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Theorem grprinvd 6771
 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 1258 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
76caovclg 6724 . . . 4 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
87adantlr 747 . . 3 (((𝜑𝜓) ∧ (𝑢𝐵𝑣𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
9 grprinvd.x . . 3 ((𝜑𝜓) → 𝑋𝐵)
10 grprinvd.n . . 3 ((𝜑𝜓) → 𝑁𝐵)
118, 9, 10caovcld 6725 . 2 ((𝜑𝜓) → (𝑋 + 𝑁) ∈ 𝐵)
124caovassg 6730 . . . . 5 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
1312adantlr 747 . . . 4 (((𝜑𝜓) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
1413, 9, 10, 11caovassd 6731 . . 3 ((𝜑𝜓) → ((𝑋 + 𝑁) + (𝑋 + 𝑁)) = (𝑋 + (𝑁 + (𝑋 + 𝑁))))
15 grprinvd.e . . . . . 6 ((𝜑𝜓) → (𝑁 + 𝑋) = 𝑂)
1615oveq1d 6564 . . . . 5 ((𝜑𝜓) → ((𝑁 + 𝑋) + 𝑁) = (𝑂 + 𝑁))
1713, 10, 9, 10caovassd 6731 . . . . 5 ((𝜑𝜓) → ((𝑁 + 𝑋) + 𝑁) = (𝑁 + (𝑋 + 𝑁)))
183ralrimiva 2949 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (𝑂 + 𝑥) = 𝑥)
19 oveq2 6557 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑂 + 𝑥) = (𝑂 + 𝑦))
20 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
2119, 20eqeq12d 2625 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑂 + 𝑥) = 𝑥 ↔ (𝑂 + 𝑦) = 𝑦))
2221cbvralv 3147 . . . . . . . 8 (∀𝑥𝐵 (𝑂 + 𝑥) = 𝑥 ↔ ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
2318, 22sylib 207 . . . . . . 7 (𝜑 → ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
2423adantr 480 . . . . . 6 ((𝜑𝜓) → ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
25 oveq2 6557 . . . . . . . 8 (𝑦 = 𝑁 → (𝑂 + 𝑦) = (𝑂 + 𝑁))
26 id 22 . . . . . . . 8 (𝑦 = 𝑁𝑦 = 𝑁)
2725, 26eqeq12d 2625 . . . . . . 7 (𝑦 = 𝑁 → ((𝑂 + 𝑦) = 𝑦 ↔ (𝑂 + 𝑁) = 𝑁))
2827rspcv 3278 . . . . . 6 (𝑁𝐵 → (∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦 → (𝑂 + 𝑁) = 𝑁))
2910, 24, 28sylc 63 . . . . 5 ((𝜑𝜓) → (𝑂 + 𝑁) = 𝑁)
3016, 17, 293eqtr3d 2652 . . . 4 ((𝜑𝜓) → (𝑁 + (𝑋 + 𝑁)) = 𝑁)
3130oveq2d 6565 . . 3 ((𝜑𝜓) → (𝑋 + (𝑁 + (𝑋 + 𝑁))) = (𝑋 + 𝑁))
3214, 31eqtrd 2644 . 2 ((𝜑𝜓) → ((𝑋 + 𝑁) + (𝑋 + 𝑁)) = (𝑋 + 𝑁))
331, 2, 3, 4, 5, 11, 32grprinvlem 6770 1 ((𝜑𝜓) → (𝑋 + 𝑁) = 𝑂)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  (class class class)co 6549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552 This theorem is referenced by:  grpridd  6772  grprcan  17278  grprinv  17292
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