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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppcnlem5 | Structured version Visualization version GIF version |
Description: Lemma for knoppcn 31664. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
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) |
Ref | Expression |
---|---|
knoppcnlem5 | ⊢ (𝜑 → (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))):ℕ0⟶(ℂ ↑𝑚 ℝ)) |
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 758 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑧 ∈ ℝ) → 𝑁 ∈ ℕ) |
5 | knoppcnlem5.1 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
6 | 5 | ad2antrr 758 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑧 ∈ ℝ) → 𝐶 ∈ ℝ) |
7 | simpr 476 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ) | |
8 | simplr 788 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑧 ∈ ℝ) → 𝑚 ∈ ℕ0) | |
9 | 1, 2, 4, 6, 7, 8 | knoppcnlem3 31655 | . . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧)‘𝑚) ∈ ℝ) |
10 | 9 | recnd 9947 | . . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑧 ∈ ℝ) → ((𝐹‘𝑧)‘𝑚) ∈ ℂ) |
11 | eqid 2610 | . . . 4 ⊢ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)) = (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)) | |
12 | 10, 11 | fmptd 6292 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)):ℝ⟶ℂ) |
13 | cnex 9896 | . . . . 5 ⊢ ℂ ∈ V | |
14 | reex 9906 | . . . . 5 ⊢ ℝ ∈ V | |
15 | 13, 14 | pm3.2i 470 | . . . 4 ⊢ (ℂ ∈ V ∧ ℝ ∈ V) |
16 | elmapg 7757 | . . . 4 ⊢ ((ℂ ∈ V ∧ ℝ ∈ V) → ((𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)) ∈ (ℂ ↑𝑚 ℝ) ↔ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)):ℝ⟶ℂ)) | |
17 | 15, 16 | ax-mp 5 | . . 3 ⊢ ((𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)) ∈ (ℂ ↑𝑚 ℝ) ↔ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)):ℝ⟶ℂ) |
18 | 12, 17 | sylibr 223 | . 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)) ∈ (ℂ ↑𝑚 ℝ)) |
19 | eqid 2610 | . 2 ⊢ (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))) = (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))) | |
20 | 18, 19 | fmptd 6292 | 1 ⊢ (𝜑 → (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))):ℕ0⟶(ℂ ↑𝑚 ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 ℂcc 9813 ℝcr 9814 1c1 9816 + caddc 9818 · cmul 9820 − cmin 10145 / cdiv 10563 ℕcn 10897 2c2 10947 ℕ0cn0 11169 ⌊cfl 12453 ↑cexp 12722 abscabs 13822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fl 12455 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 |
This theorem is referenced by: knoppcnlem6 31658 |
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