unbdqndv1 - Mathbox for Asger C. Ipsen

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Theorem unbdqndv1 37213

Description: If the difference quotient (((𝐹‘𝑧) − (𝐹‘𝐴)) / (𝑧 − 𝐴)) is unbounded near 𝐴 then 𝐹 is not differentiable at 𝐴. (Contributed by Asger C. Ipsen, 12-May-2021.)

**Hypotheses**
Ref	Expression
unbdqndv1.g	⊢ 𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐴)) / (𝑧 − 𝐴)))
unbdqndv1.1	⊢ (𝜑 → 𝑆 ⊆ ℂ)
unbdqndv1.2	⊢ (𝜑 → 𝑋 ⊆ 𝑆)
unbdqndv1.3	⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
unbdqndv1.4	⊢ (𝜑 → ∀𝑏 ∈ ℝ⁺ ∀𝑑 ∈ ℝ⁺ ∃𝑥 ∈ (𝑋 ∖ {𝐴})((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐺‘𝑥))))

**Assertion**
Ref	Expression
unbdqndv1	⊢ (𝜑 → ¬ 𝐴 ∈ dom (𝑆 D 𝐹))

Distinct variable groups: 𝐴,𝑏,𝑑,𝑥 𝑧,𝐴 𝐹,𝑏,𝑑,𝑥 𝑧,𝐹 𝐺,𝑏,𝑑,𝑥 𝑆,𝑏,𝑑,𝑥 𝑧,𝑆 𝑋,𝑏,𝑑,𝑥 𝑧,𝑋 𝜑,𝑏,𝑑,𝑥 𝜑,𝑧 Allowed substitution hint: 𝐺(𝑧)

Proof of Theorem unbdqndv1 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 4287	. . . . . . . 8 ⊢ ¬ 𝑦 ∈ ∅
2	1	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → ¬ 𝑦 ∈ ∅)
3		unbdqndv1.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
4		unbdqndv1.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
5	3, 4	sstrd 3944	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
6	5	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝑋 ⊆ ℂ)
7	6	ssdifssd 4097	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (𝑋 ∖ {𝐴}) ⊆ ℂ)
8		unbdqndv1.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
9	8	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝐹:𝑋⟶ℂ)
10	4, 8, 3	dvbss 26135	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝑋)
11	10	sselda 3934	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝐴 ∈ 𝑋)
12	9, 6, 11	dvlem 26130	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) ∧ 𝑧 ∈ (𝑋 ∖ {𝐴})) → (((𝐹‘𝑧) − (𝐹‘𝐴)) / (𝑧 − 𝐴)) ∈ ℂ)
13		unbdqndv1.g	. . . . . . . . 9 ⊢ 𝐺 = (𝑧 ∈ (𝑋 ∖ {𝐴}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐴)) / (𝑧 − 𝐴)))
14	12, 13	fmptd 7111	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝐺:(𝑋 ∖ {𝐴})⟶ℂ)
15	6, 11	sseldd 3935	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝐴 ∈ ℂ)
16		unbdqndv1.4	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑏 ∈ ℝ⁺ ∀𝑑 ∈ ℝ⁺ ∃𝑥 ∈ (𝑋 ∖ {𝐴})((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐺‘𝑥))))
17	16	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → ∀𝑏 ∈ ℝ⁺ ∀𝑑 ∈ ℝ⁺ ∃𝑥 ∈ (𝑋 ∖ {𝐴})((abs‘(𝑥 − 𝐴)) < 𝑑 ∧ 𝑏 ≤ (abs‘(𝐺‘𝑥))))
18	7, 14, 15, 17	unblimceq0 37212	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (𝐺 lim_ℂ 𝐴) = ∅)
19	2, 18	neleqtrrd 2885	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → ¬ 𝑦 ∈ (𝐺 lim_ℂ 𝐴))
20	19	intnand 494	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → ¬ (𝐴 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) ∧ 𝑦 ∈ (𝐺 lim_ℂ 𝐴)))
21		eqid 2762	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ((TopOpen‘ℂ_fld) ↾_t 𝑆)
22		eqid 2762	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
23	21, 22, 13, 4, 8, 3	eldv 26132	. . . . . . 7 ⊢ (𝜑 → (𝐴(𝑆 D 𝐹)𝑦 ↔ (𝐴 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) ∧ 𝑦 ∈ (𝐺 lim_ℂ 𝐴))))
24	23	notbid 321	. . . . . 6 ⊢ (𝜑 → (¬ 𝐴(𝑆 D 𝐹)𝑦 ↔ ¬ (𝐴 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) ∧ 𝑦 ∈ (𝐺 lim_ℂ 𝐴))))
25	24	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (¬ 𝐴(𝑆 D 𝐹)𝑦 ↔ ¬ (𝐴 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) ∧ 𝑦 ∈ (𝐺 lim_ℂ 𝐴))))
26	20, 25	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → ¬ 𝐴(𝑆 D 𝐹)𝑦)
27	26	alrimiv 1960	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → ∀𝑦 ¬ 𝐴(𝑆 D 𝐹)𝑦)
28		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝐴 ∈ dom (𝑆 D 𝐹))
29		eldmg 5886	. . . . . 6 ⊢ (𝐴 ∈ dom (𝑆 D 𝐹) → (𝐴 ∈ dom (𝑆 D 𝐹) ↔ ∃𝑦 𝐴(𝑆 D 𝐹)𝑦))
30	28, 29	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (𝐴 ∈ dom (𝑆 D 𝐹) ↔ ∃𝑦 𝐴(𝑆 D 𝐹)𝑦))
31	30	notbid 321	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (¬ 𝐴 ∈ dom (𝑆 D 𝐹) ↔ ¬ ∃𝑦 𝐴(𝑆 D 𝐹)𝑦))
32		alnex 1814	. . . . . 6 ⊢ (∀𝑦 ¬ 𝐴(𝑆 D 𝐹)𝑦 ↔ ¬ ∃𝑦 𝐴(𝑆 D 𝐹)𝑦)
33	32	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (∀𝑦 ¬ 𝐴(𝑆 D 𝐹)𝑦 ↔ ¬ ∃𝑦 𝐴(𝑆 D 𝐹)𝑦))
34	33	bicomd 226	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (¬ ∃𝑦 𝐴(𝑆 D 𝐹)𝑦 ↔ ∀𝑦 ¬ 𝐴(𝑆 D 𝐹)𝑦))
35	31, 34	bitrd 282	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (¬ 𝐴 ∈ dom (𝑆 D 𝐹) ↔ ∀𝑦 ¬ 𝐴(𝑆 D 𝐹)𝑦))
36	27, 35	mpbird 260	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → ¬ 𝐴 ∈ dom (𝑆 D 𝐹))
37	36	pm2.01da 811	1 ⊢ (𝜑 → ¬ 𝐴 ∈ dom (𝑆 D 𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∀wal 1568 = wceq 1570 ∃wex 1812 ∈ wcel 2145 ∀wral 3078 ∃wrex 3088 ∖ cdif 3899 ⊆ wss 3902 ∅c0 4282 {csn 4587 class class class wbr 5107 ↦ cmpt 5190 dom cdm 5659 ⟶wf 6533 ‘cfv 6537 (class class class)co 7417 ℂcc 11126 < clt 11271 ≤ cle 11272 − cmin 11469 / cdiv 11899 ℝ⁺crp 13046 abscabs 15325 ↾_t crest 17511 TopOpenctopn 17512 ℂ_fldccnfld 21591 intcnt 23248 lim_ℂ climc 26096 D cdv 26097

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-pm 8833 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fi 9385 df-sup 9416 df-inf 9417 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-z 12620 df-dec 12741 df-uz 12892 df-q 13002 df-rp 13047 df-xneg 13167 df-xadd 13168 df-xmul 13169 df-fz 13566 df-seq 14070 df-exp 14130 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-struct 17245 df-slot 17280 df-ndx 17292 df-base 17308 df-plusg 17361 df-mulr 17362 df-starv 17363 df-tset 17367 df-ple 17368 df-ds 17370 df-unif 17371 df-rest 17513 df-topn 17514 df-topgen 17534 df-psmet 21583 df-xmet 21584 df-met 21585 df-bl 21586 df-mopn 21587 df-cnfld 21592 df-top 23125 df-topon 23142 df-topsp 23164 df-bases 23177 df-ntr 23251 df-cnp 23459 df-xms 24552 df-ms 24553 df-limc 26100 df-dv 26101

This theorem is used by: unbdqndv2 37216

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