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Theorem cntzfval 17959
 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 3361 . . 3 (𝑀𝑉𝑀 ∈ V)
3 fveq2 6332 . . . . . . 7 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
4 cntzfval.b . . . . . . 7 𝐵 = (Base‘𝑀)
53, 4syl6eqr 2822 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
65pweqd 4300 . . . . 5 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 𝐵)
7 fveq2 6332 . . . . . . . . . 10 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
8 cntzfval.p . . . . . . . . . 10 + = (+g𝑀)
97, 8syl6eqr 2822 . . . . . . . . 9 (𝑚 = 𝑀 → (+g𝑚) = + )
109oveqd 6809 . . . . . . . 8 (𝑚 = 𝑀 → (𝑥(+g𝑚)𝑦) = (𝑥 + 𝑦))
119oveqd 6809 . . . . . . . 8 (𝑚 = 𝑀 → (𝑦(+g𝑚)𝑥) = (𝑦 + 𝑥))
1210, 11eqeq12d 2785 . . . . . . 7 (𝑚 = 𝑀 → ((𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥)))
1312ralbidv 3134 . . . . . 6 (𝑚 = 𝑀 → (∀𝑦𝑠 (𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥) ↔ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
145, 13rabeqbidv 3344 . . . . 5 (𝑚 = 𝑀 → {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦𝑠 (𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥)} = {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
156, 14mpteq12dv 4865 . . . 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 17956 . . . 4 Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦𝑠 (𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥)}))
17 fvex 6342 . . . . . . 7 (Base‘𝑀) ∈ V
184, 17eqeltri 2845 . . . . . 6 𝐵 ∈ V
1918pwex 4976 . . . . 5 𝒫 𝐵 ∈ V
2019mptex 6629 . . . 4 (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}) ∈ V
2115, 16, 20fvmpt 6424 . . 3 (𝑀 ∈ V → (Cntz‘𝑀) = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
222, 21syl 17 . 2 (𝑀𝑉 → (Cntz‘𝑀) = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
231, 22syl5eq 2816 1 (𝑀𝑉𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1630   ∈ wcel 2144  ∀wral 3060  {crab 3064  Vcvv 3349  𝒫 cpw 4295   ↦ cmpt 4861  ‘cfv 6031  (class class class)co 6792  Basecbs 16063  +gcplusg 16148  Cntzccntz 17954 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-cntz 17956 This theorem is referenced by:  cntzval  17960  cntzrcl  17966
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