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Theorem mzpclval 37814
Description: Substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Assertion
Ref Expression
mzpclval (𝑉 ∈ V → (mzPolyCld‘𝑉) = {𝑝 ∈ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝) ∧ ∀𝑓𝑝𝑔𝑝 ((𝑓𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑝))})
Distinct variable groups:   𝑉,𝑝,𝑓,𝑔   𝑖,𝑉,𝑝   𝑗,𝑉,𝑥,𝑝

Proof of Theorem mzpclval
Dummy variables 𝑣 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6801 . . . . 5 (𝑣 = 𝑉 → (ℤ ↑𝑚 𝑣) = (ℤ ↑𝑚 𝑉))
21oveq2d 6809 . . . 4 (𝑣 = 𝑉 → (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) = (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)))
32pweqd 4302 . . 3 (𝑣 = 𝑉 → 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) = 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)))
41xpeq1d 5278 . . . . . . . 8 (𝑣 = 𝑉 → ((ℤ ↑𝑚 𝑣) × {𝑎}) = ((ℤ ↑𝑚 𝑉) × {𝑎}))
54eleq1d 2835 . . . . . . 7 (𝑣 = 𝑉 → (((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝))
65ralbidv 3135 . . . . . 6 (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝))
7 sneq 4326 . . . . . . . . 9 (𝑎 = 𝑖 → {𝑎} = {𝑖})
87xpeq2d 5279 . . . . . . . 8 (𝑎 = 𝑖 → ((ℤ ↑𝑚 𝑉) × {𝑎}) = ((ℤ ↑𝑚 𝑉) × {𝑖}))
98eleq1d 2835 . . . . . . 7 (𝑎 = 𝑖 → (((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝))
109cbvralv 3320 . . . . . 6 (∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝)
116, 10syl6bb 276 . . . . 5 (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝))
121mpteq1d 4872 . . . . . . . 8 (𝑣 = 𝑉 → (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐𝑏)) = (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑏)))
1312eleq1d 2835 . . . . . . 7 (𝑣 = 𝑉 → ((𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝))
1413raleqbi1dv 3295 . . . . . 6 (𝑣 = 𝑉 → (∀𝑏𝑣 (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ ∀𝑏𝑉 (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝))
15 fveq2 6332 . . . . . . . . . 10 (𝑏 = 𝑗 → (𝑐𝑏) = (𝑐𝑗))
1615mpteq2dv 4879 . . . . . . . . 9 (𝑏 = 𝑗 → (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑏)) = (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑗)))
1716eleq1d 2835 . . . . . . . 8 (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑗)) ∈ 𝑝))
18 fveq1 6331 . . . . . . . . . 10 (𝑐 = 𝑥 → (𝑐𝑗) = (𝑥𝑗))
1918cbvmptv 4884 . . . . . . . . 9 (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑗)) = (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗))
2019eleq1i 2841 . . . . . . . 8 ((𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑗)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝)
2117, 20syl6bb 276 . . . . . . 7 (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝))
2221cbvralv 3320 . . . . . 6 (∀𝑏𝑉 (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝)
2314, 22syl6bb 276 . . . . 5 (𝑣 = 𝑉 → (∀𝑏𝑣 (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝))
2411, 23anbi12d 616 . . . 4 (𝑣 = 𝑉 → ((∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ∧ ∀𝑏𝑣 (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝) ↔ (∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝)))
2524anbi1d 615 . . 3 (𝑣 = 𝑉 → (((∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ∧ ∀𝑏𝑣 (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝) ∧ ∀𝑓𝑝𝑔𝑝 ((𝑓𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑝)) ↔ ((∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝) ∧ ∀𝑓𝑝𝑔𝑝 ((𝑓𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑝))))
263, 25rabeqbidv 3345 . 2 (𝑣 = 𝑉 → {𝑝 ∈ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∣ ((∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ∧ ∀𝑏𝑣 (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝) ∧ ∀𝑓𝑝𝑔𝑝 ((𝑓𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑝))} = {𝑝 ∈ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝) ∧ ∀𝑓𝑝𝑔𝑝 ((𝑓𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑝))})
27 df-mzpcl 37812 . 2 mzPolyCld = (𝑣 ∈ V ↦ {𝑝 ∈ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∣ ((∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ∧ ∀𝑏𝑣 (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝) ∧ ∀𝑓𝑝𝑔𝑝 ((𝑓𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑝))})
28 ovex 6823 . . . 4 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∈ V
2928pwex 4981 . . 3 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∈ V
3029rabex 4946 . 2 {𝑝 ∈ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝) ∧ ∀𝑓𝑝𝑔𝑝 ((𝑓𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑝))} ∈ V
3126, 27, 30fvmpt 6424 1 (𝑉 ∈ V → (mzPolyCld‘𝑉) = {𝑝 ∈ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝) ∧ ∀𝑓𝑝𝑔𝑝 ((𝑓𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑝))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351  𝒫 cpw 4297  {csn 4316  cmpt 4863   × cxp 5247  cfv 6031  (class class class)co 6793  𝑓 cof 7042  𝑚 cmap 8009   + caddc 10141   · cmul 10143  cz 11579  mzPolyCldcmzpcl 37810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-mzpcl 37812
This theorem is referenced by:  elmzpcl  37815
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