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Theorem cntzval 17577
 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 17576 . . . 4 (𝑀 ∈ V → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
54fveq1d 6105 . . 3 (𝑀 ∈ V → (𝑍𝑆) = ((𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})‘𝑆))
6 fvex 6113 . . . . . 6 (Base‘𝑀) ∈ V
71, 6eqeltri 2684 . . . . 5 𝐵 ∈ V
87elpw2 4755 . . . 4 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
9 raleq 3115 . . . . . 6 (𝑠 = 𝑆 → (∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
109rabbidv 3164 . . . . 5 (𝑠 = 𝑆 → {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)} = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
11 eqid 2610 . . . . 5 (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}) = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
127rabex 4740 . . . . 5 {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ∈ V
1310, 11, 12fvmpt 6191 . . . 4 (𝑆 ∈ 𝒫 𝐵 → ((𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})‘𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
148, 13sylbir 224 . . 3 (𝑆𝐵 → ((𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})‘𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
155, 14sylan9eq 2664 . 2 ((𝑀 ∈ V ∧ 𝑆𝐵) → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
16 0fv 6137 . . . 4 (∅‘𝑆) = ∅
17 fvprc 6097 . . . . . 6 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
183, 17syl5eq 2656 . . . . 5 𝑀 ∈ V → 𝑍 = ∅)
1918fveq1d 6105 . . . 4 𝑀 ∈ V → (𝑍𝑆) = (∅‘𝑆))
20 ssrab2 3650 . . . . . 6 {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ⊆ 𝐵
21 fvprc 6097 . . . . . . 7 𝑀 ∈ V → (Base‘𝑀) = ∅)
221, 21syl5eq 2656 . . . . . 6 𝑀 ∈ V → 𝐵 = ∅)
2320, 22syl5sseq 3616 . . . . 5 𝑀 ∈ V → {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ⊆ ∅)
24 ss0 3926 . . . . 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 2670 . . 3 𝑀 ∈ V → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
2726adantr 480 . 2 ((¬ 𝑀 ∈ V ∧ 𝑆𝐵) → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
2815, 27pm2.61ian 827 1 (𝑆𝐵 → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108   ↦ cmpt 4643  ‘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:  elcntz  17578  cntzsnval  17580  sscntz  17582  cntzssv  17584  cntziinsn  17590
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