Theorem elmzpcl 36307

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 36306	. . 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 2673	. 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 2677	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ↔ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑃))
4	3	ralbidv 2969	. . . . . 6 ⊢ (𝑝 = 𝑃 → (∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑃))
5		eleq2 2677	. . . . . . 7 ⊢ (𝑝 = 𝑃 → ((𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑃))
6	5	ralbidv 2969	. . . . . 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 743	. . . . 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 2677	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑓 ∘𝑓 + 𝑔) ∈ 𝑝 ↔ (𝑓 ∘𝑓 + 𝑔) ∈ 𝑃))
9		eleq2 2677	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑓 ∘𝑓 · 𝑔) ∈ 𝑝 ↔ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑃))
10	8, 9	anbi12d 743	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (((𝑓 ∘𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑝) ↔ ((𝑓 ∘𝑓 + 𝑔) ∈ 𝑃 ∧ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑃)))
11	10	raleqbi1dv 3123	. . . . . 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 3123	. . . . 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 743	. . . 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 3331	. . 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 6577	. . . . 5 ⊢ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∈ V
16	15	elpw2 4755	. . . 4 ⊢ (𝑃 ∈ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ↔ 𝑃 ⊆ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)))
17	16	anbi1i 727	. . 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 263	. 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	syl6bb 275	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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793   ↑_𝑚 cmap 7744   + caddc 9818   · cmul 9820  ℤcz 11254  mzPolyCldcmzpcl 36302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-mzpcl 36304

This theorem is referenced by:  mzpclall  36308  mzpcl1  36310  mzpcl2  36311  mzpcl34  36312  mzpincl  36315  mzpindd  36327