knoppcn2 - Mathbox for Asger C. Ipsen

	Mathbox for Asger C. Ipsen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppcn2		Structured version Visualization version GIF version

Theorem knoppcn2 34725

Description: Variant of knoppcn 34693 with different codomain. (Contributed by Asger C. Ipsen, 25-Aug-2021.)

**Hypotheses**
Ref	Expression
knoppcn2.t	⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
knoppcn2.f	⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))
knoppcn2.w	⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ₀ ((𝐹‘𝑤)‘𝑖))
knoppcn2.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
knoppcn2.c	⊢ (𝜑 → 𝐶 ∈ (-1(,)1))

**Assertion**
Ref	Expression
knoppcn2	⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℝ))

Distinct variable groups: 𝐶,𝑛,𝑦 𝑖,𝐹,𝑤 𝑛,𝑁,𝑦 𝑥,𝑁 𝑇,𝑛,𝑦 𝜑,𝑖,𝑛,𝑤,𝑦 𝑥,𝑖,𝑤 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥,𝑤,𝑖) 𝑇(𝑥,𝑤,𝑖) 𝐹(𝑥,𝑦,𝑛) 𝑁(𝑤,𝑖) 𝑊(𝑥,𝑦,𝑤,𝑖,𝑛)

**Proof of Theorem knoppcn2**
Step	Hyp	Ref	Expression
1		knoppcn2.t	. . 3 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
2		knoppcn2.f	. . 3 ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))
3		knoppcn2.w	. . 3 ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ₀ ((𝐹‘𝑤)‘𝑖))
4		knoppcn2.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))
5		knoppcn2.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6	1, 2, 3, 4, 5	knoppf 34724	. 2 ⊢ (𝜑 → 𝑊:ℝ⟶ℝ)
7		ax-resscn 10939	. . . . 5 ⊢ ℝ ⊆ ℂ
8	7	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ⊆ ℂ)
9	4	knoppndvlem3 34703	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
10	9	simpld 495	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
11	9	simprd 496	. . . . 5 ⊢ (𝜑 → (abs‘𝐶) < 1)
12	1, 2, 3, 5, 10, 11	knoppcn 34693	. . . 4 ⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℂ))
13	8, 12	jca 512	. . 3 ⊢ (𝜑 → (ℝ ⊆ ℂ ∧ 𝑊 ∈ (ℝ–cn→ℂ)))
14		cncffvrn 24072	. . 3 ⊢ ((ℝ ⊆ ℂ ∧ 𝑊 ∈ (ℝ–cn→ℂ)) → (𝑊 ∈ (ℝ–cn→ℝ) ↔ 𝑊:ℝ⟶ℝ))
15	13, 14	syl 17	. 2 ⊢ (𝜑 → (𝑊 ∈ (ℝ–cn→ℝ) ↔ 𝑊:ℝ⟶ℝ))
16	6, 15	mpbird 256	1 ⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℝ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ⊆ wss 3892 class class class wbr 5079 ↦ cmpt 5162 ⟶wf 6428 ‘cfv 6432 (class class class)co 7272 ℂcc 10880 ℝcr 10881 1c1 10883 + caddc 10885 · cmul 10887 < clt 11020 − cmin 11216 -cneg 11217 / cdiv 11643 ℕcn 11984 2c2 12039 ℕ₀cn0 12244 (,)cioo 13090 ⌊cfl 13521 ↑cexp 13793 abscabs 14956 Σcsu 15408 –cn→ccncf 24050

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-inf2 9387 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-pre-sup 10960 ax-addf 10961 ax-mulf 10962

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-of 7528 df-om 7708 df-1st 7825 df-2nd 7826 df-supp 7970 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-2o 8290 df-er 8490 df-map 8609 df-pm 8610 df-ixp 8678 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-fsupp 9117 df-fi 9158 df-sup 9189 df-inf 9190 df-oi 9257 df-card 9708 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-div 11644 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-7 12052 df-8 12053 df-9 12054 df-n0 12245 df-z 12331 df-dec 12449 df-uz 12594 df-q 12700 df-rp 12742 df-xneg 12859 df-xadd 12860 df-xmul 12861 df-ioo 13094 df-ico 13096 df-icc 13097 df-fz 13251 df-fzo 13394 df-fl 13523 df-seq 13733 df-exp 13794 df-hash 14056 df-cj 14821 df-re 14822 df-im 14823 df-sqrt 14957 df-abs 14958 df-limsup 15191 df-clim 15208 df-rlim 15209 df-sum 15409 df-struct 16859 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-ress 16953 df-plusg 16986 df-mulr 16987 df-starv 16988 df-sca 16989 df-vsca 16990 df-ip 16991 df-tset 16992 df-ple 16993 df-ds 16995 df-unif 16996 df-hom 16997 df-cco 16998 df-rest 17144 df-topn 17145 df-0g 17163 df-gsum 17164 df-topgen 17165 df-pt 17166 df-prds 17169 df-xrs 17224 df-qtop 17229 df-imas 17230 df-xps 17232 df-mre 17306 df-mrc 17307 df-acs 17309 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-submnd 18442 df-mulg 18712 df-cntz 18934 df-cmn 19399 df-psmet 20600 df-xmet 20601 df-met 20602 df-bl 20603 df-mopn 20604 df-cnfld 20609 df-top 22054 df-topon 22071 df-topsp 22093 df-bases 22107 df-cn 22389 df-cnp 22390 df-tx 22724 df-hmeo 22917 df-xms 23484 df-ms 23485 df-tms 23486 df-cncf 24052 df-ulm 25547

This theorem is referenced by: cnndvlem1 34726

W3C validator