dvmptres3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvmptres3		Structured version Visualization version GIF version

Theorem dvmptres3 25914

Description: Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)

**Hypotheses**
Ref	Expression
dvmptres3.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)
dvmptres3.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvmptres3.x	⊢ (𝜑 → 𝑋 ∈ 𝐽)
dvmptres3.y	⊢ (𝜑 → (𝑆 ∩ 𝑋) = 𝑌)
dvmptres3.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
dvmptres3.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
dvmptres3.d	⊢ (𝜑 → (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))

**Assertion**
Ref	Expression
dvmptres3	⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑌 ↦ 𝐵))

Distinct variable groups: 𝜑,𝑥 𝑥,𝑋 𝑥,𝑌 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥) 𝑆(𝑥) 𝐽(𝑥) 𝑉(𝑥)

**Proof of Theorem dvmptres3**
Step	Hyp	Ref	Expression
1		dvmptres3.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		dvmptres3.a	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
3	2	fmpttd 7058	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
4		dvmptres3.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
5		dvmptres3.d	. . . . 5 ⊢ (𝜑 → (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
6	5	dmeqd 5852	. . . 4 ⊢ (𝜑 → dom (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = dom (𝑥 ∈ 𝑋 ↦ 𝐵))
7		eqid 2734	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
8		dvmptres3.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
9	7, 8	dmmptd 6635	. . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 𝐵) = 𝑋)
10	6, 9	eqtrd 2769	. . 3 ⊢ (𝜑 → dom (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = 𝑋)
11		dvmptres3.j	. . . 4 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
12	11	dvres3a 25869	. . 3 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) ∧ (𝑋 ∈ 𝐽 ∧ dom (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = 𝑋)) → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆)) = ((ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ 𝑆))
13	1, 3, 4, 10, 12	syl22anc 838	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆)) = ((ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ 𝑆))
14		rescom 5959	. . . . . 6 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆) ↾ 𝑋)
15		resres 5949	. . . . . 6 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆) ↾ 𝑋) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ (𝑆 ∩ 𝑋))
16	14, 15	eqtri 2757	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ (𝑆 ∩ 𝑋))
17		dvmptres3.y	. . . . . 6 ⊢ (𝜑 → (𝑆 ∩ 𝑋) = 𝑌)
18	17	reseq2d 5936	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ (𝑆 ∩ 𝑋)) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌))
19	16, 18	eqtrid 2781	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌))
20		ffn 6660	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ → (𝑥 ∈ 𝑋 ↦ 𝐴) Fn 𝑋)
21		fnresdm 6609	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) Fn 𝑋 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐴))
22	3, 20, 21	3syl 18	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐴))
23	22	reseq1d 5935	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆))
24		inss2 4188	. . . . . 6 ⊢ (𝑆 ∩ 𝑋) ⊆ 𝑋
25	17, 24	eqsstrrdi 3977	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
26	25	resmptd 5997	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐴))
27	19, 23, 26	3eqtr3d 2777	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆) = (𝑥 ∈ 𝑌 ↦ 𝐴))
28	27	oveq2d 7372	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)))
29		rescom 5959	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑆) ↾ 𝑋)
30		resres 5949	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑆) ↾ 𝑋) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ (𝑆 ∩ 𝑋))
31	29, 30	eqtri 2757	. . . 4 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ (𝑆 ∩ 𝑋))
32	17	reseq2d 5936	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ (𝑆 ∩ 𝑋)) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌))
33	31, 32	eqtrid 2781	. . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌))
34	8	ralrimiva 3126	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉)
35	7	fnmpt 6630	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝑋 ↦ 𝐵) Fn 𝑋)
36		fnresdm 6609	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐵) Fn 𝑋 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐵))
37	34, 35, 36	3syl 18	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐵))
38	37, 5	eqtr4d 2772	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) = (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)))
39	38	reseq1d 5935	. . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = ((ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ 𝑆))
40	25	resmptd 5997	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐵))
41	33, 39, 40	3eqtr3d 2777	. 2 ⊢ (𝜑 → ((ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ 𝑆) = (𝑥 ∈ 𝑌 ↦ 𝐵))
42	13, 28, 41	3eqtr3d 2777	1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑌 ↦ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3049 ∩ cin 3898 {cpr 4580 ↦ cmpt 5177 dom cdm 5622 ↾ cres 5624 Fn wfn 6485 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 ℂcc 11022 ℝcr 11023 TopOpenctopn 17339 ℂ_fldccnfld 21307 D cdv 25818

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 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

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 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8763 df-pm 8764 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fi 9312 df-sup 9343 df-inf 9344 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-icc 13266 df-fz 13422 df-seq 13923 df-exp 13983 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-struct 17072 df-slot 17107 df-ndx 17119 df-base 17135 df-plusg 17188 df-mulr 17189 df-starv 17190 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-rest 17340 df-topn 17341 df-topgen 17361 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-fbas 21304 df-fg 21305 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cld 22961 df-ntr 22962 df-cls 22963 df-nei 23040 df-lp 23078 df-perf 23079 df-cnp 23170 df-haus 23257 df-fil 23788 df-fm 23880 df-flim 23881 df-flf 23882 df-xms 24262 df-ms 24263 df-limc 25821 df-dv 25822

This theorem is referenced by: dvmptid 25915 dvmptc 25916 taylthlem1 26335 taylthlem2 26336 taylthlem2OLD 26337 pige3ALT 26483 dvcxp1 26703 dvreasin 37846 dvreacos 37847 areacirclem1 37848 readvrec2 42558 readvcot 42561

W3C validator