knoppcnlem5 - Mathbox for Asger C. Ipsen

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Theorem knoppcnlem5 36492

Description: Lemma for knoppcn 36499. (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 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → 𝑁 ∈ ℕ)
5		knoppcnlem5.1	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6	5	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → 𝐶 ∈ ℝ)
7		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
8		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → 𝑚 ∈ ℕ₀)
9	1, 2, 4, 6, 7, 8	knoppcnlem3 36490	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧)‘𝑚) ∈ ℝ)
10	9	recnd 11209	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧)‘𝑚) ∈ ℂ)
11	10	fmpttd 7090	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)):ℝ⟶ℂ)
12		cnex 11156	. . . . 5 ⊢ ℂ ∈ V
13		reex 11166	. . . . 5 ⊢ ℝ ∈ V
14	12, 13	pm3.2i 470	. . . 4 ⊢ (ℂ ∈ V ∧ ℝ ∈ V)
15		elmapg 8815	. . . 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 234	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)) ∈ (ℂ ↑_m ℝ))
18	17	fmpttd 7090	1 ⊢ (𝜑 → (𝑚 ∈ ℕ₀ ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))):ℕ₀⟶(ℂ ↑_m ℝ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3450 ↦ cmpt 5191 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ↑_m cmap 8802 ℂcc 11073 ℝcr 11074 1c1 11076 + caddc 11078 · cmul 11080 − cmin 11412 / cdiv 11842 ℕcn 12193 2c2 12248 ℕ₀cn0 12449 ⌊cfl 13759 ↑cexp 14033 abscabs 15207

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-rp 12959 df-fl 13761 df-seq 13974 df-exp 14034 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209

This theorem is referenced by: knoppcnlem6 36493

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