unbdqndv1 - Mathbox for Asger C. Ipsen

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Theorem unbdqndv1 36474

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 4360	. . . . . . . 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 4019	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
6	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝑋 ⊆ ℂ)
7	6	ssdifssd 4170	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (𝑋 ∖ {𝐴}) ⊆ ℂ)
8		unbdqndv1.3	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
9	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝐹:𝑋⟶ℂ)
10	4, 8, 3	dvbss 25956	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝑋)
11	10	sselda 4008	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝐴 ∈ 𝑋)
12	9, 6, 11	dvlem 25951	. . . . . . . . 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 7148	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝐺:(𝑋 ∖ {𝐴})⟶ℂ)
15	6, 11	sseldd 4009	. . . . . . . 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 480	. . . . . . . 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 36473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (𝐺 lim_ℂ 𝐴) = ∅)
19	2, 18	neleqtrrd 2867	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → ¬ 𝑦 ∈ (𝐺 lim_ℂ 𝐴))
20	19	intnand 488	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → ¬ (𝐴 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) ∧ 𝑦 ∈ (𝐺 lim_ℂ 𝐴)))
21		eqid 2740	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ((TopOpen‘ℂ_fld) ↾_t 𝑆)
22		eqid 2740	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
23	21, 22, 13, 4, 8, 3	eldv 25953	. . . . . . 7 ⊢ (𝜑 → (𝐴(𝑆 D 𝐹)𝑦 ↔ (𝐴 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) ∧ 𝑦 ∈ (𝐺 lim_ℂ 𝐴))))
24	23	notbid 318	. . . . . 6 ⊢ (𝜑 → (¬ 𝐴(𝑆 D 𝐹)𝑦 ↔ ¬ (𝐴 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) ∧ 𝑦 ∈ (𝐺 lim_ℂ 𝐴))))
25	24	adantr 480	. . . . 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 257	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → ¬ 𝐴(𝑆 D 𝐹)𝑦)
27	26	alrimiv 1926	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → ∀𝑦 ¬ 𝐴(𝑆 D 𝐹)𝑦)
28		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝐴 ∈ dom (𝑆 D 𝐹))
29		eldmg 5923	. . . . . 6 ⊢ (𝐴 ∈ dom (𝑆 D 𝐹) → (𝐴 ∈ dom (𝑆 D 𝐹) ↔ ∃𝑦 𝐴(𝑆 D 𝐹)𝑦))
30	28, 29	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (𝐴 ∈ dom (𝑆 D 𝐹) ↔ ∃𝑦 𝐴(𝑆 D 𝐹)𝑦))
31	30	notbid 318	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (¬ 𝐴 ∈ dom (𝑆 D 𝐹) ↔ ¬ ∃𝑦 𝐴(𝑆 D 𝐹)𝑦))
32		alnex 1779	. . . . . 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 223	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (¬ ∃𝑦 𝐴(𝑆 D 𝐹)𝑦 ↔ ∀𝑦 ¬ 𝐴(𝑆 D 𝐹)𝑦))
35	31, 34	bitrd 279	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (¬ 𝐴 ∈ dom (𝑆 D 𝐹) ↔ ∀𝑦 ¬ 𝐴(𝑆 D 𝐹)𝑦))
36	27, 35	mpbird 257	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → ¬ 𝐴 ∈ dom (𝑆 D 𝐹))
37	36	pm2.01da 798	1 ⊢ (𝜑 → ¬ 𝐴 ∈ dom (𝑆 D 𝐹))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1535 = wceq 1537 ∃wex 1777 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 ∖ cdif 3973 ⊆ wss 3976 ∅c0 4352 {csn 4648 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℝ⁺crp 13057 abscabs 15283 ↾_t crest 17480 TopOpenctopn 17481 ℂ_fldccnfld 21387 intcnt 23046 lim_ℂ climc 25917 D cdv 25918

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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

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 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fi 9480 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-fz 13568 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-struct 17194 df-slot 17229 df-ndx 17241 df-base 17259 df-plusg 17324 df-mulr 17325 df-starv 17326 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-rest 17482 df-topn 17483 df-topgen 17503 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-ntr 23049 df-cnp 23257 df-xms 24351 df-ms 24352 df-limc 25921 df-dv 25922

This theorem is referenced by: unbdqndv2 36477

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