dvmptres2 - Metamath Proof Explorer

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Theorem dvmptres2 25873

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 25812	. . . 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 7090	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
6		dvmptadd.da	. . . . . 6 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
7	6	dmeqd 5872	. . . . 5 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = dom (𝑥 ∈ 𝑋 ↦ 𝐵))
8		dvmptadd.b	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
9	8	ralrimiva 3126	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉)
10		dmmptg 6218	. . . . . 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 2765	. . . 4 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = 𝑋)
13		dvbsss 25810	. . . 4 ⊢ dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ⊆ 𝑆
14	12, 13	eqsstrrdi 3995	. . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
15		dvmptres2.z	. . . 4 ⊢ (𝜑 → 𝑍 ⊆ 𝑋)
16	15, 14	sstrd 3960	. . 3 ⊢ (𝜑 → 𝑍 ⊆ 𝑆)
17		dvmptres2.k	. . . 4 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
18		dvmptres2.j	. . . 4 ⊢ 𝐽 = (𝐾 ↾_t 𝑆)
19	17, 18	dvres 25819	. . 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 6014	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑍) = (𝑥 ∈ 𝑍 ↦ 𝐴))
22	21	oveq2d 7406	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑍)) = (𝑆 D (𝑥 ∈ 𝑍 ↦ 𝐴)))
23	6	reseq1d 5952	. . 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 5953	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ ((int‘𝐽)‘𝑍)) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌))
26	17	cnfldtopon 24677	. . . . . . . . . 10 ⊢ 𝐾 ∈ (TopOn‘ℂ)
27		resttopon 23055	. . . . . . . . . 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 2833	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑆))
30		topontop 22807	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝐽 ∈ Top)
31	29, 30	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top)
32		toponuni 22808	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐽)
33	29, 32	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = ∪ 𝐽)
34	16, 33	sseqtrd 3986	. . . . . . 7 ⊢ (𝜑 → 𝑍 ⊆ ∪ 𝐽)
35		eqid 2730	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
36	35	ntrss2 22951	. . . . . . 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 3985	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑍)
39	38, 15	sstrd 3960	. . . 4 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
40	39	resmptd 6014	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐵))
41	23, 25, 40	3eqtrd 2769	. 2 ⊢ (𝜑 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑍)) = (𝑥 ∈ 𝑌 ↦ 𝐵))
42	20, 22, 41	3eqtr3d 2773	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 3045 ⊆ wss 3917 {cpr 4594 ∪ cuni 4874 ↦ cmpt 5191 dom cdm 5641 ↾ cres 5643 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 ℝcr 11074 ↾_t crest 17390 TopOpenctopn 17391 ℂ_fldccnfld 21271 Topctop 22787 TopOnctopon 22804 intcnt 22911 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-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-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-cnp 23122 df-xms 24215 df-ms 24216 df-limc 25774 df-dv 25775

This theorem is referenced by: dvmptres 25874 dvmptcmul 25875 rolle 25901 mvth 25904 itgpowd 25964 taylthlem1 26288 pige3ALT 26436 logccv 26579 lgamgulmlem2 26947 dvrelog2 42059 dvrelog3 42060 lhe4.4ex1a 44325 binomcxplemdvbinom 44349 binomcxplemnotnn0 44352 itgsinexplem1 45959 dirkeritg 46107 fourierdlem39 46151 etransclem46 46285

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