dvmptntr - Metamath Proof Explorer

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Theorem dvmptntr 25891

Description: Function-builder for derivative: expand the function from an open set to its closure. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

**Hypotheses**
Ref	Expression
dvmptntr.s	⊢ (𝜑 → 𝑆 ⊆ ℂ)
dvmptntr.x	⊢ (𝜑 → 𝑋 ⊆ 𝑆)
dvmptntr.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
dvmptntr.j	⊢ 𝐽 = (𝐾 ↾_t 𝑆)
dvmptntr.k	⊢ 𝐾 = (TopOpen‘ℂ_fld)
dvmptntr.i	⊢ (𝜑 → ((int‘𝐽)‘𝑋) = 𝑌)

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

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

**Proof of Theorem dvmptntr**
Step	Hyp	Ref	Expression
1		dvmptntr.j	. . . . . . . . 9 ⊢ 𝐽 = (𝐾 ↾_t 𝑆)
2		dvmptntr.k	. . . . . . . . . . 11 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
3	2	cnfldtopon 24686	. . . . . . . . . 10 ⊢ 𝐾 ∈ (TopOn‘ℂ)
4		dvmptntr.s	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
5		resttopon 23064	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆))
6	3, 4, 5	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆))
7	1, 6	eqeltrid 2832	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑆))
8		topontop 22816	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝐽 ∈ Top)
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top)
10		dvmptntr.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
11		toponuni 22817	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐽)
12	7, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = ∪ 𝐽)
13	10, 12	sseqtrd 3974	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ ∪ 𝐽)
14		eqid 2729	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
15	14	ntridm 22971	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ∪ 𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑋))
16	9, 13, 15	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑋))
17		dvmptntr.i	. . . . . . 7 ⊢ (𝜑 → ((int‘𝐽)‘𝑋) = 𝑌)
18	17	fveq2d 6830	. . . . . 6 ⊢ (𝜑 → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑌))
19	16, 18	eqtr3d 2766	. . . . 5 ⊢ (𝜑 → ((int‘𝐽)‘𝑋) = ((int‘𝐽)‘𝑌))
20	19	reseq2d 5934	. . . 4 ⊢ (𝜑 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑋)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑌)))
21		dvmptntr.a	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
22	21	fmpttd 7053	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
23	2, 1	dvres 25828	. . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) ∧ (𝑋 ⊆ 𝑆 ∧ 𝑋 ⊆ 𝑆)) → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑋)))
24	4, 22, 10, 10, 23	syl22anc 838	. . . 4 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑋)))
25	14	ntrss2 22960	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑋) ⊆ 𝑋)
26	9, 13, 25	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((int‘𝐽)‘𝑋) ⊆ 𝑋)
27	17, 26	eqsstrrd 3973	. . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
28	27, 10	sstrd 3948	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑆)
29	2, 1	dvres 25828	. . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) ∧ (𝑋 ⊆ 𝑆 ∧ 𝑌 ⊆ 𝑆)) → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑌)))
30	4, 22, 10, 28, 29	syl22anc 838	. . . 4 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑌)))
31	20, 24, 30	3eqtr4d 2774	. . 3 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋)) = (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)))
32		ssid 3960	. . . . 5 ⊢ 𝑋 ⊆ 𝑋
33		resmpt 5992	. . . . 5 ⊢ (𝑋 ⊆ 𝑋 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐴))
34	32, 33	mp1i 13	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐴))
35	34	oveq2d 7369	. . 3 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)))
36	31, 35	eqtr3d 2766	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)))
37	27	resmptd 5995	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐴))
38	37	oveq2d 7369	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)))
39	36, 38	eqtr3d 2766	1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ⊆ wss 3905 ∪ cuni 4861 ↦ cmpt 5176 ↾ cres 5625 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 ℂcc 11026 ↾_t crest 17342 TopOpenctopn 17343 ℂ_fldccnfld 21279 Topctop 22796 TopOnctopon 22813 intcnt 22920 D cdv 25780

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 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

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 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 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 8632 df-map 8762 df-pm 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fi 9320 df-sup 9351 df-inf 9352 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-fz 13429 df-seq 13927 df-exp 13987 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17139 df-plusg 17192 df-mulr 17193 df-starv 17194 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-rest 17344 df-topn 17345 df-topgen 17365 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-cnfld 21280 df-top 22797 df-topon 22814 df-topsp 22836 df-bases 22849 df-cld 22922 df-ntr 22923 df-cls 22924 df-cnp 23131 df-xms 24224 df-ms 24225 df-limc 25783 df-dv 25784

This theorem is referenced by: rolle 25910 cmvth 25911 cmvthOLD 25912 dvlip 25914 dvlipcn 25915 dvle 25928 dvfsumabs 25945 ftc2 25967 itgparts 25970 itgsubstlem 25971 lgamgulmlem2 26956 ftc2nc 37684 areacirc 37695 itgsin0pilem1 45935 itgsbtaddcnst 45967

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