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Theorem mzpclval 42712
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 7438 . . . . 5 (𝑣 = 𝑉 → (ℤ ↑m 𝑣) = (ℤ ↑m 𝑉))
21oveq2d 7446 . . . 4 (𝑣 = 𝑉 → (ℤ ↑m (ℤ ↑m 𝑣)) = (ℤ ↑m (ℤ ↑m 𝑉)))
32pweqd 4621 . . 3 (𝑣 = 𝑉 → 𝒫 (ℤ ↑m (ℤ ↑m 𝑣)) = 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)))
41xpeq1d 5717 . . . . . . . 8 (𝑣 = 𝑉 → ((ℤ ↑m 𝑣) × {𝑎}) = ((ℤ ↑m 𝑉) × {𝑎}))
54eleq1d 2823 . . . . . . 7 (𝑣 = 𝑉 → (((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝))
65ralbidv 3175 . . . . . 6 (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑎 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝))
7 sneq 4640 . . . . . . . . 9 (𝑎 = 𝑖 → {𝑎} = {𝑖})
87xpeq2d 5718 . . . . . . . 8 (𝑎 = 𝑖 → ((ℤ ↑m 𝑉) × {𝑎}) = ((ℤ ↑m 𝑉) × {𝑖}))
98eleq1d 2823 . . . . . . 7 (𝑎 = 𝑖 → (((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝))
109cbvralvw 3234 . . . . . 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 5242 . . . . . . . 8 (𝑣 = 𝑉 → (𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐𝑏)) = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)))
1312eleq1d 2823 . . . . . . 7 (𝑣 = 𝑉 → ((𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝))
1413raleqbi1dv 3335 . . . . . 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 5249 . . . . . . . . 9 (𝑏 = 𝑗 → (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑗)))
1716eleq1d 2823 . . . . . . . 8 (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑗)) ∈ 𝑝))
18 fveq1 6905 . . . . . . . . . 10 (𝑐 = 𝑥 → (𝑐𝑗) = (𝑥𝑗))
1918cbvmptv 5260 . . . . . . . . 9 (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑗)) = (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥𝑗))
2019eleq1i 2829 . . . . . . . 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 3234 . . . . . 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 3451 . 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 42710 . 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 7463 . . . 4 (ℤ ↑m (ℤ ↑m 𝑉)) ∈ V
2928pwex 5385 . . 3 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)) ∈ V
3029rabex 5344 . 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 7015 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 1536  wcel 2105  wral 3058  {crab 3432  Vcvv 3477  𝒫 cpw 4604  {csn 4630  cmpt 5230   × cxp 5686  cfv 6562  (class class class)co 7430  f cof 7694  m cmap 8864   + caddc 11155   · cmul 11157  cz 12610  mzPolyCldcmzpcl 42708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-mzpcl 42710
This theorem is referenced by:  elmzpcl  42713
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