dvres3a - Metamath Proof Explorer

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Theorem dvres3a 25887

Description: Restriction of a complex differentiable function to the reals. This version of dvres3 25886 assumes that 𝐹 is differentiable on its domain, but does not require 𝐹 to be differentiable on the whole real line. (Contributed by Mario Carneiro, 11-Feb-2015.)

**Hypothesis**
Ref	Expression
dvres3a.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)

**Assertion**
Ref	Expression
dvres3a	⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → (𝑆 D (𝐹 ↾ 𝑆)) = ((ℂ D 𝐹) ↾ 𝑆))

**Proof of Theorem dvres3a**
Step	Hyp	Ref	Expression
1		reldv 25843	. . 3 ⊢ Rel (𝑆 D (𝐹 ↾ 𝑆))
2		recnprss 25877	. . . . . 6 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
3	2	ad2antrr 724	. . . . 5 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → 𝑆 ⊆ ℂ)
4		simplr 767	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → 𝐹:𝐴⟶ℂ)
5		inss2 4228	. . . . . . 7 ⊢ (𝑆 ∩ 𝐴) ⊆ 𝐴
6		fssres 6763	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℂ ∧ (𝑆 ∩ 𝐴) ⊆ 𝐴) → (𝐹 ↾ (𝑆 ∩ 𝐴)):(𝑆 ∩ 𝐴)⟶ℂ)
7	4, 5, 6	sylancl 584	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → (𝐹 ↾ (𝑆 ∩ 𝐴)):(𝑆 ∩ 𝐴)⟶ℂ)
8		rescom 6008	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑆) = ((𝐹 ↾ 𝑆) ↾ 𝐴)
9		resres 5998	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝑆) ↾ 𝐴) = (𝐹 ↾ (𝑆 ∩ 𝐴))
10	8, 9	eqtri 2753	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑆) = (𝐹 ↾ (𝑆 ∩ 𝐴))
11		ffn 6723	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶ℂ → 𝐹 Fn 𝐴)
12		fnresdm 6675	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
13	4, 11, 12	3syl 18	. . . . . . . . 9 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → (𝐹 ↾ 𝐴) = 𝐹)
14	13	reseq1d 5984	. . . . . . . 8 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → ((𝐹 ↾ 𝐴) ↾ 𝑆) = (𝐹 ↾ 𝑆))
15	10, 14	eqtr3id 2779	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → (𝐹 ↾ (𝑆 ∩ 𝐴)) = (𝐹 ↾ 𝑆))
16	15	feq1d 6708	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → ((𝐹 ↾ (𝑆 ∩ 𝐴)):(𝑆 ∩ 𝐴)⟶ℂ ↔ (𝐹 ↾ 𝑆):(𝑆 ∩ 𝐴)⟶ℂ))
17	7, 16	mpbid 231	. . . . 5 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → (𝐹 ↾ 𝑆):(𝑆 ∩ 𝐴)⟶ℂ)
18		inss1 4227	. . . . . 6 ⊢ (𝑆 ∩ 𝐴) ⊆ 𝑆
19	18	a1i 11	. . . . 5 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → (𝑆 ∩ 𝐴) ⊆ 𝑆)
20	3, 17, 19	dvbss 25874	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → dom (𝑆 D (𝐹 ↾ 𝑆)) ⊆ (𝑆 ∩ 𝐴))
21		dmres 6017	. . . . 5 ⊢ dom ((ℂ D 𝐹) ↾ 𝑆) = (𝑆 ∩ dom (ℂ D 𝐹))
22		simprr 771	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → dom (ℂ D 𝐹) = 𝐴)
23	22	ineq2d 4210	. . . . 5 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → (𝑆 ∩ dom (ℂ D 𝐹)) = (𝑆 ∩ 𝐴))
24	21, 23	eqtrid 2777	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → dom ((ℂ D 𝐹) ↾ 𝑆) = (𝑆 ∩ 𝐴))
25	20, 24	sseqtrrd 4018	. . 3 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → dom (𝑆 D (𝐹 ↾ 𝑆)) ⊆ dom ((ℂ D 𝐹) ↾ 𝑆))
26		relssres 6027	. . 3 ⊢ ((Rel (𝑆 D (𝐹 ↾ 𝑆)) ∧ dom (𝑆 D (𝐹 ↾ 𝑆)) ⊆ dom ((ℂ D 𝐹) ↾ 𝑆)) → ((𝑆 D (𝐹 ↾ 𝑆)) ↾ dom ((ℂ D 𝐹) ↾ 𝑆)) = (𝑆 D (𝐹 ↾ 𝑆)))
27	1, 25, 26	sylancr 585	. 2 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → ((𝑆 D (𝐹 ↾ 𝑆)) ↾ dom ((ℂ D 𝐹) ↾ 𝑆)) = (𝑆 D (𝐹 ↾ 𝑆)))
28		dvfg 25879	. . . . 5 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D (𝐹 ↾ 𝑆)):dom (𝑆 D (𝐹 ↾ 𝑆))⟶ℂ)
29	28	ad2antrr 724	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → (𝑆 D (𝐹 ↾ 𝑆)):dom (𝑆 D (𝐹 ↾ 𝑆))⟶ℂ)
30	29	ffund 6727	. . 3 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → Fun (𝑆 D (𝐹 ↾ 𝑆)))
31		ssidd 4000	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → ℂ ⊆ ℂ)
32		dvres3a.j	. . . . . 6 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
33	32	cnfldtopon 24743	. . . . 5 ⊢ 𝐽 ∈ (TopOn‘ℂ)
34		simprl 769	. . . . 5 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → 𝐴 ∈ 𝐽)
35		toponss 22873	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ ℂ)
36	33, 34, 35	sylancr 585	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → 𝐴 ⊆ ℂ)
37		dvres2 25885	. . . 4 ⊢ (((ℂ ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ⊆ ℂ ∧ 𝑆 ⊆ ℂ)) → ((ℂ D 𝐹) ↾ 𝑆) ⊆ (𝑆 D (𝐹 ↾ 𝑆)))
38	31, 4, 36, 3, 37	syl22anc 837	. . 3 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → ((ℂ D 𝐹) ↾ 𝑆) ⊆ (𝑆 D (𝐹 ↾ 𝑆)))
39		funssres 6598	. . 3 ⊢ ((Fun (𝑆 D (𝐹 ↾ 𝑆)) ∧ ((ℂ D 𝐹) ↾ 𝑆) ⊆ (𝑆 D (𝐹 ↾ 𝑆))) → ((𝑆 D (𝐹 ↾ 𝑆)) ↾ dom ((ℂ D 𝐹) ↾ 𝑆)) = ((ℂ D 𝐹) ↾ 𝑆))
40	30, 38, 39	syl2anc 582	. 2 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → ((𝑆 D (𝐹 ↾ 𝑆)) ↾ dom ((ℂ D 𝐹) ↾ 𝑆)) = ((ℂ D 𝐹) ↾ 𝑆))
41	27, 40	eqtr3d 2767	1 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → (𝑆 D (𝐹 ↾ 𝑆)) = ((ℂ D 𝐹) ↾ 𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∩ cin 3943 ⊆ wss 3944 {cpr 4632 dom cdm 5678 ↾ cres 5680 Rel wrel 5683 Fun wfun 6543 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 ℝcr 11139 TopOpenctopn 17406 ℂ_fldccnfld 21296 TopOnctopon 22856 D cdv 25836

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9436 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-icc 13366 df-fz 13520 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-struct 17119 df-slot 17154 df-ndx 17166 df-base 17184 df-plusg 17249 df-mulr 17250 df-starv 17251 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-rest 17407 df-topn 17408 df-topgen 17428 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cnp 23176 df-haus 23263 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24270 df-ms 24271 df-limc 25839 df-dv 25840

This theorem is referenced by: dvnres 25905 dvmptres3 25932

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