dvcnp - Metamath Proof Explorer

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Theorem dvcnp 25955

Description: The difference quotient is continuous at 𝐵 when the original function is differentiable at 𝐵. (Contributed by Mario Carneiro, 8-Aug-2014.) (Revised by Mario Carneiro, 28-Dec-2016.)

**Hypotheses**
Ref	Expression
dvcnp.j	⊢ 𝐽 = (𝐾 ↾_t 𝐴)
dvcnp.k	⊢ 𝐾 = (TopOpen‘ℂ_fld)
dvcnp.g	⊢ 𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 = 𝐵, ((𝑆 D 𝐹)‘𝐵), (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))))

**Assertion**
Ref	Expression
dvcnp	⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵))

Distinct variable groups: 𝑧,𝐴 𝑧,𝐵 𝑧,𝐹 𝑧,𝐾 𝑧,𝑆 𝑧,𝐽 Allowed substitution hint: 𝐺(𝑧)

**Proof of Theorem dvcnp**
Step	Hyp	Ref	Expression
1		dvcnp.g	. 2 ⊢ 𝐺 = (𝑧 ∈ 𝐴 ↦ if(𝑧 = 𝐵, ((𝑆 D 𝐹)‘𝐵), (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))))
2		dvfg 25942	. . . . . . 7 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
3	2	3ad2ant1 1133	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
4		ffun 6738	. . . . . 6 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹))
5		funfvbrb 7070	. . . . . 6 ⊢ (Fun (𝑆 D 𝐹) → (𝐵 ∈ dom (𝑆 D 𝐹) ↔ 𝐵(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐵)))
6	3, 4, 5	3syl 18	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → (𝐵 ∈ dom (𝑆 D 𝐹) ↔ 𝐵(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐵)))
7		eqid 2736	. . . . . 6 ⊢ (𝐾 ↾_t 𝑆) = (𝐾 ↾_t 𝑆)
8		dvcnp.k	. . . . . 6 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
9		eqid 2736	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))) = (𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)))
10		recnprss 25940	. . . . . . 7 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
11	10	3ad2ant1 1133	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → 𝑆 ⊆ ℂ)
12		simp2 1137	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → 𝐹:𝐴⟶ℂ)
13		simp3 1138	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → 𝐴 ⊆ 𝑆)
14	7, 8, 9, 11, 12, 13	eldv 25934	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → (𝐵(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐵) ↔ (𝐵 ∈ ((int‘(𝐾 ↾_t 𝑆))‘𝐴) ∧ ((𝑆 D 𝐹)‘𝐵) ∈ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))) lim_ℂ 𝐵))))
15	6, 14	bitrd 279	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → (𝐵 ∈ dom (𝑆 D 𝐹) ↔ (𝐵 ∈ ((int‘(𝐾 ↾_t 𝑆))‘𝐴) ∧ ((𝑆 D 𝐹)‘𝐵) ∈ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))) lim_ℂ 𝐵))))
16	15	simplbda 499	. . 3 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → ((𝑆 D 𝐹)‘𝐵) ∈ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))) lim_ℂ 𝐵))
17	13, 11	sstrd 3993	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → 𝐴 ⊆ ℂ)
18	17	adantr 480	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐴 ⊆ ℂ)
19	11, 12, 13	dvbss 25937	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → dom (𝑆 D 𝐹) ⊆ 𝐴)
20	19	sselda 3982	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐵 ∈ 𝐴)
21		eldifsn 4785	. . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧 ≠ 𝐵))
22	12	adantr 480	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐹:𝐴⟶ℂ)
23	22, 18, 20	dvlem 25932	. . . . 5 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) ∧ 𝑧 ∈ (𝐴 ∖ {𝐵})) → (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)) ∈ ℂ)
24	21, 23	sylan2br 595	. . . 4 ⊢ ((((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) ∧ (𝑧 ∈ 𝐴 ∧ 𝑧 ≠ 𝐵)) → (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)) ∈ ℂ)
25		dvcnp.j	. . . 4 ⊢ 𝐽 = (𝐾 ↾_t 𝐴)
26	18, 20, 24, 25, 8	limcmpt2 25920	. . 3 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → (((𝑆 D 𝐹)‘𝐵) ∈ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))) lim_ℂ 𝐵) ↔ (𝑧 ∈ 𝐴 ↦ if(𝑧 = 𝐵, ((𝑆 D 𝐹)‘𝐵), (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)))) ∈ ((𝐽 CnP 𝐾)‘𝐵)))
27	16, 26	mpbid 232	. 2 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → (𝑧 ∈ 𝐴 ↦ if(𝑧 = 𝐵, ((𝑆 D 𝐹)‘𝐵), (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)))) ∈ ((𝐽 CnP 𝐾)‘𝐵))
28	1, 27	eqeltrid 2844	1 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∖ cdif 3947 ⊆ wss 3950 ifcif 4524 {csn 4625 {cpr 4627 class class class wbr 5142 ↦ cmpt 5224 dom cdm 5684 Fun wfun 6554 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 ℝcr 11155 − cmin 11493 / cdiv 11921 ↾_t crest 17466 TopOpenctopn 17467 ℂ_fldccnfld 21365 intcnt 23026 CnP ccnp 23234 lim_ℂ climc 25898 D cdv 25899

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fi 9452 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-icc 13395 df-fz 13549 df-seq 14044 df-exp 14104 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-struct 17185 df-slot 17220 df-ndx 17232 df-base 17249 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-rest 17468 df-topn 17469 df-topgen 17489 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cnp 23237 df-haus 23324 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-limc 25902 df-dv 25903

This theorem is referenced by: efrlim 27013 efrlimOLD 27014

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