dvcnp - Metamath Proof Explorer

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Theorem dvcnp 25878

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 25865	. . . . . . 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 6665	. . . . . 6 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹))
5		funfvbrb 6996	. . . . . 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 25863	. . . . . . 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 25857	. . . . 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 3944	. . . . 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 25860	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → dom (𝑆 D 𝐹) ⊆ 𝐴)
20	19	sselda 3933	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐵 ∈ 𝐴)
21		eldifsn 4742	. . . . 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 25855	. . . . 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 25843	. . 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 2840	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 1541 ∈ wcel 2113 ≠ wne 2932 ∖ cdif 3898 ⊆ wss 3901 ifcif 4479 {csn 4580 {cpr 4582 class class class wbr 5098 ↦ cmpt 5179 dom cdm 5624 Fun wfun 6486 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ℂcc 11026 ℝcr 11027 − cmin 11366 / cdiv 11796 ↾_t crest 17342 TopOpenctopn 17343 ℂ_fldccnfld 21311 intcnt 22963 CnP ccnp 23171 lim_ℂ climc 25821 D cdv 25822

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 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 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8767 df-pm 8768 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fi 9316 df-sup 9347 df-inf 9348 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-icc 13270 df-fz 13426 df-seq 13927 df-exp 13987 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17139 df-plusg 17192 df-mulr 17193 df-starv 17194 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-rest 17344 df-topn 17345 df-topgen 17365 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-fbas 21308 df-fg 21309 df-cnfld 21312 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22892 df-cld 22965 df-ntr 22966 df-cls 22967 df-nei 23044 df-lp 23082 df-perf 23083 df-cnp 23174 df-haus 23261 df-fil 23792 df-fm 23884 df-flim 23885 df-flf 23886 df-xms 24266 df-ms 24267 df-limc 25825 df-dv 25826

This theorem is referenced by: efrlim 26937 efrlimOLD 26938

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