dvmptntr - Metamath Proof Explorer

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Theorem dvmptntr 26021

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 24830	. . . . . . . . . 10 ⊢ 𝐾 ∈ (TopOn‘ℂ)
4		dvmptntr.s	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
5		resttopon 23209	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆))
6	3, 4, 5	sylancr 596	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ↾_t 𝑆) ∈ (TopOn‘𝑆))
7	1, 6	eqeltrid 2865	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑆))
8		topontop 22961	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝐽 ∈ Top)
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top)
10		dvmptntr.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
11		toponuni 22962	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐽)
12	7, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = ∪ 𝐽)
13	10, 12	sseqtrd 3970	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ ∪ 𝐽)
14		eqid 2761	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
15	14	ntridm 23116	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ∪ 𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑋))
16	9, 13, 15	syl2anc 593	. . . . . 6 ⊢ (𝜑 → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑋))
17		dvmptntr.i	. . . . . . 7 ⊢ (𝜑 → ((int‘𝐽)‘𝑋) = 𝑌)
18	17	fveq2d 6866	. . . . . 6 ⊢ (𝜑 → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑌))
19	16, 18	eqtr3d 2798	. . . . 5 ⊢ (𝜑 → ((int‘𝐽)‘𝑋) = ((int‘𝐽)‘𝑌))
20	19	reseq2d 5961	. . . 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 7091	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
23	2, 1	dvres 25961	. . . . 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 849	. . . 4 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑋)))
25	14	ntrss2 23105	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑋) ⊆ 𝑋)
26	9, 13, 25	syl2anc 593	. . . . . . 7 ⊢ (𝜑 → ((int‘𝐽)‘𝑋) ⊆ 𝑋)
27	17, 26	eqsstrrd 3969	. . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
28	27, 10	sstrd 3944	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑆)
29	2, 1	dvres 25961	. . . . 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 849	. . . 4 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑌)))
31	20, 24, 30	3eqtr4d 2806	. . 3 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋)) = (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)))
32		ssid 3956	. . . . 5 ⊢ 𝑋 ⊆ 𝑋
33		resmpt 6022	. . . . 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 7407	. . 3 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)))
36	31, 35	eqtr3d 2798	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)))
37	27	resmptd 6025	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐴))
38	37	oveq2d 7407	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)))
39	36, 38	eqtr3d 2798	1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ⊆ wss 3902 ∪ cuni 4862 ↦ cmpt 5178 ↾ cres 5645 ⟶wf 6512 ‘cfv 6516 (class class class)co 7391 ℂcc 11065 ↾_t crest 17440 TopOpenctopn 17441 ℂ_fldccnfld 21412 Topctop 22941 TopOnctopon 22958 intcnt 23065 D cdv 25913

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-er 8672 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fi 9351 df-sup 9382 df-inf 9383 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-z 12563 df-dec 12683 df-uz 12834 df-q 12944 df-rp 12988 df-xneg 13108 df-xadd 13109 df-xmul 13110 df-fz 13507 df-seq 14009 df-exp 14069 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254 df-struct 17174 df-slot 17209 df-ndx 17221 df-base 17237 df-plusg 17290 df-mulr 17291 df-starv 17292 df-tset 17296 df-ple 17297 df-ds 17299 df-unif 17300 df-rest 17442 df-topn 17443 df-topgen 17463 df-psmet 21404 df-xmet 21405 df-met 21406 df-bl 21407 df-mopn 21408 df-cnfld 21413 df-top 22942 df-topon 22959 df-topsp 22981 df-bases 22994 df-cld 23067 df-ntr 23068 df-cls 23069 df-cnp 23276 df-xms 24368 df-ms 24369 df-limc 25916 df-dv 25917

This theorem is referenced by: rolle 26040 cmvth 26041 dvlip 26043 dvlipcn 26044 dvle 26057 dvfsumabs 26073 ftc2 26094 itgparts 26097 itgsubstlem 26098 lgamgulmlem2 27082 ftc2nc 38162 areacirc 38173 itgsin0pilem1 46485 itgsbtaddcnst 46517

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