cnndvlem2 - Mathbox for Asger C. Ipsen

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Theorem cnndvlem2 36473

Description: Lemma for cnndv 36474. (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 36472	. 2 ⊢ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅)
5		reex 11227	. . . . 5 ⊢ ℝ ∈ V
6	5	mptex 7224	. . . 4 ⊢ (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ₀ ((𝐹‘𝑤)‘𝑖)) ∈ V
7	3, 6	eqeltri 2829	. . 3 ⊢ 𝑊 ∈ V
8		eleq1 2821	. . . 4 ⊢ (𝑓 = 𝑊 → (𝑓 ∈ (ℝ–cn→ℝ) ↔ 𝑊 ∈ (ℝ–cn→ℝ)))
9		oveq2 7420	. . . . . 6 ⊢ (𝑓 = 𝑊 → (ℝ D 𝑓) = (ℝ D 𝑊))
10	9	dmeqd 5896	. . . . 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 632	. . 3 ⊢ (𝑓 = 𝑊 → ((𝑓 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑓) = ∅) ↔ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅)))
13	7, 12	spcev 3589	. 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 1539 ∃wex 1778 ∈ wcel 2107 Vcvv 3463 ∅c0 4313 ↦ cmpt 5205 dom cdm 5665 ‘cfv 6540 (class class class)co 7412 ℝcr 11135 1c1 11137 + caddc 11139 · cmul 11141 − cmin 11473 / cdiv 11901 2c2 12302 3c3 12303 ℕ₀cn0 12508 ⌊cfl 13811 ↑cexp 14083 abscabs 15254 Σcsu 15703 –cn→ccncf 24837 D cdv 25833

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7736 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 ax-addf 11215

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4888 df-int 4927 df-iun 4973 df-iin 4974 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-se 5618 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6493 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7678 df-om 7869 df-1st 7995 df-2nd 7996 df-supp 8167 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8726 df-map 8849 df-pm 8850 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9383 df-fi 9432 df-sup 9463 df-inf 9464 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11475 df-neg 11476 df-div 11902 df-nn 12248 df-2 12310 df-3 12311 df-4 12312 df-5 12313 df-6 12314 df-7 12315 df-8 12316 df-9 12317 df-n0 12509 df-z 12596 df-dec 12716 df-uz 12860 df-q 12972 df-rp 13016 df-xneg 13135 df-xadd 13136 df-xmul 13137 df-ioo 13372 df-ico 13374 df-icc 13375 df-fz 13529 df-fzo 13676 df-fl 13813 df-seq 14024 df-exp 14084 df-hash 14351 df-cj 15119 df-re 15120 df-im 15121 df-sqrt 15255 df-abs 15256 df-limsup 15488 df-clim 15505 df-rlim 15506 df-sum 15704 df-dvds 16272 df-struct 17165 df-sets 17182 df-slot 17200 df-ndx 17212 df-base 17229 df-ress 17252 df-plusg 17285 df-mulr 17286 df-starv 17287 df-sca 17288 df-vsca 17289 df-ip 17290 df-tset 17291 df-ple 17292 df-ds 17294 df-unif 17295 df-hom 17296 df-cco 17297 df-rest 17437 df-topn 17438 df-0g 17456 df-gsum 17457 df-topgen 17458 df-pt 17459 df-prds 17462 df-xrs 17517 df-qtop 17522 df-imas 17523 df-xps 17525 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18621 df-sgrp 18700 df-mnd 18716 df-submnd 18765 df-mulg 19054 df-cntz 19303 df-cmn 19767 df-psmet 21317 df-xmet 21318 df-met 21319 df-bl 21320 df-mopn 21321 df-cnfld 21326 df-top 22847 df-topon 22864 df-topsp 22886 df-bases 22899 df-ntr 22973 df-cn 23180 df-cnp 23181 df-tx 23515 df-hmeo 23708 df-xms 24274 df-ms 24275 df-tms 24276 df-cncf 24839 df-limc 25836 df-dv 25837 df-ulm 26355

This theorem is referenced by: cnndv 36474

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