dvmptres2 - Metamath Proof Explorer

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Theorem dvmptres2 25864

Description: Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

**Hypotheses**
Ref	Expression
dvmptadd.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvmptadd.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
dvmptadd.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
dvmptadd.da	⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
dvmptres2.z	⊢ (𝜑 → 𝑍 ⊆ 𝑋)
dvmptres2.j	⊢ 𝐽 = (𝐾 ↾_t 𝑆)
dvmptres2.k	⊢ 𝐾 = (TopOpen‘ℂ_fld)
dvmptres2.i	⊢ (𝜑 → ((int‘𝐽)‘𝑍) = 𝑌)

**Assertion**
Ref	Expression
dvmptres2	⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑍 ↦ 𝐴)) = (𝑥 ∈ 𝑌 ↦ 𝐵))

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

**Proof of Theorem dvmptres2**
Step	Hyp	Ref	Expression
1		dvmptadd.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		recnprss 25803	. . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
4		dvmptadd.a	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
5	4	fmpttd 7049	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
6		dvmptadd.da	. . . . . 6 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
7	6	dmeqd 5848	. . . . 5 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = dom (𝑥 ∈ 𝑋 ↦ 𝐵))
8		dvmptadd.b	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
9	8	ralrimiva 3121	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉)
10		dmmptg 6191	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉 → dom (𝑥 ∈ 𝑋 ↦ 𝐵) = 𝑋)
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 𝐵) = 𝑋)
12	7, 11	eqtrd 2764	. . . 4 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = 𝑋)
13		dvbsss 25801	. . . 4 ⊢ dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ⊆ 𝑆
14	12, 13	eqsstrrdi 3981	. . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
15		dvmptres2.z	. . . 4 ⊢ (𝜑 → 𝑍 ⊆ 𝑋)
16	15, 14	sstrd 3946	. . 3 ⊢ (𝜑 → 𝑍 ⊆ 𝑆)
17		dvmptres2.k	. . . 4 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
18		dvmptres2.j	. . . 4 ⊢ 𝐽 = (𝐾 ↾_t 𝑆)
19	17, 18	dvres 25810	. . 3 ⊢ (((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) ∧ (𝑋 ⊆ 𝑆 ∧ 𝑍 ⊆ 𝑆)) → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑍)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑍)))
20	3, 5, 14, 16, 19	syl22anc 838	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑍)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑍)))
21	15	resmptd 5991	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑍) = (𝑥 ∈ 𝑍 ↦ 𝐴))
22	21	oveq2d 7365	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑍)) = (𝑆 D (𝑥 ∈ 𝑍 ↦ 𝐴)))
23	6	reseq1d 5929	. . 3 ⊢ (𝜑 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑍)) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ ((int‘𝐽)‘𝑍)))
24		dvmptres2.i	. . . 4 ⊢ (𝜑 → ((int‘𝐽)‘𝑍) = 𝑌)
25	24	reseq2d 5930	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ ((int‘𝐽)‘𝑍)) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌))
26	17	cnfldtopon 24668	. . . . . . . . . 10 ⊢ 𝐾 ∈ (TopOn‘ℂ)
27		resttopon 23046	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆))
28	26, 3, 27	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆))
29	18, 28	eqeltrid 2832	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑆))
30		topontop 22798	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝐽 ∈ Top)
31	29, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top)
32		toponuni 22799	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐽)
33	29, 32	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = ∪ 𝐽)
34	16, 33	sseqtrd 3972	. . . . . . 7 ⊢ (𝜑 → 𝑍 ⊆ ∪ 𝐽)
35		eqid 2729	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
36	35	ntrss2 22942	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑍 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑍) ⊆ 𝑍)
37	31, 34, 36	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((int‘𝐽)‘𝑍) ⊆ 𝑍)
38	24, 37	eqsstrrd 3971	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑍)
39	38, 15	sstrd 3946	. . . 4 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
40	39	resmptd 5991	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐵))
41	23, 25, 40	3eqtrd 2768	. 2 ⊢ (𝜑 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑍)) = (𝑥 ∈ 𝑌 ↦ 𝐵))
42	20, 22, 41	3eqtr3d 2772	1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑍 ↦ 𝐴)) = (𝑥 ∈ 𝑌 ↦ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ⊆ wss 3903 {cpr 4579 ∪ cuni 4858 ↦ cmpt 5173 dom cdm 5619 ↾ cres 5621 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ℝcr 11008 ↾_t crest 17324 TopOpenctopn 17325 ℂ_fldccnfld 21261 Topctop 22778 TopOnctopon 22795 intcnt 22902 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-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-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-cnp 23113 df-xms 24206 df-ms 24207 df-limc 25765 df-dv 25766

This theorem is referenced by: dvmptres 25865 dvmptcmul 25866 rolle 25892 mvth 25895 itgpowd 25955 taylthlem1 26279 pige3ALT 26427 logccv 26570 lgamgulmlem2 26938 dvrelog2 42037 dvrelog3 42038 lhe4.4ex1a 44302 binomcxplemdvbinom 44326 binomcxplemnotnn0 44329 itgsinexplem1 45935 dirkeritg 46083 fourierdlem39 46127 etransclem46 46261

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