cnndvlem2 - Mathbox for Asger C. Ipsen

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Theorem cnndvlem2 36453

Description: Lemma for cnndv 36454. (Contributed by Asger C. Ipsen, 26-Aug-2021.)

**Hypotheses**
Ref	Expression
cnndvlem2.t	⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
cnndvlem2.f	⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ₀ ↦ (((1 / 2)↑𝑛) · (𝑇‘(((2 · 3)↑𝑛) · 𝑦)))))
cnndvlem2.w	⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ₀ ((𝐹‘𝑤)‘𝑖))

**Assertion**
Ref	Expression
cnndvlem2	⊢ ∃𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅)

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

**Proof of Theorem cnndvlem2**
Step	Hyp	Ref	Expression
1		cnndvlem2.t	. . 3 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
2		cnndvlem2.f	. . 3 ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ₀ ↦ (((1 / 2)↑𝑛) · (𝑇‘(((2 · 3)↑𝑛) · 𝑦)))))
3		cnndvlem2.w	. . 3 ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ₀ ((𝐹‘𝑤)‘𝑖))
4	1, 2, 3	cnndvlem1 36452	. 2 ⊢ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅)
5		reex 11271	. . . . 5 ⊢ ℝ ∈ V
6	5	mptex 7258	. . . 4 ⊢ (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ₀ ((𝐹‘𝑤)‘𝑖)) ∈ V
7	3, 6	eqeltri 2834	. . 3 ⊢ 𝑊 ∈ V
8		eleq1 2826	. . . 4 ⊢ (𝑓 = 𝑊 → (𝑓 ∈ (ℝ–cn→ℝ) ↔ 𝑊 ∈ (ℝ–cn→ℝ)))
9		oveq2 7453	. . . . . 6 ⊢ (𝑓 = 𝑊 → (ℝ D 𝑓) = (ℝ D 𝑊))
10	9	dmeqd 5929	. . . . 5 ⊢ (𝑓 = 𝑊 → dom (ℝ D 𝑓) = dom (ℝ D 𝑊))
11	10	eqeq1d 2736	. . . 4 ⊢ (𝑓 = 𝑊 → (dom (ℝ D 𝑓) = ∅ ↔ dom (ℝ D 𝑊) = ∅))
12	8, 11	anbi12d 631	. . 3 ⊢ (𝑓 = 𝑊 → ((𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅) ↔ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅)))
13	7, 12	spcev 3615	. 2 ⊢ ((𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅) → ∃𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅))
14	4, 13	ax-mp 5	1 ⊢ ∃𝑓(𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1537 ∃wex 1777 ∈ wcel 2103 Vcvv 3482 ∅c0 4347 ↦ cmpt 5252 dom cdm 5699 ‘cfv 6572 (class class class)co 7445 ℝcr 11179 1c1 11181 + caddc 11183 · cmul 11185 − cmin 11516 / cdiv 11943 2c2 12344 3c3 12345 ℕ₀cn0 12549 ⌊cfl 13837 ↑cexp 14108 abscabs 15279 Σcsu 15730 –cn→ccncf 24914 D cdv 25910

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-inf2 9706 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 ax-addf 11259

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-iin 5022 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-of 7710 df-om 7900 df-1st 8026 df-2nd 8027 df-supp 8198 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-er 8759 df-map 8882 df-pm 8883 df-ixp 8952 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-fsupp 9428 df-fi 9476 df-sup 9507 df-inf 9508 df-oi 9575 df-card 10004 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-q 13010 df-rp 13054 df-xneg 13171 df-xadd 13172 df-xmul 13173 df-ioo 13407 df-ico 13409 df-icc 13410 df-fz 13564 df-fzo 13708 df-fl 13839 df-seq 14049 df-exp 14109 df-hash 14376 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-limsup 15513 df-clim 15530 df-rlim 15531 df-sum 15731 df-dvds 16297 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-hom 17330 df-cco 17331 df-rest 17477 df-topn 17478 df-0g 17496 df-gsum 17497 df-topgen 17498 df-pt 17499 df-prds 17502 df-xrs 17557 df-qtop 17562 df-imas 17563 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-submnd 18814 df-mulg 19103 df-cntz 19352 df-cmn 19819 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22914 df-topon 22931 df-topsp 22953 df-bases 22967 df-ntr 23042 df-cn 23249 df-cnp 23250 df-tx 23584 df-hmeo 23777 df-xms 24344 df-ms 24345 df-tms 24346 df-cncf 24916 df-limc 25913 df-dv 25914 df-ulm 26430

This theorem is referenced by: cnndv 36454

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