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Theorem mzpclval 42736
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 7439 . . . . 5 (𝑣 = 𝑉 → (ℤ ↑m 𝑣) = (ℤ ↑m 𝑉))
21oveq2d 7447 . . . 4 (𝑣 = 𝑉 → (ℤ ↑m (ℤ ↑m 𝑣)) = (ℤ ↑m (ℤ ↑m 𝑉)))
32pweqd 4617 . . 3 (𝑣 = 𝑉 → 𝒫 (ℤ ↑m (ℤ ↑m 𝑣)) = 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)))
41xpeq1d 5714 . . . . . . . 8 (𝑣 = 𝑉 → ((ℤ ↑m 𝑣) × {𝑎}) = ((ℤ ↑m 𝑉) × {𝑎}))
54eleq1d 2826 . . . . . . 7 (𝑣 = 𝑉 → (((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝))
65ralbidv 3178 . . . . . 6 (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑎 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝))
7 sneq 4636 . . . . . . . . 9 (𝑎 = 𝑖 → {𝑎} = {𝑖})
87xpeq2d 5715 . . . . . . . 8 (𝑎 = 𝑖 → ((ℤ ↑m 𝑉) × {𝑎}) = ((ℤ ↑m 𝑉) × {𝑖}))
98eleq1d 2826 . . . . . . 7 (𝑎 = 𝑖 → (((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝))
109cbvralvw 3237 . . . . . 6 (∀𝑎 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝)
116, 10bitrdi 287 . . . . 5 (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝))
121mpteq1d 5237 . . . . . . . 8 (𝑣 = 𝑉 → (𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐𝑏)) = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)))
1312eleq1d 2826 . . . . . . 7 (𝑣 = 𝑉 → ((𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝))
1413raleqbi1dv 3338 . . . . . 6 (𝑣 = 𝑉 → (∀𝑏𝑣 (𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ ∀𝑏𝑉 (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝))
15 fveq2 6906 . . . . . . . . . 10 (𝑏 = 𝑗 → (𝑐𝑏) = (𝑐𝑗))
1615mpteq2dv 5244 . . . . . . . . 9 (𝑏 = 𝑗 → (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑗)))
1716eleq1d 2826 . . . . . . . 8 (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑗)) ∈ 𝑝))
18 fveq1 6905 . . . . . . . . . 10 (𝑐 = 𝑥 → (𝑐𝑗) = (𝑥𝑗))
1918cbvmptv 5255 . . . . . . . . 9 (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑗)) = (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥𝑗))
2019eleq1i 2832 . . . . . . . 8 ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑗)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝)
2117, 20bitrdi 287 . . . . . . 7 (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝))
2221cbvralvw 3237 . . . . . 6 (∀𝑏𝑉 (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝)
2314, 22bitrdi 287 . . . . 5 (𝑣 = 𝑉 → (∀𝑏𝑣 (𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝))
2411, 23anbi12d 632 . . . 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 631 . . 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 3455 . 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 42734 . 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 7464 . . . 4 (ℤ ↑m (ℤ ↑m 𝑉)) ∈ V
2928pwex 5380 . . 3 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)) ∈ V
3029rabex 5339 . 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 7016 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 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  𝒫 cpw 4600  {csn 4626  cmpt 5225   × cxp 5683  cfv 6561  (class class class)co 7431  f cof 7695  m cmap 8866   + caddc 11158   · cmul 11160  cz 12613  mzPolyCldcmzpcl 42732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-mzpcl 42734
This theorem is referenced by:  elmzpcl  42737
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