dvmptres3 - Metamath Proof Explorer

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Theorem dvmptres3 25867

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 7090	. . 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 5872	. . . 4 ⊢ (𝜑 → dom (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = dom (𝑥 ∈ 𝑋 ↦ 𝐵))
7		eqid 2730	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
8		dvmptres3.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
9	7, 8	dmmptd 6666	. . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 𝐵) = 𝑋)
10	6, 9	eqtrd 2765	. . 3 ⊢ (𝜑 → dom (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = 𝑋)
11		dvmptres3.j	. . . 4 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
12	11	dvres3a 25822	. . 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 5976	. . . . . 6 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆) ↾ 𝑋)
15		resres 5966	. . . . . 6 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆) ↾ 𝑋) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ (𝑆 ∩ 𝑋))
16	14, 15	eqtri 2753	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ (𝑆 ∩ 𝑋))
17		dvmptres3.y	. . . . . 6 ⊢ (𝜑 → (𝑆 ∩ 𝑋) = 𝑌)
18	17	reseq2d 5953	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ (𝑆 ∩ 𝑋)) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌))
19	16, 18	eqtrid 2777	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌))
20		ffn 6691	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ → (𝑥 ∈ 𝑋 ↦ 𝐴) Fn 𝑋)
21		fnresdm 6640	. . . . . 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 5952	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆))
24		inss2 4204	. . . . . 6 ⊢ (𝑆 ∩ 𝑋) ⊆ 𝑋
25	17, 24	eqsstrrdi 3995	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
26	25	resmptd 6014	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐴))
27	19, 23, 26	3eqtr3d 2773	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆) = (𝑥 ∈ 𝑌 ↦ 𝐴))
28	27	oveq2d 7406	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)))
29		rescom 5976	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑆) ↾ 𝑋)
30		resres 5966	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑆) ↾ 𝑋) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ (𝑆 ∩ 𝑋))
31	29, 30	eqtri 2753	. . . 4 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ (𝑆 ∩ 𝑋))
32	17	reseq2d 5953	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ (𝑆 ∩ 𝑋)) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌))
33	31, 32	eqtrid 2777	. . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌))
34	8	ralrimiva 3126	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉)
35	7	fnmpt 6661	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝑋 ↦ 𝐵) Fn 𝑋)
36		fnresdm 6640	. . . . . 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 2768	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) = (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)))
39	38	reseq1d 5952	. . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = ((ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ 𝑆))
40	25	resmptd 6014	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐵))
41	33, 39, 40	3eqtr3d 2773	. 2 ⊢ (𝜑 → ((ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ 𝑆) = (𝑥 ∈ 𝑌 ↦ 𝐵))
42	13, 28, 41	3eqtr3d 2773	1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑌 ↦ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3045 ∩ cin 3916 {cpr 4594 ↦ cmpt 5191 dom cdm 5641 ↾ cres 5643 Fn wfn 6509 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 ℝcr 11074 TopOpenctopn 17391 ℂ_fldccnfld 21271 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: dvmptid 25868 dvmptc 25869 taylthlem1 26288 taylthlem2 26289 taylthlem2OLD 26290 pige3ALT 26436 dvcxp1 26656 dvreasin 37707 dvreacos 37708 areacirclem1 37709 readvrec2 42356 readvcot 42359

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