unbdqndv1 - Mathbox for Asger C. Ipsen

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Theorem unbdqndv1 36784

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 4279	. . . . . . . 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 3933	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
6	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝑋 ⊆ ℂ)
7	6	ssdifssd 4088	. . . . . . . 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 25878	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝑋)
11	10	sselda 3922	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝐴 ∈ 𝑋)
12	9, 6, 11	dvlem 25873	. . . . . . . . 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 7060	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝐺:(𝑋 ∖ {𝐴})⟶ℂ)
15	6, 11	sseldd 3923	. . . . . . . 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 36783	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (𝐺 lim_ℂ 𝐴) = ∅)
19	2, 18	neleqtrrd 2860	. . . . . 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 2737	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ((TopOpen‘ℂ_fld) ↾_t 𝑆)
22		eqid 2737	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
23	21, 22, 13, 4, 8, 3	eldv 25875	. . . . . . 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 1929	. . 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 5847	. . . . . 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 1783	. . . . . 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 799	1 ⊢ (𝜑 → ¬ 𝐴 ∈ dom (𝑆 D 𝐹))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1540 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 ∖ cdif 3887 ⊆ wss 3890 ∅c0 4274 {csn 4568 class class class wbr 5086 ↦ cmpt 5167 dom cdm 5624 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 < clt 11170 ≤ cle 11171 − cmin 11368 / cdiv 11798 ℝ⁺crp 12933 abscabs 15187 ↾_t crest 17374 TopOpenctopn 17375 ℂ_fldccnfld 21344 intcnt 22992 lim_ℂ climc 25839 D cdv 25840

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-map 8768 df-pm 8769 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fi 9317 df-sup 9348 df-inf 9349 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-fz 13453 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17108 df-slot 17143 df-ndx 17155 df-base 17171 df-plusg 17224 df-mulr 17225 df-starv 17226 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-rest 17376 df-topn 17377 df-topgen 17397 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-ntr 22995 df-cnp 23203 df-xms 24295 df-ms 24296 df-limc 25843 df-dv 25844

This theorem is referenced by: unbdqndv2 36787

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