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Theorem mzpclval 39661
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 7143 . . . . 5 (𝑣 = 𝑉 → (ℤ ↑m 𝑣) = (ℤ ↑m 𝑉))
21oveq2d 7151 . . . 4 (𝑣 = 𝑉 → (ℤ ↑m (ℤ ↑m 𝑣)) = (ℤ ↑m (ℤ ↑m 𝑉)))
32pweqd 4516 . . 3 (𝑣 = 𝑉 → 𝒫 (ℤ ↑m (ℤ ↑m 𝑣)) = 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)))
41xpeq1d 5548 . . . . . . . 8 (𝑣 = 𝑉 → ((ℤ ↑m 𝑣) × {𝑎}) = ((ℤ ↑m 𝑉) × {𝑎}))
54eleq1d 2874 . . . . . . 7 (𝑣 = 𝑉 → (((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝))
65ralbidv 3162 . . . . . 6 (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑎 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝))
7 sneq 4535 . . . . . . . . 9 (𝑎 = 𝑖 → {𝑎} = {𝑖})
87xpeq2d 5549 . . . . . . . 8 (𝑎 = 𝑖 → ((ℤ ↑m 𝑉) × {𝑎}) = ((ℤ ↑m 𝑉) × {𝑖}))
98eleq1d 2874 . . . . . . 7 (𝑎 = 𝑖 → (((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝))
109cbvralvw 3396 . . . . . 6 (∀𝑎 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝)
116, 10syl6bb 290 . . . . 5 (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝))
121mpteq1d 5119 . . . . . . . 8 (𝑣 = 𝑉 → (𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐𝑏)) = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)))
1312eleq1d 2874 . . . . . . 7 (𝑣 = 𝑉 → ((𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝))
1413raleqbi1dv 3356 . . . . . 6 (𝑣 = 𝑉 → (∀𝑏𝑣 (𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ ∀𝑏𝑉 (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝))
15 fveq2 6645 . . . . . . . . . 10 (𝑏 = 𝑗 → (𝑐𝑏) = (𝑐𝑗))
1615mpteq2dv 5126 . . . . . . . . 9 (𝑏 = 𝑗 → (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑗)))
1716eleq1d 2874 . . . . . . . 8 (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑗)) ∈ 𝑝))
18 fveq1 6644 . . . . . . . . . 10 (𝑐 = 𝑥 → (𝑐𝑗) = (𝑥𝑗))
1918cbvmptv 5133 . . . . . . . . 9 (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑗)) = (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥𝑗))
2019eleq1i 2880 . . . . . . . 8 ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑗)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝)
2117, 20syl6bb 290 . . . . . . 7 (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝))
2221cbvralvw 3396 . . . . . 6 (∀𝑏𝑉 (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝)
2314, 22syl6bb 290 . . . . 5 (𝑣 = 𝑉 → (∀𝑏𝑣 (𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝))
2411, 23anbi12d 633 . . . 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 632 . . 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 3433 . 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 39659 . 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 7168 . . . 4 (ℤ ↑m (ℤ ↑m 𝑉)) ∈ V
2928pwex 5246 . . 3 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)) ∈ V
3029rabex 5199 . 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 6745 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 399   = wceq 1538  wcel 2111  wral 3106  {crab 3110  Vcvv 3441  𝒫 cpw 4497  {csn 4525  cmpt 5110   × cxp 5517  cfv 6324  (class class class)co 7135  f cof 7387  m cmap 8389   + caddc 10529   · cmul 10531  cz 11969  mzPolyCldcmzpcl 39657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-mzpcl 39659
This theorem is referenced by:  elmzpcl  39662
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