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

Theorem cntzval 17670
Description: Definition substitution for a centralizer. (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
cntzval (𝑆𝐵 → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
Distinct variable groups:   𝑥,𝑦, +   𝑥,𝐵   𝑥,𝑀,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐵(𝑦)   𝑍(𝑥,𝑦)

Proof of Theorem cntzval
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 cntzfval.b . . . . 5 𝐵 = (Base‘𝑀)
2 cntzfval.p . . . . 5 + = (+g𝑀)
3 cntzfval.z . . . . 5 𝑍 = (Cntz‘𝑀)
41, 2, 3cntzfval 17669 . . . 4 (𝑀 ∈ V → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
54fveq1d 6152 . . 3 (𝑀 ∈ V → (𝑍𝑆) = ((𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})‘𝑆))
6 fvex 6160 . . . . . 6 (Base‘𝑀) ∈ V
71, 6eqeltri 2700 . . . . 5 𝐵 ∈ V
87elpw2 4793 . . . 4 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
9 raleq 3132 . . . . . 6 (𝑠 = 𝑆 → (∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
109rabbidv 3182 . . . . 5 (𝑠 = 𝑆 → {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)} = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
11 eqid 2626 . . . . 5 (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}) = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
127rabex 4778 . . . . 5 {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ∈ V
1310, 11, 12fvmpt 6240 . . . 4 (𝑆 ∈ 𝒫 𝐵 → ((𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})‘𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
148, 13sylbir 225 . . 3 (𝑆𝐵 → ((𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})‘𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
155, 14sylan9eq 2680 . 2 ((𝑀 ∈ V ∧ 𝑆𝐵) → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
16 0fv 6185 . . . 4 (∅‘𝑆) = ∅
17 fvprc 6144 . . . . . 6 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
183, 17syl5eq 2672 . . . . 5 𝑀 ∈ V → 𝑍 = ∅)
1918fveq1d 6152 . . . 4 𝑀 ∈ V → (𝑍𝑆) = (∅‘𝑆))
20 ssrab2 3671 . . . . . 6 {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ⊆ 𝐵
21 fvprc 6144 . . . . . . 7 𝑀 ∈ V → (Base‘𝑀) = ∅)
221, 21syl5eq 2672 . . . . . 6 𝑀 ∈ V → 𝐵 = ∅)
2320, 22syl5sseq 3637 . . . . 5 𝑀 ∈ V → {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ⊆ ∅)
24 ss0 3951 . . . . 5 ({𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ⊆ ∅ → {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} = ∅)
2523, 24syl 17 . . . 4 𝑀 ∈ V → {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} = ∅)
2616, 19, 253eqtr4a 2686 . . 3 𝑀 ∈ V → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
2726adantr 481 . 2 ((¬ 𝑀 ∈ V ∧ 𝑆𝐵) → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
2815, 27pm2.61ian 830 1 (𝑆𝐵 → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wcel 1992  wral 2912  {crab 2916  Vcvv 3191  wss 3560  c0 3896  𝒫 cpw 4135  cmpt 4678  cfv 5850  (class class class)co 6605  Basecbs 15776  +gcplusg 15857  Cntzccntz 17664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-cntz 17666
This theorem is referenced by:  elcntz  17671  cntzsnval  17673  sscntz  17675  cntzssv  17677  cntziinsn  17683
  Copyright terms: Public domain W3C validator