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Theorem elcntzsn 17581
 Description: Value of the centralizer of a singleton. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
cntzfval.b 𝐵 = (Base‘𝑀)
cntzfval.p + = (+g𝑀)
cntzfval.z 𝑍 = (Cntz‘𝑀)
Assertion
Ref Expression
elcntzsn (𝑌𝐵 → (𝑋 ∈ (𝑍‘{𝑌}) ↔ (𝑋𝐵 ∧ (𝑋 + 𝑌) = (𝑌 + 𝑋))))

Proof of Theorem elcntzsn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cntzfval.b . . . 4 𝐵 = (Base‘𝑀)
2 cntzfval.p . . . 4 + = (+g𝑀)
3 cntzfval.z . . . 4 𝑍 = (Cntz‘𝑀)
41, 2, 3cntzsnval 17580 . . 3 (𝑌𝐵 → (𝑍‘{𝑌}) = {𝑥𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})
54eleq2d 2673 . 2 (𝑌𝐵 → (𝑋 ∈ (𝑍‘{𝑌}) ↔ 𝑋 ∈ {𝑥𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)}))
6 oveq1 6556 . . . 4 (𝑥 = 𝑋 → (𝑥 + 𝑌) = (𝑋 + 𝑌))
7 oveq2 6557 . . . 4 (𝑥 = 𝑋 → (𝑌 + 𝑥) = (𝑌 + 𝑋))
86, 7eqeq12d 2625 . . 3 (𝑥 = 𝑋 → ((𝑥 + 𝑌) = (𝑌 + 𝑥) ↔ (𝑋 + 𝑌) = (𝑌 + 𝑋)))
98elrab 3331 . 2 (𝑋 ∈ {𝑥𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)} ↔ (𝑋𝐵 ∧ (𝑋 + 𝑌) = (𝑌 + 𝑋)))
105, 9syl6bb 275 1 (𝑌𝐵 → (𝑋 ∈ (𝑍‘{𝑌}) ↔ (𝑋𝐵 ∧ (𝑋 + 𝑌) = (𝑌 + 𝑋))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  {csn 4125  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Cntzccntz 17571 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-cntz 17573 This theorem is referenced by:  gsumconst  18157  gsumpt  18184
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