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Theorem mzpclval 41778
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 7420 . . . . 5 (𝑣 = 𝑉 → (ℤ ↑m 𝑣) = (ℤ ↑m 𝑉))
21oveq2d 7428 . . . 4 (𝑣 = 𝑉 → (ℤ ↑m (ℤ ↑m 𝑣)) = (ℤ ↑m (ℤ ↑m 𝑉)))
32pweqd 4619 . . 3 (𝑣 = 𝑉 → 𝒫 (ℤ ↑m (ℤ ↑m 𝑣)) = 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)))
41xpeq1d 5705 . . . . . . . 8 (𝑣 = 𝑉 → ((ℤ ↑m 𝑣) × {𝑎}) = ((ℤ ↑m 𝑉) × {𝑎}))
54eleq1d 2817 . . . . . . 7 (𝑣 = 𝑉 → (((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝))
65ralbidv 3176 . . . . . 6 (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑎 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝))
7 sneq 4638 . . . . . . . . 9 (𝑎 = 𝑖 → {𝑎} = {𝑖})
87xpeq2d 5706 . . . . . . . 8 (𝑎 = 𝑖 → ((ℤ ↑m 𝑉) × {𝑎}) = ((ℤ ↑m 𝑉) × {𝑖}))
98eleq1d 2817 . . . . . . 7 (𝑎 = 𝑖 → (((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝))
109cbvralvw 3233 . . . . . 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 5243 . . . . . . . 8 (𝑣 = 𝑉 → (𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐𝑏)) = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)))
1312eleq1d 2817 . . . . . . 7 (𝑣 = 𝑉 → ((𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝))
1413raleqbi1dv 3332 . . . . . 6 (𝑣 = 𝑉 → (∀𝑏𝑣 (𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ ∀𝑏𝑉 (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝))
15 fveq2 6891 . . . . . . . . . 10 (𝑏 = 𝑗 → (𝑐𝑏) = (𝑐𝑗))
1615mpteq2dv 5250 . . . . . . . . 9 (𝑏 = 𝑗 → (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑗)))
1716eleq1d 2817 . . . . . . . 8 (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑗)) ∈ 𝑝))
18 fveq1 6890 . . . . . . . . . 10 (𝑐 = 𝑥 → (𝑐𝑗) = (𝑥𝑗))
1918cbvmptv 5261 . . . . . . . . 9 (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐𝑗)) = (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥𝑗))
2019eleq1i 2823 . . . . . . . 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 3233 . . . . . 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 630 . . . 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 629 . . 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 3448 . 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 41776 . 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 7445 . . . 4 (ℤ ↑m (ℤ ↑m 𝑉)) ∈ V
2928pwex 5378 . . 3 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)) ∈ V
3029rabex 5332 . 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 6998 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 2105  wral 3060  {crab 3431  Vcvv 3473  𝒫 cpw 4602  {csn 4628  cmpt 5231   × cxp 5674  cfv 6543  (class class class)co 7412  f cof 7672  m cmap 8826   + caddc 11119   · cmul 11121  cz 12565  mzPolyCldcmzpcl 41774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-mzpcl 41776
This theorem is referenced by:  elmzpcl  41779
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