knoppcnlem5 - Mathbox for Asger C. Ipsen

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Theorem knoppcnlem5 36716

Description: Lemma for knoppcn 36723. (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 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → 𝑁 ∈ ℕ)
5		knoppcnlem5.1	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6	5	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → 𝐶 ∈ ℝ)
7		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
8		simplr 769	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → 𝑚 ∈ ℕ₀)
9	1, 2, 4, 6, 7, 8	knoppcnlem3 36714	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧)‘𝑚) ∈ ℝ)
10	9	recnd 11172	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧)‘𝑚) ∈ ℂ)
11	10	fmpttd 7069	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)):ℝ⟶ℂ)
12		cnex 11119	. . . . 5 ⊢ ℂ ∈ V
13		reex 11129	. . . . 5 ⊢ ℝ ∈ V
14	12, 13	pm3.2i 470	. . . 4 ⊢ (ℂ ∈ V ∧ ℝ ∈ V)
15		elmapg 8788	. . . 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 7069	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 1542 ∈ wcel 2114 Vcvv 3442 ↦ cmpt 5181 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ↑_m cmap 8775 ℂcc 11036 ℝcr 11037 1c1 11039 + caddc 11041 · cmul 11043 − cmin 11376 / cdiv 11806 ℕcn 12157 2c2 12212 ℕ₀cn0 12413 ⌊cfl 13722 ↑cexp 13996 abscabs 15169

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-z 12501 df-uz 12764 df-rp 12918 df-fl 13724 df-seq 13937 df-exp 13997 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171

This theorem is referenced by: knoppcnlem6 36717

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