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Theorem elcntzsn 18393
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 18392 . . 3 (𝑌𝐵 → (𝑍‘{𝑌}) = {𝑥𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})
54eleq2d 2895 . 2 (𝑌𝐵 → (𝑋 ∈ (𝑍‘{𝑌}) ↔ 𝑋 ∈ {𝑥𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)}))
6 oveq1 7152 . . . 4 (𝑥 = 𝑋 → (𝑥 + 𝑌) = (𝑋 + 𝑌))
7 oveq2 7153 . . . 4 (𝑥 = 𝑋 → (𝑌 + 𝑥) = (𝑌 + 𝑋))
86, 7eqeq12d 2834 . . 3 (𝑥 = 𝑋 → ((𝑥 + 𝑌) = (𝑌 + 𝑥) ↔ (𝑋 + 𝑌) = (𝑌 + 𝑋)))
98elrab 3677 . 2 (𝑋 ∈ {𝑥𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)} ↔ (𝑋𝐵 ∧ (𝑋 + 𝑌) = (𝑌 + 𝑋)))
105, 9syl6bb 288 1 (𝑌𝐵 → (𝑋 ∈ (𝑍‘{𝑌}) ↔ (𝑋𝐵 ∧ (𝑋 + 𝑌) = (𝑌 + 𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  {crab 3139  {csn 4557  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  Cntzccntz 18383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-cntz 18385
This theorem is referenced by:  gsumconst  18983  gsumpt  19011  cntzsnid  30623
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