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Theorem cntzfval 18924
Description: First level 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
cntzfval (𝑀𝑉𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
Distinct variable groups:   𝑥,𝑠,𝑦, +   𝐵,𝑠,𝑥   𝑀,𝑠,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑦)   𝑉(𝑥,𝑦,𝑠)   𝑍(𝑥,𝑦,𝑠)

Proof of Theorem cntzfval
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 cntzfval.z . 2 𝑍 = (Cntz‘𝑀)
2 elex 3449 . . 3 (𝑀𝑉𝑀 ∈ V)
3 fveq2 6776 . . . . . . 7 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
4 cntzfval.b . . . . . . 7 𝐵 = (Base‘𝑀)
53, 4eqtr4di 2796 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
65pweqd 4554 . . . . 5 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 𝐵)
7 fveq2 6776 . . . . . . . . . 10 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
8 cntzfval.p . . . . . . . . . 10 + = (+g𝑀)
97, 8eqtr4di 2796 . . . . . . . . 9 (𝑚 = 𝑀 → (+g𝑚) = + )
109oveqd 7294 . . . . . . . 8 (𝑚 = 𝑀 → (𝑥(+g𝑚)𝑦) = (𝑥 + 𝑦))
119oveqd 7294 . . . . . . . 8 (𝑚 = 𝑀 → (𝑦(+g𝑚)𝑥) = (𝑦 + 𝑥))
1210, 11eqeq12d 2754 . . . . . . 7 (𝑚 = 𝑀 → ((𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥)))
1312ralbidv 3119 . . . . . 6 (𝑚 = 𝑀 → (∀𝑦𝑠 (𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥) ↔ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
145, 13rabeqbidv 3419 . . . . 5 (𝑚 = 𝑀 → {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦𝑠 (𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥)} = {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
156, 14mpteq12dv 5167 . . . 4 (𝑚 = 𝑀 → (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦𝑠 (𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥)}) = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
16 df-cntz 18921 . . . 4 Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦𝑠 (𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥)}))
174fvexi 6790 . . . . . 6 𝐵 ∈ V
1817pwex 5305 . . . . 5 𝒫 𝐵 ∈ V
1918mptex 7101 . . . 4 (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}) ∈ V
2015, 16, 19fvmpt 6877 . . 3 (𝑀 ∈ V → (Cntz‘𝑀) = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
212, 20syl 17 . 2 (𝑀𝑉 → (Cntz‘𝑀) = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
221, 21eqtrid 2790 1 (𝑀𝑉𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  wral 3064  {crab 3068  Vcvv 3431  𝒫 cpw 4535  cmpt 5159  cfv 6435  (class class class)co 7277  Basecbs 16910  +gcplusg 16960  Cntzccntz 18919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5211  ax-sep 5225  ax-nul 5232  ax-pow 5290  ax-pr 5354
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5077  df-opab 5139  df-mpt 5160  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6393  df-fun 6437  df-fn 6438  df-f 6439  df-f1 6440  df-fo 6441  df-f1o 6442  df-fv 6443  df-ov 7280  df-cntz 18921
This theorem is referenced by:  cntzval  18925  cntzrcl  18931
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