unbdqndv1 - Mathbox for Asger C. Ipsen

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Theorem unbdqndv1 37138

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 4294	. . . . . . . 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 3950	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
6	5	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝑋 ⊆ ℂ)
7	6	ssdifssd 4104	. . . . . . . 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 26097	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝑋)
11	10	sselda 3940	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝐴 ∈ 𝑋)
12	9, 6, 11	dvlem 26092	. . . . . . . . 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 7116	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝐺:(𝑋 ∖ {𝐴})⟶ℂ)
15	6, 11	sseldd 3941	. . . . . . . 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 37137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (𝐺 lim_ℂ 𝐴) = ∅)
19	2, 18	neleqtrrd 2889	. . . . . 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 2766	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ((TopOpen‘ℂ_fld) ↾_t 𝑆)
22		eqid 2766	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
23	21, 22, 13, 4, 8, 3	eldv 26094	. . . . . . 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 5893	. . . . . 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 2146 ∀wral 3082 ∃wrex 3092 ∖ cdif 3905 ⊆ wss 3908 ∅c0 4289 {csn 4594 class class class wbr 5114 ↦ cmpt 5197 dom cdm 5666 ⟶wf 6539 ‘cfv 6543 (class class class)co 7423 ℂcc 11116 < clt 11261 ≤ cle 11262 − cmin 11459 / cdiv 11889 ℝ⁺crp 13034 abscabs 15311 ↾_t crest 17498 TopOpenctopn 17499 ℂ_fldccnfld 21559 intcnt 23211 lim_ℂ climc 26058 D cdv 26059

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-map 8835 df-pm 8836 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fi 9381 df-sup 9412 df-inf 9413 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-fz 13554 df-seq 14058 df-exp 14118 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-struct 17232 df-slot 17267 df-ndx 17279 df-base 17295 df-plusg 17348 df-mulr 17349 df-starv 17350 df-tset 17354 df-ple 17355 df-ds 17357 df-unif 17358 df-rest 17500 df-topn 17501 df-topgen 17521 df-psmet 21551 df-xmet 21552 df-met 21553 df-bl 21554 df-mopn 21555 df-cnfld 21560 df-top 23088 df-topon 23105 df-topsp 23127 df-bases 23140 df-ntr 23214 df-cnp 23422 df-xms 24514 df-ms 24515 df-limc 26062 df-dv 26063

This theorem is used by: unbdqndv2 37141

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