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Theorem grprinvlem 5722
Description: Lemma for grprinvd 5723. (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 2409 . . . . 5 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 𝑂)
4 oveq2 5547 . . . . . . . 8 (𝑥 = 𝑧 → (𝑦 + 𝑥) = (𝑦 + 𝑧))
54eqeq1d 2064 . . . . . . 7 (𝑥 = 𝑧 → ((𝑦 + 𝑥) = 𝑂 ↔ (𝑦 + 𝑧) = 𝑂))
65rexbidv 2344 . . . . . 6 (𝑥 = 𝑧 → (∃𝑦𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∃𝑦𝐵 (𝑦 + 𝑧) = 𝑂))
76cbvralv 2550 . . . . 5 (∀𝑥𝐵𝑦𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∀𝑧𝐵𝑦𝐵 (𝑦 + 𝑧) = 𝑂)
83, 7sylib 131 . . . 4 (𝜑 → ∀𝑧𝐵𝑦𝐵 (𝑦 + 𝑧) = 𝑂)
9 oveq2 5547 . . . . . . 7 (𝑧 = 𝑋 → (𝑦 + 𝑧) = (𝑦 + 𝑋))
109eqeq1d 2064 . . . . . 6 (𝑧 = 𝑋 → ((𝑦 + 𝑧) = 𝑂 ↔ (𝑦 + 𝑋) = 𝑂))
1110rexbidv 2344 . . . . 5 (𝑧 = 𝑋 → (∃𝑦𝐵 (𝑦 + 𝑧) = 𝑂 ↔ ∃𝑦𝐵 (𝑦 + 𝑋) = 𝑂))
1211rspccva 2672 . . . 4 ((∀𝑧𝐵𝑦𝐵 (𝑦 + 𝑧) = 𝑂𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 𝑂)
138, 12sylan 271 . . 3 ((𝜑𝑋𝐵) → ∃𝑦𝐵 (𝑦 + 𝑋) = 𝑂)
141, 13syldan 270 . 2 ((𝜑𝜓) → ∃𝑦𝐵 (𝑦 + 𝑋) = 𝑂)
15 grprinvlem.e . . . . 5 ((𝜑𝜓) → (𝑋 + 𝑋) = 𝑋)
1615oveq2d 5555 . . . 4 ((𝜑𝜓) → (𝑦 + (𝑋 + 𝑋)) = (𝑦 + 𝑋))
1716adantr 265 . . 3 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + (𝑋 + 𝑋)) = (𝑦 + 𝑋))
18 simprr 492 . . . . 5 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + 𝑋) = 𝑂)
1918oveq1d 5554 . . . 4 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → ((𝑦 + 𝑋) + 𝑋) = (𝑂 + 𝑋))
20 simpll 489 . . . . . 6 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝜑)
21 grprinvlem.a . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
2221caovassg 5686 . . . . . 6 ((𝜑 ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
2320, 22sylan 271 . . . . 5 ((((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
24 simprl 491 . . . . 5 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑦𝐵)
251adantr 265 . . . . 5 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑋𝐵)
2623, 24, 25, 25caovassd 5687 . . . 4 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → ((𝑦 + 𝑋) + 𝑋) = (𝑦 + (𝑋 + 𝑋)))
27 grprinvlem.i . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑂 + 𝑥) = 𝑥)
2827ralrimiva 2409 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (𝑂 + 𝑥) = 𝑥)
29 oveq2 5547 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑂 + 𝑥) = (𝑂 + 𝑦))
30 id 19 . . . . . . . . . 10 (𝑥 = 𝑦𝑥 = 𝑦)
3129, 30eqeq12d 2070 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑂 + 𝑥) = 𝑥 ↔ (𝑂 + 𝑦) = 𝑦))
3231cbvralv 2550 . . . . . . . 8 (∀𝑥𝐵 (𝑂 + 𝑥) = 𝑥 ↔ ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
3328, 32sylib 131 . . . . . . 7 (𝜑 → ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
3433adantr 265 . . . . . 6 ((𝜑𝜓) → ∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦)
35 oveq2 5547 . . . . . . . 8 (𝑦 = 𝑋 → (𝑂 + 𝑦) = (𝑂 + 𝑋))
36 id 19 . . . . . . . 8 (𝑦 = 𝑋𝑦 = 𝑋)
3735, 36eqeq12d 2070 . . . . . . 7 (𝑦 = 𝑋 → ((𝑂 + 𝑦) = 𝑦 ↔ (𝑂 + 𝑋) = 𝑋))
3837rspcv 2669 . . . . . 6 (𝑋𝐵 → (∀𝑦𝐵 (𝑂 + 𝑦) = 𝑦 → (𝑂 + 𝑋) = 𝑋))
391, 34, 38sylc 60 . . . . 5 ((𝜑𝜓) → (𝑂 + 𝑋) = 𝑋)
4039adantr 265 . . . 4 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑂 + 𝑋) = 𝑋)
4119, 26, 403eqtr3d 2096 . . 3 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + (𝑋 + 𝑋)) = 𝑋)
4217, 41, 183eqtr3d 2096 . 2 (((𝜑𝜓) ∧ (𝑦𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑋 = 𝑂)
4314, 42rexlimddv 2454 1 ((𝜑𝜓) → 𝑋 = 𝑂)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 896   = wceq 1259  wcel 1409  wral 2323  wrex 2324  (class class class)co 5539
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-iota 4894  df-fv 4937  df-ov 5542
This theorem is referenced by:  grprinvd  5723
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