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Theorem grprinvlem 5747
Description: Lemma for grprinvd 5748. (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.x . . 3 ((𝜑𝜓) → 𝑋𝐵)
2 grprinvlem.n . . . . . 6 ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂)
32ralrimiva 2439 . . . . 5 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 𝑂)
4 oveq2 5572 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 + 𝑥) = (𝑦 + 𝑧))
54eqeq1d 2091 . . . . . . 7 (𝑥 = 𝑧 → ((𝑦 + 𝑥) = 𝑂 ↔ (𝑦 + 𝑧) = 𝑂))
65rexbidv 2374 . . . . . 6 (𝑥 = 𝑧 → (∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∃𝑦𝐵 (𝑦 + 𝑧) = 𝑂))
76cbvralv 2582 . . . . 5 (∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∀𝑧𝐵𝑦𝐵 (𝑦 + 𝑧) = 𝑂)
83, 7sylib 120 . . . 4 (𝜑 → ∀𝑧𝐵𝑦𝐵 (𝑦 + 𝑧) = 𝑂)
9 oveq2 5572 . . . . . . 7 (𝑧 = 𝑋 → (𝑦 + 𝑧) = (𝑦 + 𝑋))
109eqeq1d 2091 . . . . . 6 (𝑧 = 𝑋 → ((𝑦 + 𝑧) = 𝑂 ↔ (𝑦 + 𝑋) = 𝑂))
1110rexbidv 2374 . . . . 5 (𝑧 = 𝑋 → (∃𝑦𝐵 (𝑦 + 𝑧) = 𝑂 ↔ ∃𝑦𝐵 (𝑦 + 𝑋) = 𝑂))
1211rspccva 2709 . . . 4 ((∀𝑧𝐵𝑦𝐵 (𝑦 + 𝑧) = 𝑂𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 𝑂)
138, 12sylan 277 . . 3 ((𝜑𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 𝑂)
141, 13syldan 276 . 2 ((𝜑𝜓) → ∃𝑦𝐵 (𝑦 + 𝑋) = 𝑂)
15 grprinvlem.e . . . . 5 ((𝜑𝜓) → (𝑋 + 𝑋) = 𝑋)
1615oveq2d 5580 . . . 4 ((𝜑𝜓) → (𝑦 + (𝑋 + 𝑋)) = (𝑦 + 𝑋))
1716adantr 270 . . 3 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + (𝑋 + 𝑋)) = (𝑦 + 𝑋))
18 simprr 499 . . . . 5 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + 𝑋) = 𝑂)
1918oveq1d 5579 . . . 4 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → ((𝑦 + 𝑋) + 𝑋) = (𝑂 + 𝑋))
20 simpll 496 . . . . . 6 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝜑)
21 grprinvlem.a . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
2221caovassg 5711 . . . . . 6 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
2320, 22sylan 277 . . . . 5 ((((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
24 simprl 498 . . . . 5 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑦𝐵)
251adantr 270 . . . . 5 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑋𝐵)
2623, 24, 25, 25caovassd 5712 . . . 4 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → ((𝑦 + 𝑋) + 𝑋) = (𝑦 + (𝑋 + 𝑋)))
27 oveq2 5572 . . . . . . 7 (𝑦 = 𝑋 → (𝑂 + 𝑦) = (𝑂 + 𝑋))
28 id 19 . . . . . . 7 (𝑦 = 𝑋𝑦 = 𝑋)
2927, 28eqeq12d 2097 . . . . . 6 (𝑦 = 𝑋 → ((𝑂 + 𝑦) = 𝑦 ↔ (𝑂 + 𝑋) = 𝑋))
30 grprinvlem.i . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = 𝑥)
3130ralrimiva 2439 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (𝑂 + 𝑥) = 𝑥)
32 oveq2 5572 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑂 + 𝑥) = (𝑂 + 𝑦))
33 id 19 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
3432, 33eqeq12d 2097 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑂 + 𝑥) = 𝑥 ↔ (𝑂 + 𝑦) = 𝑦))
3534cbvralv 2582 . . . . . . . 8 (∀𝑥𝐵 (𝑂 + 𝑥) = 𝑥 ↔ ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
3631, 35sylib 120 . . . . . . 7 (𝜑 → ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
3736adantr 270 . . . . . 6 ((𝜑𝜓) → ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
3829, 37, 1rspcdva 2715 . . . . 5 ((𝜑𝜓) → (𝑂 + 𝑋) = 𝑋)
3938adantr 270 . . . 4 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑂 + 𝑋) = 𝑋)
4019, 26, 393eqtr3d 2123 . . 3 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + (𝑋 + 𝑋)) = 𝑋)
4117, 40, 183eqtr3d 2123 . 2 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑋 = 𝑂)
4214, 41rexlimddv 2486 1 ((𝜑𝜓) → 𝑋 = 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:  grprinvd  5748
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