Theorem elmzpcl 42713

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 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 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.

Step	Hyp	Ref	Expression
1		mzpclval 42712	. . 3 ⊢ (𝑉 ∈ V → (mzPolyCld‘𝑉) = {𝑝 ∈ 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘f + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑝))})
2	1	eleq2d 2824	. 2 ⊢ (𝑉 ∈ V → (𝑃 ∈ (mzPolyCld‘𝑉) ↔ 𝑃 ∈ {𝑝 ∈ 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘f + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑝))}))
3		eleq2 2827	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝 ↔ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑃))
4	3	ralbidv 3175	. . . . . 6 ⊢ (𝑝 = 𝑃 → (∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑃))
5		eleq2 2827	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑃))
6	5	ralbidv 3175	. . . . . 6 ⊢ (𝑝 = 𝑃 → (∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝 ↔ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑃))
7	4, 6	anbi12d 632	. . . . 5 ⊢ (𝑝 = 𝑃 → ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝) ↔ (∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑃 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑃)))
8		eleq2 2827	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑓 ∘f + 𝑔) ∈ 𝑝 ↔ (𝑓 ∘f + 𝑔) ∈ 𝑃))
9		eleq2 2827	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑓 ∘f · 𝑔) ∈ 𝑝 ↔ (𝑓 ∘f · 𝑔) ∈ 𝑃))
10	8, 9	anbi12d 632	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (((𝑓 ∘f + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑝) ↔ ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃)))
11	10	raleqbi1dv 3335	. . . . . 6 ⊢ (𝑝 = 𝑃 → (∀𝑔 ∈ 𝑝 ((𝑓 ∘f + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑝) ↔ ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃)))
12	11	raleqbi1dv 3335	. . . . 5 ⊢ (𝑝 = 𝑃 → (∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘f + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑝) ↔ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃)))
13	7, 12	anbi12d 632	. . . 4 ⊢ (𝑝 = 𝑃 → (((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘f + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑝)) ↔ ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑃 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃))))
14	13	elrab 3694	. . 3 ⊢ (𝑃 ∈ {𝑝 ∈ 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘f + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑝))} ↔ (𝑃 ∈ 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)) ∧ ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑃 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃))))
15		ovex 7463	. . . . 5 ⊢ (ℤ ↑m (ℤ ↑m 𝑉)) ∈ V
16	15	elpw2 5339	. . . 4 ⊢ (𝑃 ∈ 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)) ↔ 𝑃 ⊆ (ℤ ↑m (ℤ ↑m 𝑉)))
17	16	anbi1i 624	. . 3 ⊢ ((𝑃 ∈ 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)) ∧ ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑃 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃))) ↔ (𝑃 ⊆ (ℤ ↑m (ℤ ↑m 𝑉)) ∧ ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑃 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃))))
18	14, 17	bitri 275	. 2 ⊢ (𝑃 ∈ {𝑝 ∈ 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘f + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑝))} ↔ (𝑃 ⊆ (ℤ ↑m (ℤ ↑m 𝑉)) ∧ ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑃 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃))))
19	2, 18	bitrdi 287	1 ⊢ (𝑉 ∈ V → (𝑃 ∈ (mzPolyCld‘𝑉) ↔ (𝑃 ⊆ (ℤ ↑m (ℤ ↑m 𝑉)) ∧ ((∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑃 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑃) ∧ ∀𝑓 ∈ 𝑃 ∀𝑔 ∈ 𝑃 ((𝑓 ∘f + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘f · 𝑔) ∈ 𝑃)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∀wral 3058  {crab 3432  Vcvv 3477   ⊆ wss 3962  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  ℤcz 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:  mzpclall  42714  mzpcl1  42716  mzpcl2  42717  mzpcl34  42718  mzpincl  42721  mzpindd  42733