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Theorem grprinvlem 7019
Description: Lemma for grprinvd 7020. (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 3115 . . . 4 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 𝑂)
3 oveq2 6801 . . . . . . 7 (𝑥 = 𝑧 → (𝑦 + 𝑥) = (𝑦 + 𝑧))
43eqeq1d 2773 . . . . . 6 (𝑥 = 𝑧 → ((𝑦 + 𝑥) = 𝑂 ↔ (𝑦 + 𝑧) = 𝑂))
54rexbidv 3200 . . . . 5 (𝑥 = 𝑧 → (∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∃𝑦𝐵 (𝑦 + 𝑧) = 𝑂))
65cbvralv 3320 . . . 4 (∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∀𝑧𝐵𝑦𝐵 (𝑦 + 𝑧) = 𝑂)
72, 6sylib 208 . . 3 (𝜑 → ∀𝑧𝐵𝑦𝐵 (𝑦 + 𝑧) = 𝑂)
8 grprinvlem.x . . 3 ((𝜑𝜓) → 𝑋𝐵)
9 oveq2 6801 . . . . . 6 (𝑧 = 𝑋 → (𝑦 + 𝑧) = (𝑦 + 𝑋))
109eqeq1d 2773 . . . . 5 (𝑧 = 𝑋 → ((𝑦 + 𝑧) = 𝑂 ↔ (𝑦 + 𝑋) = 𝑂))
1110rexbidv 3200 . . . 4 (𝑧 = 𝑋 → (∃𝑦𝐵 (𝑦 + 𝑧) = 𝑂 ↔ ∃𝑦𝐵 (𝑦 + 𝑋) = 𝑂))
1211rspccva 3459 . . 3 ((∀𝑧𝐵𝑦𝐵 (𝑦 + 𝑧) = 𝑂𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 𝑂)
137, 8, 12syl2an2r 664 . 2 ((𝜑𝜓) → ∃𝑦𝐵 (𝑦 + 𝑋) = 𝑂)
14 grprinvlem.e . . . . 5 ((𝜑𝜓) → (𝑋 + 𝑋) = 𝑋)
1514oveq2d 6809 . . . 4 ((𝜑𝜓) → (𝑦 + (𝑋 + 𝑋)) = (𝑦 + 𝑋))
1615adantr 466 . . 3 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + (𝑋 + 𝑋)) = (𝑦 + 𝑋))
17 simprr 756 . . . . 5 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + 𝑋) = 𝑂)
1817oveq1d 6808 . . . 4 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → ((𝑦 + 𝑋) + 𝑋) = (𝑂 + 𝑋))
19 grprinvlem.a . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
2019caovassg 6979 . . . . . 6 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
2120ad4ant14 1208 . . . . 5 ((((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
22 simprl 754 . . . . 5 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑦𝐵)
238adantr 466 . . . . 5 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑋𝐵)
2421, 22, 23, 23caovassd 6980 . . . 4 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → ((𝑦 + 𝑋) + 𝑋) = (𝑦 + (𝑋 + 𝑋)))
25 oveq2 6801 . . . . . . 7 (𝑦 = 𝑋 → (𝑂 + 𝑦) = (𝑂 + 𝑋))
26 id 22 . . . . . . 7 (𝑦 = 𝑋𝑦 = 𝑋)
2725, 26eqeq12d 2786 . . . . . 6 (𝑦 = 𝑋 → ((𝑂 + 𝑦) = 𝑦 ↔ (𝑂 + 𝑋) = 𝑋))
28 grprinvlem.i . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = 𝑥)
2928ralrimiva 3115 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (𝑂 + 𝑥) = 𝑥)
30 oveq2 6801 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑂 + 𝑥) = (𝑂 + 𝑦))
31 id 22 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
3230, 31eqeq12d 2786 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑂 + 𝑥) = 𝑥 ↔ (𝑂 + 𝑦) = 𝑦))
3332cbvralv 3320 . . . . . . . 8 (∀𝑥𝐵 (𝑂 + 𝑥) = 𝑥 ↔ ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
3429, 33sylib 208 . . . . . . 7 (𝜑 → ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
3534adantr 466 . . . . . 6 ((𝜑𝜓) → ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
3627, 35, 8rspcdva 3466 . . . . 5 ((𝜑𝜓) → (𝑂 + 𝑋) = 𝑋)
3736adantr 466 . . . 4 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑂 + 𝑋) = 𝑋)
3818, 24, 373eqtr3d 2813 . . 3 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + (𝑋 + 𝑋)) = 𝑋)
3916, 38, 173eqtr3d 2813 . 2 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑋 = 𝑂)
4013, 39rexlimddv 3183 1 ((𝜑𝜓) → 𝑋 = 𝑂)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wrex 3062  (class class class)co 6793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796
This theorem is referenced by:  grprinvd  7020
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