dvres3a - Metamath Proof Explorer

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Theorem dvres3a 25822

Description: Restriction of a complex differentiable function to the reals. This version of dvres3 25821 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 25778	. . 3 ⊢ Rel (𝑆 D (𝐹 ↾ 𝑆))
2		recnprss 25812	. . . . . 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 4204	. . . . . . 7 ⊢ (𝑆 ∩ 𝐴) ⊆ 𝐴
6		fssres 6729	. . . . . . 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 5976	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑆) = ((𝐹 ↾ 𝑆) ↾ 𝐴)
9		resres 5966	. . . . . . . . 9 ⊢ ((𝐹 ↾ 𝑆) ↾ 𝐴) = (𝐹 ↾ (𝑆 ∩ 𝐴))
10	8, 9	eqtri 2753	. . . . . . . 8 ⊢ ((𝐹 ↾ 𝐴) ↾ 𝑆) = (𝐹 ↾ (𝑆 ∩ 𝐴))
11		ffn 6691	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶ℂ → 𝐹 Fn 𝐴)
12		fnresdm 6640	. . . . . . . . . 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 5952	. . . . . . . 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 6673	. . . . . 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 4203	. . . . . 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 25809	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → dom (𝑆 D (𝐹 ↾ 𝑆)) ⊆ (𝑆 ∩ 𝐴))
21		dmres 5986	. . . . 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 4186	. . . . 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 3987	. . 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 5996	. . 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 25814	. . . . 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 6695	. . 3 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹:𝐴⟶ℂ) ∧ (𝐴 ∈ 𝐽 ∧ dom (ℂ D 𝐹) = 𝐴)) → Fun (𝑆 D (𝐹 ↾ 𝑆)))
31		ssidd 3973	. . . 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 24677	. . . . 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 22821	. . . . 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 25820	. . . 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 6563	. . 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 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 395 = wceq 1540 ∈ wcel 2109 ∩ cin 3916 ⊆ wss 3917 {cpr 4594 dom cdm 5641 ↾ cres 5643 Rel wrel 5646 Fun wfun 6508 Fn wfn 6509 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 ℝcr 11074 TopOpenctopn 17391 ℂ_fldccnfld 21271 TopOnctopon 22804 D cdv 25771

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fi 9369 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-icc 13320 df-fz 13476 df-seq 13974 df-exp 14034 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17124 df-slot 17159 df-ndx 17171 df-base 17187 df-plusg 17240 df-mulr 17241 df-starv 17242 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-rest 17392 df-topn 17393 df-topgen 17413 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-perf 23031 df-cnp 23122 df-haus 23209 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-limc 25774 df-dv 25775

This theorem is referenced by: dvnres 25840 dvmptres3 25867

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