knoppcnlem5 - Mathbox for Asger C. Ipsen

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Theorem knoppcnlem5 36613

Description: Lemma for knoppcn 36620. (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 36611	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧)‘𝑚) ∈ ℝ)
10	9	recnd 11151	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧)‘𝑚) ∈ ℂ)
11	10	fmpttd 7057	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)):ℝ⟶ℂ)
12		cnex 11098	. . . . 5 ⊢ ℂ ∈ V
13		reex 11108	. . . . 5 ⊢ ℝ ∈ V
14	12, 13	pm3.2i 470	. . . 4 ⊢ (ℂ ∈ V ∧ ℝ ∈ V)
15		elmapg 8772	. . . 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 7057	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 1541 ∈ wcel 2113 Vcvv 3437 ↦ cmpt 5176 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 ↑_m cmap 8759 ℂcc 11015 ℝcr 11016 1c1 11018 + caddc 11020 · cmul 11022 − cmin 11355 / cdiv 11785 ℕcn 12136 2c2 12191 ℕ₀cn0 12392 ⌊cfl 13701 ↑cexp 13975 abscabs 15148

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-er 8631 df-map 8761 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9337 df-inf 9338 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-n0 12393 df-z 12480 df-uz 12743 df-rp 12897 df-fl 13703 df-seq 13916 df-exp 13976 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150

This theorem is referenced by: knoppcnlem6 36614

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