knoppcnlem5 - Mathbox for Asger C. Ipsen

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Theorem knoppcnlem5 34604

Description: Lemma for knoppcn 34611. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)

**Hypotheses**
Ref	Expression
knoppcnlem5.t	⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
knoppcnlem5.f	⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))
knoppcnlem5.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
knoppcnlem5.1	⊢ (𝜑 → 𝐶 ∈ ℝ)

**Assertion**
Ref	Expression
knoppcnlem5	⊢ (𝜑 → (𝑚 ∈ ℕ₀ ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))):ℕ₀⟶(ℂ ↑_m ℝ))

Distinct variable groups: 𝐶,𝑛,𝑦 𝑛,𝑁,𝑦 𝑥,𝑁 𝑇,𝑛,𝑦 𝜑,𝑚,𝑛,𝑦,𝑧 𝑥,𝑚,𝑧 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥,𝑧,𝑚) 𝑇(𝑥,𝑧,𝑚) 𝐹(𝑥,𝑦,𝑧,𝑚,𝑛) 𝑁(𝑧,𝑚)

**Proof of Theorem knoppcnlem5**
Step	Hyp	Ref	Expression
1		knoppcnlem5.t	. . . . . 6 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
2		knoppcnlem5.f	. . . . . 6 ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))
3		knoppcnlem5.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4	3	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → 𝑁 ∈ ℕ)
5		knoppcnlem5.1	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6	5	ad2antrr 722	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → 𝐶 ∈ ℝ)
7		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
8		simplr 765	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → 𝑚 ∈ ℕ₀)
9	1, 2, 4, 6, 7, 8	knoppcnlem3 34602	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧)‘𝑚) ∈ ℝ)
10	9	recnd 10934	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧)‘𝑚) ∈ ℂ)
11	10	fmpttd 6971	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)):ℝ⟶ℂ)
12		cnex 10883	. . . . 5 ⊢ ℂ ∈ V
13		reex 10893	. . . . 5 ⊢ ℝ ∈ V
14	12, 13	pm3.2i 470	. . . 4 ⊢ (ℂ ∈ V ∧ ℝ ∈ V)
15		elmapg 8586	. . . 4 ⊢ ((ℂ ∈ V ∧ ℝ ∈ V) → ((𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)) ∈ (ℂ ↑_m ℝ) ↔ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)):ℝ⟶ℂ))
16	14, 15	ax-mp 5	. . 3 ⊢ ((𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)) ∈ (ℂ ↑_m ℝ) ↔ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)):ℝ⟶ℂ)
17	11, 16	sylibr 233	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)) ∈ (ℂ ↑_m ℝ))
18	17	fmpttd 6971	1 ⊢ (𝜑 → (𝑚 ∈ ℕ₀ ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))):ℕ₀⟶(ℂ ↑_m ℝ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 Vcvv 3422 ↦ cmpt 5153 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 ℂcc 10800 ℝcr 10801 1c1 10803 + caddc 10805 · cmul 10807 − cmin 11135 / cdiv 11562 ℕcn 11903 2c2 11958 ℕ₀cn0 12163 ⌊cfl 13438 ↑cexp 13710 abscabs 14873

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-fl 13440 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875

This theorem is referenced by: knoppcnlem6 34605

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