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Theorem rngoideu 36412
Description: The unity 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 36408 . . 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 773 . 2 (𝑅 ∈ RingOps → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
6 simpl 484 . . . . . . 7 (((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) → (𝑢𝐻𝑥) = 𝑥)
76ralimi 3083 . . . . . 6 (∀𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) → ∀𝑥𝑋 (𝑢𝐻𝑥) = 𝑥)
8 oveq2 7369 . . . . . . . 8 (𝑥 = 𝑦 → (𝑢𝐻𝑥) = (𝑢𝐻𝑦))
9 id 22 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
108, 9eqeq12d 2749 . . . . . . 7 (𝑥 = 𝑦 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑢𝐻𝑦) = 𝑦))
1110rspcv 3579 . . . . . 6 (𝑦𝑋 → (∀𝑥𝑋 (𝑢𝐻𝑥) = 𝑥 → (𝑢𝐻𝑦) = 𝑦))
127, 11syl5 34 . . . . 5 (𝑦𝑋 → (∀𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) → (𝑢𝐻𝑦) = 𝑦))
13 simpr 486 . . . . . . 7 (((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥) → (𝑥𝐻𝑦) = 𝑥)
1413ralimi 3083 . . . . . 6 (∀𝑥𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥) → ∀𝑥𝑋 (𝑥𝐻𝑦) = 𝑥)
15 oveq1 7368 . . . . . . . 8 (𝑥 = 𝑢 → (𝑥𝐻𝑦) = (𝑢𝐻𝑦))
16 id 22 . . . . . . . 8 (𝑥 = 𝑢𝑥 = 𝑢)
1715, 16eqeq12d 2749 . . . . . . 7 (𝑥 = 𝑢 → ((𝑥𝐻𝑦) = 𝑥 ↔ (𝑢𝐻𝑦) = 𝑢))
1817rspcv 3579 . . . . . 6 (𝑢𝑋 → (∀𝑥𝑋 (𝑥𝐻𝑦) = 𝑥 → (𝑢𝐻𝑦) = 𝑢))
1914, 18syl5 34 . . . . 5 (𝑢𝑋 → (∀𝑥𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥) → (𝑢𝐻𝑦) = 𝑢))
2012, 19im2anan9r 622 . . . 4 ((𝑢𝑋𝑦𝑋) → ((∀𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ ∀𝑥𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥)) → ((𝑢𝐻𝑦) = 𝑦 ∧ (𝑢𝐻𝑦) = 𝑢)))
21 eqtr2 2757 . . . . 5 (((𝑢𝐻𝑦) = 𝑦 ∧ (𝑢𝐻𝑦) = 𝑢) → 𝑦 = 𝑢)
2221equcomd 2023 . . . 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 3191 . 2 𝑢𝑋𝑦𝑋 ((∀𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ ∀𝑥𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥)) → 𝑢 = 𝑦)
25 oveq1 7368 . . . . 5 (𝑢 = 𝑦 → (𝑢𝐻𝑥) = (𝑦𝐻𝑥))
2625eqeq1d 2735 . . . 4 (𝑢 = 𝑦 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑦𝐻𝑥) = 𝑥))
2726ovanraleqv 7385 . . 3 (𝑢 = 𝑦 → (∀𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∀𝑥𝑋 ((𝑦𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑦) = 𝑥)))
2827reu4 3693 . 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 591 1 (𝑅 ∈ RingOps → ∃!𝑢𝑋𝑥𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wrex 3070  ∃!wreu 3350   × cxp 5635  ran crn 5638  wf 6496  cfv 6500  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924  AbelOpcablo 29535  RingOpscrngo 36403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-1st 7925  df-2nd 7926  df-rngo 36404
This theorem is referenced by: (None)
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