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Theorem cntzval 19339
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 19338 . . . 4 (𝑀 ∈ V → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
54fveq1d 6908 . . 3 (𝑀 ∈ V → (𝑍𝑆) = ((𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})‘𝑆))
61fvexi 6920 . . . . 5 𝐵 ∈ V
76elpw2 5334 . . . 4 (𝑆 ∈ 𝒫 𝐵𝑆𝐵)
8 raleq 3323 . . . . . 6 (𝑠 = 𝑆 → (∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
98rabbidv 3444 . . . . 5 (𝑠 = 𝑆 → {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)} = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
10 eqid 2737 . . . . 5 (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}) = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
116rabex 5339 . . . . 5 {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ∈ V
129, 10, 11fvmpt 7016 . . . 4 (𝑆 ∈ 𝒫 𝐵 → ((𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})‘𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
137, 12sylbir 235 . . 3 (𝑆𝐵 → ((𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})‘𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
145, 13sylan9eq 2797 . 2 ((𝑀 ∈ V ∧ 𝑆𝐵) → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
15 0fv 6950 . . . 4 (∅‘𝑆) = ∅
16 fvprc 6898 . . . . . 6 𝑀 ∈ V → (Cntz‘𝑀) = ∅)
173, 16eqtrid 2789 . . . . 5 𝑀 ∈ V → 𝑍 = ∅)
1817fveq1d 6908 . . . 4 𝑀 ∈ V → (𝑍𝑆) = (∅‘𝑆))
19 ssrab2 4080 . . . . . 6 {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ⊆ 𝐵
20 fvprc 6898 . . . . . . 7 𝑀 ∈ V → (Base‘𝑀) = ∅)
211, 20eqtrid 2789 . . . . . 6 𝑀 ∈ V → 𝐵 = ∅)
2219, 21sseqtrid 4026 . . . . 5 𝑀 ∈ V → {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ⊆ ∅)
23 ss0 4402 . . . . 5 ({𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ⊆ ∅ → {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} = ∅)
2422, 23syl 17 . . . 4 𝑀 ∈ V → {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} = ∅)
2515, 18, 243eqtr4a 2803 . . 3 𝑀 ∈ V → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
2625adantr 480 . 2 ((¬ 𝑀 ∈ V ∧ 𝑆𝐵) → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
2714, 26pm2.61ian 812 1 (𝑆𝐵 → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  wss 3951  c0 4333  𝒫 cpw 4600  cmpt 5225  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Cntzccntz 19333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-cntz 19335
This theorem is referenced by:  elcntz  19340  cntzsnval  19342  sscntz  19344  cntzssv  19346  cntziinsn  19355  cmnbascntr  19823
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