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Theorem mzpclval 36306
 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 6557 . . . . 5 (𝑣 = 𝑉 → (ℤ ↑𝑚 𝑣) = (ℤ ↑𝑚 𝑉))
21oveq2d 6565 . . . 4 (𝑣 = 𝑉 → (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) = (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)))
32pweqd 4113 . . 3 (𝑣 = 𝑉 → 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) = 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)))
41xpeq1d 5062 . . . . . . . 8 (𝑣 = 𝑉 → ((ℤ ↑𝑚 𝑣) × {𝑎}) = ((ℤ ↑𝑚 𝑉) × {𝑎}))
54eleq1d 2672 . . . . . . 7 (𝑣 = 𝑉 → (((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝))
65ralbidv 2969 . . . . . 6 (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝))
7 sneq 4135 . . . . . . . . 9 (𝑎 = 𝑖 → {𝑎} = {𝑖})
87xpeq2d 5063 . . . . . . . 8 (𝑎 = 𝑖 → ((ℤ ↑𝑚 𝑉) × {𝑎}) = ((ℤ ↑𝑚 𝑉) × {𝑖}))
98eleq1d 2672 . . . . . . 7 (𝑎 = 𝑖 → (((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝))
109cbvralv 3147 . . . . . 6 (∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝)
116, 10syl6bb 275 . . . . 5 (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝))
121mpteq1d 4666 . . . . . . . 8 (𝑣 = 𝑉 → (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐𝑏)) = (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑏)))
1312eleq1d 2672 . . . . . . 7 (𝑣 = 𝑉 → ((𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝))
1413raleqbi1dv 3123 . . . . . 6 (𝑣 = 𝑉 → (∀𝑏𝑣 (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ ∀𝑏𝑉 (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝))
15 fveq2 6103 . . . . . . . . . 10 (𝑏 = 𝑗 → (𝑐𝑏) = (𝑐𝑗))
1615mpteq2dv 4673 . . . . . . . . 9 (𝑏 = 𝑗 → (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑏)) = (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑗)))
1716eleq1d 2672 . . . . . . . 8 (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑗)) ∈ 𝑝))
18 fveq1 6102 . . . . . . . . . 10 (𝑐 = 𝑥 → (𝑐𝑗) = (𝑥𝑗))
1918cbvmptv 4678 . . . . . . . . 9 (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑗)) = (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗))
2019eleq1i 2679 . . . . . . . 8 ((𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑗)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝)
2117, 20syl6bb 275 . . . . . . 7 (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝))
2221cbvralv 3147 . . . . . 6 (∀𝑏𝑉 (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝)
2314, 22syl6bb 275 . . . . 5 (𝑣 = 𝑉 → (∀𝑏𝑣 (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝))
2411, 23anbi12d 743 . . . 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 737 . . 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 3168 . 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 36304 . 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 6577 . . . 4 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∈ V
2928pwex 4774 . . 3 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∈ V
3029rabex 4740 . 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 6191 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 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  𝒫 cpw 4108  {csn 4125   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793   ↑𝑚 cmap 7744   + caddc 9818   · cmul 9820  ℤcz 11254  mzPolyCldcmzpcl 36302 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-mzpcl 36304 This theorem is referenced by:  elmzpcl  36307
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