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

Theorem cntzfval 18441
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 3497 . . 3 (𝑀𝑉𝑀 ∈ V)
3 fveq2 6653 . . . . . . 7 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
4 cntzfval.b . . . . . . 7 𝐵 = (Base‘𝑀)
53, 4syl6eqr 2877 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
65pweqd 4539 . . . . 5 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 𝐵)
7 fveq2 6653 . . . . . . . . . 10 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
8 cntzfval.p . . . . . . . . . 10 + = (+g𝑀)
97, 8syl6eqr 2877 . . . . . . . . 9 (𝑚 = 𝑀 → (+g𝑚) = + )
109oveqd 7157 . . . . . . . 8 (𝑚 = 𝑀 → (𝑥(+g𝑚)𝑦) = (𝑥 + 𝑦))
119oveqd 7157 . . . . . . . 8 (𝑚 = 𝑀 → (𝑦(+g𝑚)𝑥) = (𝑦 + 𝑥))
1210, 11eqeq12d 2840 . . . . . . 7 (𝑚 = 𝑀 → ((𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥)))
1312ralbidv 3191 . . . . . 6 (𝑚 = 𝑀 → (∀𝑦𝑠 (𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥) ↔ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
145, 13rabeqbidv 3470 . . . . 5 (𝑚 = 𝑀 → {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦𝑠 (𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥)} = {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
156, 14mpteq12dv 5134 . . . 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 18438 . . . 4 Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦𝑠 (𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥)}))
174fvexi 6667 . . . . . 6 𝐵 ∈ V
1817pwex 5264 . . . . 5 𝒫 𝐵 ∈ V
1918mptex 6969 . . . 4 (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}) ∈ V
2015, 16, 19fvmpt 6751 . . 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, 21syl5eq 2871 1 (𝑀𝑉𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wral 3132  {crab 3136  Vcvv 3479  𝒫 cpw 4520  cmpt 5129  cfv 6338  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  Cntzccntz 18436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-id 5443  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-ov 7143  df-cntz 18438
This theorem is referenced by:  cntzval  18442  cntzrcl  18448
  Copyright terms: Public domain W3C validator