dvcnp - Metamath Proof Explorer

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Theorem dvcnp 25900

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 25887	. . . . . . 7 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
3	2	3ad2ant1 1134	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
4		ffun 6667	. . . . . 6 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹))
5		funfvbrb 6999	. . . . . 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 2737	. . . . . 6 ⊢ (𝐾 ↾_t 𝑆) = (𝐾 ↾_t 𝑆)
8		dvcnp.k	. . . . . 6 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
9		eqid 2737	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))) = (𝑧 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)))
10		recnprss 25885	. . . . . . 7 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
11	10	3ad2ant1 1134	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → 𝑆 ⊆ ℂ)
12		simp2 1138	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → 𝐹:𝐴⟶ℂ)
13		simp3 1139	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → 𝐴 ⊆ 𝑆)
14	7, 8, 9, 11, 12, 13	eldv 25879	. . . . 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 3933	. . . . 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 25882	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → dom (𝑆 D 𝐹) ⊆ 𝐴)
20	19	sselda 3922	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) ∧ 𝐵 ∈ dom (𝑆 D 𝐹)) → 𝐵 ∈ 𝐴)
21		eldifsn 4730	. . . . 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 25877	. . . . 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 596	. . . 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 25865	. . 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 2841	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 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3887 ⊆ wss 3890 ifcif 4467 {csn 4568 {cpr 4570 class class class wbr 5086 ↦ cmpt 5167 dom cdm 5626 Fun wfun 6488 ⟶wf 6490 ‘cfv 6494 (class class class)co 7362 ℂcc 11031 ℝcr 11032 − cmin 11372 / cdiv 11802 ↾_t crest 17378 TopOpenctopn 17379 ℂ_fldccnfld 21348 intcnt 22996 CnP ccnp 23204 lim_ℂ climc 25843 D cdv 25844

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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111

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-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-er 8638 df-map 8770 df-pm 8771 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fi 9319 df-sup 9350 df-inf 9351 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-icc 13300 df-fz 13457 df-seq 13959 df-exp 14019 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-struct 17112 df-slot 17147 df-ndx 17159 df-base 17175 df-plusg 17228 df-mulr 17229 df-starv 17230 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-rest 17380 df-topn 17381 df-topgen 17401 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-fbas 21345 df-fg 21346 df-cnfld 21349 df-top 22873 df-topon 22890 df-topsp 22912 df-bases 22925 df-cld 22998 df-ntr 22999 df-cls 23000 df-nei 23077 df-lp 23115 df-perf 23116 df-cnp 23207 df-haus 23294 df-fil 23825 df-fm 23917 df-flim 23918 df-flf 23919 df-xms 24299 df-ms 24300 df-limc 25847 df-dv 25848

This theorem is referenced by: efrlim 26950 efrlimOLD 26951

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