knoppcn2 - Mathbox for Asger C. Ipsen

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Theorem knoppcn2 36652

Description: Variant of knoppcn 36620 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 36651	. 2 ⊢ (𝜑 → 𝑊:ℝ⟶ℝ)
7		ax-resscn 11074	. . . . 5 ⊢ ℝ ⊆ ℂ
8	7	a1i 11	. . . 4 ⊢ (𝜑 → ℝ ⊆ ℂ)
9	4	knoppndvlem3 36630	. . . . . 6 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
10	9	simpld 494	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ)
11	9	simprd 495	. . . . 5 ⊢ (𝜑 → (abs‘𝐶) < 1)
12	1, 2, 3, 5, 10, 11	knoppcn 36620	. . . 4 ⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℂ))
13	8, 12	jca 511	. . 3 ⊢ (𝜑 → (ℝ ⊆ ℂ ∧ 𝑊 ∈ (ℝ–cn→ℂ)))
14		cncfcdm 24838	. . 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 257	1 ⊢ (𝜑 → 𝑊 ∈ (ℝ–cn→ℝ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ⊆ wss 3898 class class class wbr 5095 ↦ cmpt 5176 ⟶wf 6485 ‘cfv 6489 (class class class)co 7355 ℂcc 11015 ℝcr 11016 1c1 11018 + caddc 11020 · cmul 11022 < clt 11157 − cmin 11355 -cneg 11356 / cdiv 11785 ℕcn 12136 2c2 12191 ℕ₀cn0 12392 (,)cioo 13252 ⌊cfl 13701 ↑cexp 13975 abscabs 15148 Σcsu 15600 –cn→ccncf 24816

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-inf2 9542 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 ax-addf 11096

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-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 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-se 5575 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-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-map 8761 df-pm 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9257 df-fi 9306 df-sup 9337 df-inf 9338 df-oi 9407 df-card 9843 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-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-z 12480 df-dec 12599 df-uz 12743 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ioo 13256 df-ico 13258 df-icc 13259 df-fz 13415 df-fzo 13562 df-fl 13703 df-seq 13916 df-exp 13976 df-hash 14245 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-limsup 15385 df-clim 15402 df-rlim 15403 df-sum 15601 df-struct 17065 df-sets 17082 df-slot 17100 df-ndx 17112 df-base 17128 df-ress 17149 df-plusg 17181 df-mulr 17182 df-starv 17183 df-sca 17184 df-vsca 17185 df-ip 17186 df-tset 17187 df-ple 17188 df-ds 17190 df-unif 17191 df-hom 17192 df-cco 17193 df-rest 17333 df-topn 17334 df-0g 17352 df-gsum 17353 df-topgen 17354 df-pt 17355 df-prds 17358 df-xrs 17414 df-qtop 17419 df-imas 17420 df-xps 17422 df-mre 17496 df-mrc 17497 df-acs 17499 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-submnd 18700 df-mulg 18989 df-cntz 19237 df-cmn 19702 df-psmet 21292 df-xmet 21293 df-met 21294 df-bl 21295 df-mopn 21296 df-cnfld 21301 df-top 22829 df-topon 22846 df-topsp 22868 df-bases 22881 df-cn 23162 df-cnp 23163 df-tx 23497 df-hmeo 23690 df-xms 24255 df-ms 24256 df-tms 24257 df-cncf 24818 df-ulm 26333

This theorem is referenced by: cnndvlem1 36653

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