)
4 grprinvlem.a . 2 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
5 grprinvlem.n . 2 ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂)
613expb 1140 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
76caovclg 5705 . . . 4 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
87adantlr 461 . . 3 (((𝜑𝜓) ∧ (𝑢𝐵𝑣𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
9 grprinvd.x . . 3 ((𝜑𝜓) → 𝑋𝐵)
10 grprinvd.n . . 3 ((𝜑𝜓) → 𝑁𝐵)
118, 9, 10caovcld 5706 . 2 ((𝜑𝜓) → (𝑋 + 𝑁) ∈ 𝐵)
124caovassg 5711 . . . . 5 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
1312adantlr 461 . . . 4 (((𝜑𝜓) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
1413, 9, 10, 11caovassd 5712 . . 3 ((𝜑𝜓) → ((𝑋 + 𝑁) + (𝑋 + 𝑁)) = (𝑋 + (𝑁 + (𝑋 + 𝑁))))
15 grprinvd.e . . . . . 6 ((𝜑𝜓) → (𝑁 + 𝑋) = 𝑂)
1615oveq1d 5579 . . . . 5 ((𝜑𝜓) → ((𝑁 + 𝑋) + 𝑁) = (𝑂 + 𝑁))
1713, 10, 9, 10caovassd 5712 . . . . 5 ((𝜑𝜓) → ((𝑁 + 𝑋) + 𝑁) = (𝑁 + (𝑋 + 𝑁)))
18 oveq2 5572 . . . . . . 7 (𝑦 = 𝑁 → (𝑂 + 𝑦) = (𝑂 + 𝑁))
19 id 19 . . . . . . 7 (𝑦 = 𝑁𝑦 = 𝑁)
2018, 19eqeq12d 2097 . . . . . 6 (𝑦 = 𝑁 → ((𝑂 + 𝑦) = 𝑦 ↔ (𝑂 + 𝑁) = 𝑁))
213ralrimiva 2439 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (𝑂 + 𝑥) = 𝑥)
22 oveq2 5572 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑂 + 𝑥) = (𝑂 + 𝑦))
23 id 19 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
2422, 23eqeq12d 2097 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑂 + 𝑥) = 𝑥 ↔ (𝑂 + 𝑦) = 𝑦))
2524cbvralv 2582 . . . . . . . 8 (∀𝑥𝐵 (𝑂 + 𝑥) = 𝑥 ↔ ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
2621, 25sylib 120 . . . . . . 7 (𝜑 → ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
2726adantr 270 . . . . . 6 ((𝜑𝜓) → ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
2820, 27, 10rspcdva 2715 . . . . 5 ((𝜑𝜓) → (𝑂 + 𝑁) = 𝑁)
2916, 17, 283eqtr3d 2123 . . . 4 ((𝜑𝜓) → (𝑁 + (𝑋 + 𝑁)) = 𝑁)
3029oveq2d 5580 . . 3 ((𝜑𝜓) → (𝑋 + (𝑁 + (𝑋 + 𝑁))) = (𝑋 + 𝑁))
3114, 30eqtrd 2115 . 2 ((𝜑𝜓) → ((𝑋 + 𝑁) + (𝑋 + 𝑁)) = (𝑋 + 𝑁))
321, 2, 3, 4, 5, 11, 31grprinvlem 5747 1 ((𝜑𝜓) → (𝑋 + 𝑁) = 𝑂)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920   = wceq 1285  wcel 1434  wral 2353  wrex 2354  (class class class)co 5564
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-iota 4917  df-fv 4960  df-ov 5567
This theorem is referenced by:  grpridd  5749
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