dvres3a - Metamath Proof Explorer

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Theorem dvres3a 25871

Description: Restriction of a complex differentiable function to the reals. This version of dvres3 25870 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 25827	. . 3 ⊢ Rel (𝑆 D (𝐹 ↾ 𝑆))
2		recnprss 25861	. . . . . 6 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
3	2	ad2antrr 726	. . . . 5 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → 𝑆 ⊆ ℂ)
4		simplr 768	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → 𝐹:𝐴⟶ℂ)
5		inss2 4190	. . . . . . 7 ⊢ (𝑆 ∩ 𝐴) ⊆ 𝐴
6		fssres 6700	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℂ ∧ (𝑆 ∩ 𝐴) ⊆ 𝐴) → (𝐹 ↾ (𝑆 ∩ 𝐴)):(𝑆 ∩ 𝐴)⟶ℂ)
7	4, 5, 6	sylancl 586	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → (𝐹 ↾ (𝑆 ∩ 𝐴)):(𝑆 ∩ 𝐴)⟶ℂ)
8		rescom 5961	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑆) = ((𝐹 ↾ 𝑆) ↾ 𝐴)
9		resres 5951	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝑆) ↾ 𝐴) = (𝐹 ↾ (𝑆 ∩ 𝐴))
10	8, 9	eqtri 2759	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑆) = (𝐹 ↾ (𝑆 ∩ 𝐴))
11		ffn 6662	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶ℂ → 𝐹 Fn 𝐴)
12		fnresdm 6611	. . . . . . . . . 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 5937	. . . . . . . 8 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → ((𝐹 ↾ 𝐴) ↾ 𝑆) = (𝐹 ↾ 𝑆))
15	10, 14	eqtr3id 2785	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → (𝐹 ↾ (𝑆 ∩ 𝐴)) = (𝐹 ↾ 𝑆))
16	15	feq1d 6644	. . . . . 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 232	. . . . 5 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → (𝐹 ↾ 𝑆):(𝑆 ∩ 𝐴)⟶ℂ)
18		inss1 4189	. . . . . 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 25858	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → dom (𝑆 D (𝐹 ↾ 𝑆)) ⊆ (𝑆 ∩ 𝐴))
21		dmres 5971	. . . . 5 ⊢ dom ((ℂ D 𝐹) ↾ 𝑆) = (𝑆 ∩ dom (ℂ D 𝐹))
22		simprr 772	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → dom (ℂ D 𝐹) = 𝐴)
23	22	ineq2d 4172	. . . . 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 2783	. . . 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 3971	. . 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 5981	. . 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 587	. 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 25863	. . . . 5 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D (𝐹 ↾ 𝑆)):dom (𝑆 D (𝐹 ↾ 𝑆))⟶ℂ)
29	28	ad2antrr 726	. . . 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 6666	. . 3 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → Fun (𝑆 D (𝐹 ↾ 𝑆)))
31		ssidd 3957	. . . 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 24726	. . . . 5 ⊢ 𝐽 ∈ (TopOn‘ℂ)
34		simprl 770	. . . . 5 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → 𝐴 ∈ 𝐽)
35		toponss 22871	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ ℂ)
36	33, 34, 35	sylancr 587	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → 𝐴 ⊆ ℂ)
37		dvres2 25869	. . . 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 838	. . 3 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → ((ℂ D 𝐹) ↾ 𝑆) ⊆ (𝑆 D (𝐹 ↾ 𝑆)))
39		funssres 6536	. . 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 584	. 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 2773	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 395 = wceq 1541 ∈ wcel 2113 ∩ cin 3900 ⊆ wss 3901 {cpr 4582 dom cdm 5624 ↾ cres 5626 Rel wrel 5629 Fun wfun 6486 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 ℝcr 11025 TopOpenctopn 17341 ℂ_fldccnfld 21309 TopOnctopon 22854 D cdv 25820

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 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

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 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fi 9314 df-sup 9345 df-inf 9346 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-icc 13268 df-fz 13424 df-seq 13925 df-exp 13985 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-struct 17074 df-slot 17109 df-ndx 17121 df-base 17137 df-plusg 17190 df-mulr 17191 df-starv 17192 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-rest 17342 df-topn 17343 df-topgen 17363 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cnp 23172 df-haus 23259 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-limc 25823 df-dv 25824

This theorem is referenced by: dvnres 25889 dvmptres3 25916

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