ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  grprinvlem GIF version

Theorem grprinvlem 12639
Description: Lemma for grprinvd 12640. (Contributed by NM, 9-Aug-2013.)
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)
grprinvlem.x ((𝜑𝜓) → 𝑋𝐵)
grprinvlem.e ((𝜑𝜓) → (𝑋 + 𝑋) = 𝑋)
Assertion
Ref Expression
grprinvlem ((𝜑𝜓) → 𝑋 = 𝑂)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝑂,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥, + ,𝑦,𝑧   𝑦,𝑋,𝑧   𝜓,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑧)   𝑋(𝑥)

Proof of Theorem grprinvlem
Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grprinvlem.n . . . . 5 ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂)
21ralrimiva 2543 . . . 4 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 𝑂)
3 oveq2 5861 . . . . . . 7 (𝑥 = 𝑧 → (𝑦 + 𝑥) = (𝑦 + 𝑧))
43eqeq1d 2179 . . . . . 6 (𝑥 = 𝑧 → ((𝑦 + 𝑥) = 𝑂 ↔ (𝑦 + 𝑧) = 𝑂))
54rexbidv 2471 . . . . 5 (𝑥 = 𝑧 → (∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∃𝑦𝐵 (𝑦 + 𝑧) = 𝑂))
65cbvralvw 2700 . . . 4 (∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∀𝑧𝐵𝑦𝐵 (𝑦 + 𝑧) = 𝑂)
72, 6sylib 121 . . 3 (𝜑 → ∀𝑧𝐵𝑦𝐵 (𝑦 + 𝑧) = 𝑂)
8 grprinvlem.x . . 3 ((𝜑𝜓) → 𝑋𝐵)
9 oveq2 5861 . . . . . 6 (𝑧 = 𝑋 → (𝑦 + 𝑧) = (𝑦 + 𝑋))
109eqeq1d 2179 . . . . 5 (𝑧 = 𝑋 → ((𝑦 + 𝑧) = 𝑂 ↔ (𝑦 + 𝑋) = 𝑂))
1110rexbidv 2471 . . . 4 (𝑧 = 𝑋 → (∃𝑦𝐵 (𝑦 + 𝑧) = 𝑂 ↔ ∃𝑦𝐵 (𝑦 + 𝑋) = 𝑂))
1211rspccva 2833 . . 3 ((∀𝑧𝐵𝑦𝐵 (𝑦 + 𝑧) = 𝑂𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 𝑂)
137, 8, 12syl2an2r 590 . 2 ((𝜑𝜓) → ∃𝑦𝐵 (𝑦 + 𝑋) = 𝑂)
14 grprinvlem.e . . . . 5 ((𝜑𝜓) → (𝑋 + 𝑋) = 𝑋)
1514oveq2d 5869 . . . 4 ((𝜑𝜓) → (𝑦 + (𝑋 + 𝑋)) = (𝑦 + 𝑋))
1615adantr 274 . . 3 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + (𝑋 + 𝑋)) = (𝑦 + 𝑋))
17 simprr 527 . . . . 5 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + 𝑋) = 𝑂)
1817oveq1d 5868 . . . 4 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → ((𝑦 + 𝑋) + 𝑋) = (𝑂 + 𝑋))
19 grprinvlem.a . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
2019caovassg 6011 . . . . . 6 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
2120ad4ant14 511 . . . . 5 ((((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
22 simprl 526 . . . . 5 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑦𝐵)
238adantr 274 . . . . 5 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑋𝐵)
2421, 22, 23, 23caovassd 6012 . . . 4 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → ((𝑦 + 𝑋) + 𝑋) = (𝑦 + (𝑋 + 𝑋)))
25 oveq2 5861 . . . . . . 7 (𝑦 = 𝑋 → (𝑂 + 𝑦) = (𝑂 + 𝑋))
26 id 19 . . . . . . 7 (𝑦 = 𝑋𝑦 = 𝑋)
2725, 26eqeq12d 2185 . . . . . 6 (𝑦 = 𝑋 → ((𝑂 + 𝑦) = 𝑦 ↔ (𝑂 + 𝑋) = 𝑋))
28 grprinvlem.i . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = 𝑥)
2928ralrimiva 2543 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (𝑂 + 𝑥) = 𝑥)
30 oveq2 5861 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑂 + 𝑥) = (𝑂 + 𝑦))
31 id 19 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
3230, 31eqeq12d 2185 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑂 + 𝑥) = 𝑥 ↔ (𝑂 + 𝑦) = 𝑦))
3332cbvralvw 2700 . . . . . . . 8 (∀𝑥𝐵 (𝑂 + 𝑥) = 𝑥 ↔ ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
3429, 33sylib 121 . . . . . . 7 (𝜑 → ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
3534adantr 274 . . . . . 6 ((𝜑𝜓) → ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
3627, 35, 8rspcdva 2839 . . . . 5 ((𝜑𝜓) → (𝑂 + 𝑋) = 𝑋)
3736adantr 274 . . . 4 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑂 + 𝑋) = 𝑋)
3818, 24, 373eqtr3d 2211 . . 3 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + (𝑋 + 𝑋)) = 𝑋)
3916, 38, 173eqtr3d 2211 . 2 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑋 = 𝑂)
4013, 39rexlimddv 2592 1 ((𝜑𝜓) → 𝑋 = 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 5853
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 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856
This theorem is referenced by:  grprinvd  12640
  Copyright terms: Public domain W3C validator