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Theorem rngoideu 35824
Description: The unit element of a ring is unique. (Contributed by NM, 4-Apr-2009.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringi.1 𝐺 = (1st𝑅)
ringi.2 𝐻 = (2nd𝑅)
ringi.3 𝑋 = ran 𝐺
Assertion
Ref Expression
rngoideu (𝑅 ∈ RingOps → ∃!𝑢𝑋𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
Distinct variable groups:   𝑥,𝑢,𝐺   𝑢,𝐻,𝑥   𝑢,𝑋,𝑥   𝑢,𝑅,𝑥

Proof of Theorem rngoideu
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ringi.1 . . . 4 𝐺 = (1st𝑅)
2 ringi.2 . . . 4 𝐻 = (2nd𝑅)
3 ringi.3 . . . 4 𝑋 = ran 𝐺
41, 2, 3rngoi 35820 . . 3 (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑢𝑋𝑥𝑋𝑦𝑋 (((𝑢𝐻𝑥)𝐻𝑦) = (𝑢𝐻(𝑥𝐻𝑦)) ∧ (𝑢𝐻(𝑥𝐺𝑦)) = ((𝑢𝐻𝑥)𝐺(𝑢𝐻𝑦)) ∧ ((𝑢𝐺𝑥)𝐻𝑦) = ((𝑢𝐻𝑦)𝐺(𝑥𝐻𝑦))) ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))))
54simprrd 774 . 2 (𝑅 ∈ RingOps → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
6 simpl 486 . . . . . . 7 (((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) → (𝑢𝐻𝑥) = 𝑥)
76ralimi 3084 . . . . . 6 (∀𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) → ∀𝑥𝑋 (𝑢𝐻𝑥) = 𝑥)
8 oveq2 7239 . . . . . . . 8 (𝑥 = 𝑦 → (𝑢𝐻𝑥) = (𝑢𝐻𝑦))
9 id 22 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
108, 9eqeq12d 2754 . . . . . . 7 (𝑥 = 𝑦 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑢𝐻𝑦) = 𝑦))
1110rspcv 3544 . . . . . 6 (𝑦𝑋 → (∀𝑥𝑋 (𝑢𝐻𝑥) = 𝑥 → (𝑢𝐻𝑦) = 𝑦))
127, 11syl5 34 . . . . 5 (𝑦𝑋 → (∀𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) → (𝑢𝐻𝑦) = 𝑦))
13 simpr 488 . . . . . . 7 (((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥) → (𝑥𝐻𝑦) = 𝑥)
1413ralimi 3084 . . . . . 6 (∀𝑥𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥) → ∀𝑥𝑋 (𝑥𝐻𝑦) = 𝑥)
15 oveq1 7238 . . . . . . . 8 (𝑥 = 𝑢 → (𝑥𝐻𝑦) = (𝑢𝐻𝑦))
16 id 22 . . . . . . . 8 (𝑥 = 𝑢𝑥 = 𝑢)
1715, 16eqeq12d 2754 . . . . . . 7 (𝑥 = 𝑢 → ((𝑥𝐻𝑦) = 𝑥 ↔ (𝑢𝐻𝑦) = 𝑢))
1817rspcv 3544 . . . . . 6 (𝑢𝑋 → (∀𝑥𝑋 (𝑥𝐻𝑦) = 𝑥 → (𝑢𝐻𝑦) = 𝑢))
1914, 18syl5 34 . . . . 5 (𝑢𝑋 → (∀𝑥𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥) → (𝑢𝐻𝑦) = 𝑢))
2012, 19im2anan9r 624 . . . 4 ((𝑢𝑋𝑦𝑋) → ((∀𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ ∀𝑥𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥)) → ((𝑢𝐻𝑦) = 𝑦 ∧ (𝑢𝐻𝑦) = 𝑢)))
21 eqtr2 2762 . . . . 5 (((𝑢𝐻𝑦) = 𝑦 ∧ (𝑢𝐻𝑦) = 𝑢) → 𝑦 = 𝑢)
2221equcomd 2027 . . . 4 (((𝑢𝐻𝑦) = 𝑦 ∧ (𝑢𝐻𝑦) = 𝑢) → 𝑢 = 𝑦)
2320, 22syl6 35 . . 3 ((𝑢𝑋𝑦𝑋) → ((∀𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ ∀𝑥𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥)) → 𝑢 = 𝑦))
2423rgen2 3125 . 2 𝑢𝑋𝑦𝑋 ((∀𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ ∀𝑥𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥)) → 𝑢 = 𝑦)
25 oveq1 7238 . . . . 5 (𝑢 = 𝑦 → (𝑢𝐻𝑥) = (𝑦𝐻𝑥))
2625eqeq1d 2740 . . . 4 (𝑢 = 𝑦 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑦𝐻𝑥) = 𝑥))
2726ovanraleqv 7255 . . 3 (𝑢 = 𝑦 → (∀𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∀𝑥𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥)))
2827reu4 3658 . 2 (∃!𝑢𝑋𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ (∃𝑢𝑋𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ ∀𝑢𝑋𝑦𝑋 ((∀𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ ∀𝑥𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥)) → 𝑢 = 𝑦)))
295, 24, 28sylanblrc 593 1 (𝑅 ∈ RingOps → ∃!𝑢𝑋𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2111  wral 3062  wrex 3063  ∃!wreu 3064   × cxp 5563  ran crn 5566  wf 6393  cfv 6397  (class class class)co 7231  1st c1st 7777  2nd c2nd 7778  AbelOpcablo 28649  RingOpscrngo 35815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pr 5336  ax-un 7541
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rmo 3070  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-br 5068  df-opab 5130  df-mpt 5150  df-id 5469  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-rn 5576  df-iota 6355  df-fun 6399  df-fn 6400  df-f 6401  df-fv 6405  df-ov 7234  df-1st 7779  df-2nd 7780  df-rngo 35816
This theorem is referenced by: (None)
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