knoppcnlem5 - Mathbox for Asger C. Ipsen

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Theorem knoppcnlem5 36896

Description: Lemma for knoppcn 36903. (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 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → 𝑁 ∈ ℕ)
5		knoppcnlem5.1	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
6	5	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → 𝐶 ∈ ℝ)
7		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
8		simplr 778	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → 𝑚 ∈ ℕ₀)
9	1, 2, 4, 6, 7, 8	knoppcnlem3 36894	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧)‘𝑚) ∈ ℝ)
10	9	recnd 11204	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ₀) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧)‘𝑚) ∈ ℂ)
11	10	fmpttd 7091	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)):ℝ⟶ℂ)
12		cnex 11148	. . . . 5 ⊢ ℂ ∈ V
13		reex 11158	. . . . 5 ⊢ ℝ ∈ V
14	12, 13	pm3.2i 474	. . . 4 ⊢ (ℂ ∈ V ∧ ℝ ∈ V)
15		elmapg 8814	. . . 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 236	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ₀) → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)) ∈ (ℂ ↑_m ℝ))
18	17	fmpttd 7091	1 ⊢ (𝜑 → (𝑚 ∈ ℕ₀ ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))):ℕ₀⟶(ℂ ↑_m ℝ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 Vcvv 3453 ↦ cmpt 5178 ⟶wf 6512 ‘cfv 6516 (class class class)co 7391 ↑_m cmap 8802 ℂcc 11065 ℝcr 11066 1c1 11068 + caddc 11070 · cmul 11072 − cmin 11408 / cdiv 11838 ℕcn 12204 2c2 12266 ℕ₀cn0 12475 ⌊cfl 13794 ↑cexp 14068 abscabs 15252

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-er 8672 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-sup 9382 df-inf 9383 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-n0 12476 df-z 12563 df-uz 12834 df-rp 12988 df-fl 13796 df-seq 14009 df-exp 14069 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254

This theorem is referenced by: knoppcnlem6 36897

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