unbdqndv1 - Mathbox for Asger C. Ipsen

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Theorem unbdqndv1 35688

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 4331	. . . . . . . 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 3993	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
6	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝑋 ⊆ ℂ)
7	6	ssdifssd 4143	. . . . . . . 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 25651	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝑋)
11	10	sselda 3983	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → 𝐴 ∈ 𝑋)
12	9, 6, 11	dvlem 25646	. . . . . . . . 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 3984	. . . . . . . 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 35687	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (𝐺 lim_ℂ 𝐴) = ∅)
19	2, 18	neleqtrrd 2855	. . . . . 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 2731	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ((TopOpen‘ℂ_fld) ↾_t 𝑆)
22		eqid 2731	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
23	21, 22, 13, 4, 8, 3	eldv 25648	. . . . . . 7 ⊢ (𝜑 → (𝐴(𝑆 D 𝐹)𝑦 ↔ (𝐴 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝑋) ∧ 𝑦 ∈ (𝐺 lim_ℂ 𝐴))))
24	23	notbid 317	. . . . . 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 256	. . . 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 5899	. . . . . 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 317	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (¬ 𝐴 ∈ dom (𝑆 D 𝐹) ↔ ¬ ∃𝑦 𝐴(𝑆 D 𝐹)𝑦))
32		alnex 1782	. . . . . 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 222	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (¬ ∃𝑦 𝐴(𝑆 D 𝐹)𝑦 ↔ ∀𝑦 ¬ 𝐴(𝑆 D 𝐹)𝑦))
35	31, 34	bitrd 278	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → (¬ 𝐴 ∈ dom (𝑆 D 𝐹) ↔ ∀𝑦 ¬ 𝐴(𝑆 D 𝐹)𝑦))
36	27, 35	mpbird 256	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ dom (𝑆 D 𝐹)) → ¬ 𝐴 ∈ dom (𝑆 D 𝐹))
37	36	pm2.01da 796	1 ⊢ (𝜑 → ¬ 𝐴 ∈ dom (𝑆 D 𝐹))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1538 = wceq 1540 ∃wex 1780 ∈ wcel 2105 ∀wral 3060 ∃wrex 3069 ∖ cdif 3946 ⊆ wss 3949 ∅c0 4323 {csn 4629 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5677 ⟶wf 6540 ‘cfv 6544 (class class class)co 7412 ℂcc 11111 < clt 11253 ≤ cle 11254 − cmin 11449 / cdiv 11876 ℝ⁺crp 12979 abscabs 15186 ↾_t crest 17371 TopOpenctopn 17372 ℂ_fldccnfld 21145 intcnt 22742 lim_ℂ climc 25612 D cdv 25613

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-pm 8826 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fi 9409 df-sup 9440 df-inf 9441 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-fz 13490 df-seq 13972 df-exp 14033 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-struct 17085 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-mulr 17216 df-starv 17217 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-rest 17373 df-topn 17374 df-topgen 17394 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-ntr 22745 df-cnp 22953 df-xms 24047 df-ms 24048 df-limc 25616 df-dv 25617

This theorem is referenced by: unbdqndv2 35691

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