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Theorem elmzpcl 37809
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 37808 . . 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 2825 . 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 2828 . . . . . . 7 (𝑝 = 𝑃 → (((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ↔ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑃))
43ralbidv 3124 . . . . . 6 (𝑝 = 𝑃 → (∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑃))
5 eleq2 2828 . . . . . . 7 (𝑝 = 𝑃 → ((𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑃))
65ralbidv 3124 . . . . . 6 (𝑝 = 𝑃 → (∀𝑗𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑝 ↔ ∀𝑗𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥𝑗)) ∈ 𝑃))
74, 6anbi12d 749 . . . . 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 2828 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑓𝑓 + 𝑔) ∈ 𝑝 ↔ (𝑓𝑓 + 𝑔) ∈ 𝑃))
9 eleq2 2828 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑓𝑓 · 𝑔) ∈ 𝑝 ↔ (𝑓𝑓 · 𝑔) ∈ 𝑃))
108, 9anbi12d 749 . . . . . . 7 (𝑝 = 𝑃 → (((𝑓𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑝) ↔ ((𝑓𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑃)))
1110raleqbi1dv 3285 . . . . . 6 (𝑝 = 𝑃 → (∀𝑔𝑝 ((𝑓𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑝) ↔ ∀𝑔𝑃 ((𝑓𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑃)))
1211raleqbi1dv 3285 . . . . 5 (𝑝 = 𝑃 → (∀𝑓𝑝𝑔𝑝 ((𝑓𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑝) ↔ ∀𝑓𝑃𝑔𝑃 ((𝑓𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓𝑓 · 𝑔) ∈ 𝑃)))
137, 12anbi12d 749 . . . 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 3504 . . 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 6842 . . . . 5 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∈ V
1615elpw2 4977 . . . 4 (𝑃 ∈ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ↔ 𝑃 ⊆ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)))
1716anbi1i 733 . . 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 264 . 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 276 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 196  wa 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054  Vcvv 3340  wss 3715  𝒫 cpw 4302  {csn 4321  cmpt 4881   × cxp 5264  cfv 6049  (class class class)co 6814  𝑓 cof 7061  𝑚 cmap 8025   + caddc 10151   · cmul 10153  cz 11589  mzPolyCldcmzpcl 37804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-mzpcl 37806
This theorem is referenced by:  mzpclall  37810  mzpcl1  37812  mzpcl2  37813  mzpcl34  37814  mzpincl  37817  mzpindd  37829
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