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Theorem grpinvalem 12968
Description: Lemma for grpinva 12969. (Contributed by NM, 9-Aug-2013.)
Hypotheses
Ref Expression
grpinva.c ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
grpinva.o (𝜑𝑂𝐵)
grpinva.i ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = 𝑥)
grpinva.a ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
grpinva.r ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂)
grpinvalem.x ((𝜑𝜓) → 𝑋𝐵)
grpinvalem.e ((𝜑𝜓) → (𝑋 + 𝑋) = 𝑋)
Assertion
Ref Expression
grpinvalem ((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 grpinvalem
Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpinva.r . . . . 5 ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂)
21ralrimiva 2567 . . . 4 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 𝑂)
3 oveq2 5926 . . . . . . 7 (𝑥 = 𝑧 → (𝑦 + 𝑥) = (𝑦 + 𝑧))
43eqeq1d 2202 . . . . . 6 (𝑥 = 𝑧 → ((𝑦 + 𝑥) = 𝑂 ↔ (𝑦 + 𝑧) = 𝑂))
54rexbidv 2495 . . . . 5 (𝑥 = 𝑧 → (∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∃𝑦𝐵 (𝑦 + 𝑧) = 𝑂))
65cbvralvw 2730 . . . 4 (∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∀𝑧𝐵𝑦𝐵 (𝑦 + 𝑧) = 𝑂)
72, 6sylib 122 . . 3 (𝜑 → ∀𝑧𝐵𝑦𝐵 (𝑦 + 𝑧) = 𝑂)
8 grpinvalem.x . . 3 ((𝜑𝜓) → 𝑋𝐵)
9 oveq2 5926 . . . . . 6 (𝑧 = 𝑋 → (𝑦 + 𝑧) = (𝑦 + 𝑋))
109eqeq1d 2202 . . . . 5 (𝑧 = 𝑋 → ((𝑦 + 𝑧) = 𝑂 ↔ (𝑦 + 𝑋) = 𝑂))
1110rexbidv 2495 . . . 4 (𝑧 = 𝑋 → (∃𝑦𝐵 (𝑦 + 𝑧) = 𝑂 ↔ ∃𝑦𝐵 (𝑦 + 𝑋) = 𝑂))
1211rspccva 2863 . . 3 ((∀𝑧𝐵𝑦𝐵 (𝑦 + 𝑧) = 𝑂𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 𝑂)
137, 8, 12syl2an2r 595 . 2 ((𝜑𝜓) → ∃𝑦𝐵 (𝑦 + 𝑋) = 𝑂)
14 grpinvalem.e . . . . 5 ((𝜑𝜓) → (𝑋 + 𝑋) = 𝑋)
1514oveq2d 5934 . . . 4 ((𝜑𝜓) → (𝑦 + (𝑋 + 𝑋)) = (𝑦 + 𝑋))
1615adantr 276 . . 3 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + (𝑋 + 𝑋)) = (𝑦 + 𝑋))
17 simprr 531 . . . . 5 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + 𝑋) = 𝑂)
1817oveq1d 5933 . . . 4 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → ((𝑦 + 𝑋) + 𝑋) = (𝑂 + 𝑋))
19 grpinva.a . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
2019caovassg 6077 . . . . . 6 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
2120ad4ant14 514 . . . . 5 ((((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
22 simprl 529 . . . . 5 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑦𝐵)
238adantr 276 . . . . 5 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑋𝐵)
2421, 22, 23, 23caovassd 6078 . . . 4 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → ((𝑦 + 𝑋) + 𝑋) = (𝑦 + (𝑋 + 𝑋)))
25 oveq2 5926 . . . . . . 7 (𝑦 = 𝑋 → (𝑂 + 𝑦) = (𝑂 + 𝑋))
26 id 19 . . . . . . 7 (𝑦 = 𝑋𝑦 = 𝑋)
2725, 26eqeq12d 2208 . . . . . 6 (𝑦 = 𝑋 → ((𝑂 + 𝑦) = 𝑦 ↔ (𝑂 + 𝑋) = 𝑋))
28 grpinva.i . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = 𝑥)
2928ralrimiva 2567 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (𝑂 + 𝑥) = 𝑥)
30 oveq2 5926 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑂 + 𝑥) = (𝑂 + 𝑦))
31 id 19 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
3230, 31eqeq12d 2208 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑂 + 𝑥) = 𝑥 ↔ (𝑂 + 𝑦) = 𝑦))
3332cbvralvw 2730 . . . . . . . 8 (∀𝑥𝐵 (𝑂 + 𝑥) = 𝑥 ↔ ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
3429, 33sylib 122 . . . . . . 7 (𝜑 → ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
3534adantr 276 . . . . . 6 ((𝜑𝜓) → ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
3627, 35, 8rspcdva 2869 . . . . 5 ((𝜑𝜓) → (𝑂 + 𝑋) = 𝑋)
3736adantr 276 . . . 4 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑂 + 𝑋) = 𝑋)
3818, 24, 373eqtr3d 2234 . . 3 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + (𝑋 + 𝑋)) = 𝑋)
3916, 38, 173eqtr3d 2234 . 2 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑋 = 𝑂)
4013, 39rexlimddv 2616 1 ((𝜑𝜓) → 𝑋 = 𝑂)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  wcel 2164  wral 2472  wrex 2473  (class class class)co 5918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921
This theorem is referenced by:  grpinva  12969
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