MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cntzsnval Structured version   Visualization version   GIF version

Theorem cntzsnval 18392
Description: Special substitution for the centralizer of a singleton. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
cntzfval.b 𝐵 = (Base‘𝑀)
cntzfval.p + = (+g𝑀)
cntzfval.z 𝑍 = (Cntz‘𝑀)
Assertion
Ref Expression
cntzsnval (𝑌𝐵 → (𝑍‘{𝑌}) = {𝑥𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})
Distinct variable groups:   𝑥, +   𝑥,𝐵   𝑥,𝑀   𝑥,𝑌
Allowed substitution hint:   𝑍(𝑥)

Proof of Theorem cntzsnval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 snssi 4733 . . 3 (𝑌𝐵 → {𝑌} ⊆ 𝐵)
2 cntzfval.b . . . 4 𝐵 = (Base‘𝑀)
3 cntzfval.p . . . 4 + = (+g𝑀)
4 cntzfval.z . . . 4 𝑍 = (Cntz‘𝑀)
52, 3, 4cntzval 18389 . . 3 ({𝑌} ⊆ 𝐵 → (𝑍‘{𝑌}) = {𝑥𝐵 ∣ ∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥)})
61, 5syl 17 . 2 (𝑌𝐵 → (𝑍‘{𝑌}) = {𝑥𝐵 ∣ ∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥)})
7 oveq2 7153 . . . . 5 (𝑦 = 𝑌 → (𝑥 + 𝑦) = (𝑥 + 𝑌))
8 oveq1 7152 . . . . 5 (𝑦 = 𝑌 → (𝑦 + 𝑥) = (𝑌 + 𝑥))
97, 8eqeq12d 2834 . . . 4 (𝑦 = 𝑌 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥 + 𝑌) = (𝑌 + 𝑥)))
109ralsng 4605 . . 3 (𝑌𝐵 → (∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥 + 𝑌) = (𝑌 + 𝑥)))
1110rabbidv 3478 . 2 (𝑌𝐵 → {𝑥𝐵 ∣ ∀𝑦 ∈ {𝑌} (𝑥 + 𝑦) = (𝑦 + 𝑥)} = {𝑥𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})
126, 11eqtrd 2853 1 (𝑌𝐵 → (𝑍‘{𝑌}) = {𝑥𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wral 3135  {crab 3139  wss 3933  {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:  elcntzsn  18393  cntziinsn  18403
  Copyright terms: Public domain W3C validator