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Theorem elmzpcl 38132
Description: Double substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Assertion
Ref Expression
elmzpcl (𝑉 ∈ V → (𝑃 ∈ (mzPolyCld‘𝑉) ↔ (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:   𝑓,𝑉,𝑔   𝑖,𝑉   𝑗,𝑉,𝑥   𝑃,𝑓,𝑔   𝑃,𝑖   𝑃,𝑗,𝑥

Proof of Theorem elmzpcl
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 mzpclval 38131 . . 3 (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))})
21eleq2d 2892 . 2 (𝑉 ∈ V → (𝑃 ∈ (mzPolyCld‘𝑉) ↔ 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))}))
3 eleq2 2895 . . . . . . 7 (𝑝 = 𝑃 → (((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ↔ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑃))
43ralbidv 3195 . . . . . 6 (𝑝 = 𝑃 → (∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑃))
5 eleq2 2895 . . . . . . 7 (𝑝 = 𝑃 → ((𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑃))
65ralbidv 3195 . . . . . 6 (𝑝 = 𝑃 → (∀𝑗𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝 ↔ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑃))
74, 6anbi12d 624 . . . . 5 (𝑝 = 𝑃 → ((∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝) ↔ (∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑃 ∧ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑃)))
8 eleq2 2895 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑓𝑓 + 𝑔) ∈ 𝑝 ↔ (𝑓𝑓 + 𝑔) ∈ 𝑃))
9 eleq2 2895 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑓𝑓 · 𝑔) ∈ 𝑝 ↔ (𝑓𝑓 · 𝑔) ∈ 𝑃))
108, 9anbi12d 624 . . . . . . 7 (𝑝 = 𝑃 → (((𝑓𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑝) ↔ ((𝑓𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑃)))
1110raleqbi1dv 3358 . . . . . 6 (𝑝 = 𝑃 → (∀𝑔𝑝 ((𝑓𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑝) ↔ ∀𝑔𝑃 ((𝑓𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑃)))
1211raleqbi1dv 3358 . . . . 5 (𝑝 = 𝑃 → (∀𝑓𝑝𝑔𝑝 ((𝑓𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑝) ↔ ∀𝑓𝑃𝑔𝑃 ((𝑓𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑃)))
137, 12anbi12d 624 . . . 4 (𝑝 = 𝑃 → (((∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 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))))
1413elrab 3585 . . 3 (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))} ↔ (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))))
15 ovex 6942 . . . . 5 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∈ V
1615elpw2 5052 . . . 4 (𝑃 ∈ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ↔ 𝑃 ⊆ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)))
1716anbi1i 617 . . 3 ((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))) ↔ (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))))
1814, 17bitri 267 . 2 (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))} ↔ (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))))
192, 18syl6bb 279 1 (𝑉 ∈ V → (𝑃 ∈ (mzPolyCld‘𝑉) ↔ (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  wb 198  wa 386   = wceq 1656  wcel 2164  wral 3117  {crab 3121  Vcvv 3414  wss 3798  𝒫 cpw 4380  {csn 4399  cmpt 4954   × cxp 5344  cfv 6127  (class class class)co 6910  𝑓 cof 7160  𝑚 cmap 8127   + caddc 10262   · cmul 10264  cz 11711  mzPolyCldcmzpcl 38127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fv 6135  df-ov 6913  df-mzpcl 38129
This theorem is referenced by:  mzpclall  38133  mzpcl1  38135  mzpcl2  38136  mzpcl34  38137  mzpincl  38140  mzpindd  38152
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