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Theorem grprinvd 12633
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 1199 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
76caovclg 6003 . . . 4 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
87adantlr 474 . . 3 (((𝜑𝜓) ∧ (𝑢𝐵𝑣𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
9 grprinvd.x . . 3 ((𝜑𝜓) → 𝑋𝐵)
10 grprinvd.n . . 3 ((𝜑𝜓) → 𝑁𝐵)
118, 9, 10caovcld 6004 . 2 ((𝜑𝜓) → (𝑋 + 𝑁) ∈ 𝐵)
124caovassg 6009 . . . . 5 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
1312adantlr 474 . . . 4 (((𝜑𝜓) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
1413, 9, 10, 11caovassd 6010 . . 3 ((𝜑𝜓) → ((𝑋 + 𝑁) + (𝑋 + 𝑁)) = (𝑋 + (𝑁 + (𝑋 + 𝑁))))
15 grprinvd.e . . . . . 6 ((𝜑𝜓) → (𝑁 + 𝑋) = 𝑂)
1615oveq1d 5866 . . . . 5 ((𝜑𝜓) → ((𝑁 + 𝑋) + 𝑁) = (𝑂 + 𝑁))
1713, 10, 9, 10caovassd 6010 . . . . 5 ((𝜑𝜓) → ((𝑁 + 𝑋) + 𝑁) = (𝑁 + (𝑋 + 𝑁)))
18 oveq2 5859 . . . . . . 7 (𝑦 = 𝑁 → (𝑂 + 𝑦) = (𝑂 + 𝑁))
19 id 19 . . . . . . 7 (𝑦 = 𝑁𝑦 = 𝑁)
2018, 19eqeq12d 2185 . . . . . 6 (𝑦 = 𝑁 → ((𝑂 + 𝑦) = 𝑦 ↔ (𝑂 + 𝑁) = 𝑁))
213ralrimiva 2543 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (𝑂 + 𝑥) = 𝑥)
22 oveq2 5859 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑂 + 𝑥) = (𝑂 + 𝑦))
23 id 19 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
2422, 23eqeq12d 2185 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑂 + 𝑥) = 𝑥 ↔ (𝑂 + 𝑦) = 𝑦))
2524cbvralvw 2700 . . . . . . . 8 (∀𝑥𝐵 (𝑂 + 𝑥) = 𝑥 ↔ ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
2621, 25sylib 121 . . . . . . 7 (𝜑 → ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
2726adantr 274 . . . . . 6 ((𝜑𝜓) → ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
2820, 27, 10rspcdva 2839 . . . . 5 ((𝜑𝜓) → (𝑂 + 𝑁) = 𝑁)
2916, 17, 283eqtr3d 2211 . . . 4 ((𝜑𝜓) → (𝑁 + (𝑋 + 𝑁)) = 𝑁)
3029oveq2d 5867 . . 3 ((𝜑𝜓) → (𝑋 + (𝑁 + (𝑋 + 𝑁))) = (𝑋 + 𝑁))
3114, 30eqtrd 2203 . 2 ((𝜑𝜓) → ((𝑋 + 𝑁) + (𝑋 + 𝑁)) = (𝑋 + 𝑁))
321, 2, 3, 4, 5, 11, 31grprinvlem 12632 1 ((𝜑𝜓) → (𝑋 + 𝑁) = 𝑂)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 973   = wceq 1348  wcel 2141  wral 2448  wrex 2449  (class class class)co 5851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-iota 5158  df-fv 5204  df-ov 5854
This theorem is referenced by:  grpridd  12634  grprcan  12733
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