dvres3a - Metamath Proof Explorer

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Theorem dvres3a 25813

Description: Restriction of a complex differentiable function to the reals. This version of dvres3 25812 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 25769	. . 3 ⊢ Rel (𝑆 D (𝐹 ↾ 𝑆))
2		recnprss 25803	. . . . . 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 4189	. . . . . . 7 ⊢ (𝑆 ∩ 𝐴) ⊆ 𝐴
6		fssres 6690	. . . . . . 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 5953	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑆) = ((𝐹 ↾ 𝑆) ↾ 𝐴)
9		resres 5943	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝑆) ↾ 𝐴) = (𝐹 ↾ (𝑆 ∩ 𝐴))
10	8, 9	eqtri 2752	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑆) = (𝐹 ↾ (𝑆 ∩ 𝐴))
11		ffn 6652	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶ℂ → 𝐹 Fn 𝐴)
12		fnresdm 6601	. . . . . . . . . 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 5929	. . . . . . . 8 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → ((𝐹 ↾ 𝐴) ↾ 𝑆) = (𝐹 ↾ 𝑆))
15	10, 14	eqtr3id 2778	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → (𝐹 ↾ (𝑆 ∩ 𝐴)) = (𝐹 ↾ 𝑆))
16	15	feq1d 6634	. . . . . 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 4188	. . . . . 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 25800	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → dom (𝑆 D (𝐹 ↾ 𝑆)) ⊆ (𝑆 ∩ 𝐴))
21		dmres 5963	. . . . 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 4171	. . . . 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 2776	. . . 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 3973	. . 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 5973	. . 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 25805	. . . . 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 6656	. . 3 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → Fun (𝑆 D (𝐹 ↾ 𝑆)))
31		ssidd 3959	. . . 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 24668	. . . . 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 22812	. . . . 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 25811	. . . 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 6526	. . 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 2766	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 1540 ∈ wcel 2109 ∩ cin 3902 ⊆ wss 3903 {cpr 4579 dom cdm 5619 ↾ cres 5621 Rel wrel 5624 Fun wfun 6476 Fn wfn 6477 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ℝcr 11008 TopOpenctopn 17325 ℂ_fldccnfld 21261 TopOnctopon 22795 D cdv 25762

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-map 8755 df-pm 8756 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fi 9301 df-sup 9332 df-inf 9333 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-icc 13255 df-fz 13411 df-seq 13909 df-exp 13969 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-struct 17058 df-slot 17093 df-ndx 17105 df-base 17121 df-plusg 17174 df-mulr 17175 df-starv 17176 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-rest 17326 df-topn 17327 df-topgen 17347 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cnp 23113 df-haus 23200 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-limc 25765 df-dv 25766

This theorem is referenced by: dvnres 25831 dvmptres3 25858

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