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Theorem cntzfval 17747
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 3210 . . 3 (𝑀𝑉𝑀 ∈ V)
3 fveq2 6189 . . . . . . 7 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
4 cntzfval.b . . . . . . 7 𝐵 = (Base‘𝑀)
53, 4syl6eqr 2673 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
65pweqd 4161 . . . . 5 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 𝐵)
7 fveq2 6189 . . . . . . . . . 10 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
8 cntzfval.p . . . . . . . . . 10 + = (+g𝑀)
97, 8syl6eqr 2673 . . . . . . . . 9 (𝑚 = 𝑀 → (+g𝑚) = + )
109oveqd 6664 . . . . . . . 8 (𝑚 = 𝑀 → (𝑥(+g𝑚)𝑦) = (𝑥 + 𝑦))
119oveqd 6664 . . . . . . . 8 (𝑚 = 𝑀 → (𝑦(+g𝑚)𝑥) = (𝑦 + 𝑥))
1210, 11eqeq12d 2636 . . . . . . 7 (𝑚 = 𝑀 → ((𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥)))
1312ralbidv 2985 . . . . . 6 (𝑚 = 𝑀 → (∀𝑦𝑠 (𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥) ↔ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
145, 13rabeqbidv 3193 . . . . 5 (𝑚 = 𝑀 → {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦𝑠 (𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥)} = {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
156, 14mpteq12dv 4731 . . . 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 17744 . . . 4 Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦𝑠 (𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥)}))
17 fvex 6199 . . . . . . 7 (Base‘𝑀) ∈ V
184, 17eqeltri 2696 . . . . . 6 𝐵 ∈ V
1918pwex 4846 . . . . 5 𝒫 𝐵 ∈ V
2019mptex 6483 . . . 4 (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}) ∈ V
2115, 16, 20fvmpt 6280 . . 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 2667 1 (𝑀𝑉𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  wral 2911  {crab 2915  Vcvv 3198  𝒫 cpw 4156  cmpt 4727  cfv 5886  (class class class)co 6647  Basecbs 15851  +gcplusg 15935  Cntzccntz 17742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-cntz 17744
This theorem is referenced by:  cntzval  17748  cntzrcl  17754
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