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Mirrors > Home > MPE Home > Th. List > dvmptntr | Structured version Visualization version GIF version |
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.) |
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‘ℂfld) |
dvmptntr.i | ⊢ (𝜑 → ((int‘𝐽)‘𝑋) = 𝑌) |
Ref | Expression |
---|---|
dvmptntr | ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvmptntr.j | . . . . . . . . 9 ⊢ 𝐽 = (𝐾 ↾t 𝑆) | |
2 | dvmptntr.k | . . . . . . . . . . 11 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
3 | 2 | cnfldtopon 23394 | . . . . . . . . . 10 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
4 | dvmptntr.s | . . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ ℂ) | |
5 | resttopon 21772 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐾 ↾t 𝑆) ∈ (TopOn‘𝑆)) | |
6 | 3, 4, 5 | sylancr 589 | . . . . . . . . 9 ⊢ (𝜑 → (𝐾 ↾t 𝑆) ∈ (TopOn‘𝑆)) |
7 | 1, 6 | eqeltrid 2920 | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑆)) |
8 | topontop 21524 | . . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝐽 ∈ Top) | |
9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top) |
10 | dvmptntr.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
11 | toponuni 21525 | . . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐽) | |
12 | 7, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 = ∪ 𝐽) |
13 | 10, 12 | sseqtrd 4010 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ ∪ 𝐽) |
14 | eqid 2824 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
15 | 14 | ntridm 21679 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ∪ 𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑋)) |
16 | 9, 13, 15 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑋)) |
17 | dvmptntr.i | . . . . . . 7 ⊢ (𝜑 → ((int‘𝐽)‘𝑋) = 𝑌) | |
18 | 17 | fveq2d 6677 | . . . . . 6 ⊢ (𝜑 → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑌)) |
19 | 16, 18 | eqtr3d 2861 | . . . . 5 ⊢ (𝜑 → ((int‘𝐽)‘𝑋) = ((int‘𝐽)‘𝑌)) |
20 | 19 | reseq2d 5856 | . . . 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 6882 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) |
23 | 2, 1 | dvres 24512 | . . . . 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 836 | . . . 4 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑋))) |
25 | 14 | ntrss2 21668 | . . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑋) ⊆ 𝑋) |
26 | 9, 13, 25 | syl2anc 586 | . . . . . . 7 ⊢ (𝜑 → ((int‘𝐽)‘𝑋) ⊆ 𝑋) |
27 | 17, 26 | eqsstrrd 4009 | . . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
28 | 27, 10 | sstrd 3980 | . . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑆) |
29 | 2, 1 | dvres 24512 | . . . . 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 836 | . . . 4 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑌))) |
31 | 20, 24, 30 | 3eqtr4d 2869 | . . 3 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋)) = (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌))) |
32 | ssid 3992 | . . . . 5 ⊢ 𝑋 ⊆ 𝑋 | |
33 | resmpt 5908 | . . . . 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 7175 | . . 3 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴))) |
36 | 31, 35 | eqtr3d 2861 | . 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴))) |
37 | 27 | resmptd 5911 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐴)) |
38 | 37 | oveq2d 7175 | . 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴))) |
39 | 36, 38 | eqtr3d 2861 | 1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 ∪ cuni 4841 ↦ cmpt 5149 ↾ cres 5560 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ↾t crest 16697 TopOpenctopn 16698 ℂfldccnfld 20548 Topctop 21504 TopOnctopon 21521 intcnt 21628 D cdv 24464 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fi 8878 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-plusg 16581 df-mulr 16582 df-starv 16583 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-rest 16699 df-topn 16700 df-topgen 16720 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cld 21630 df-ntr 21631 df-cls 21632 df-cnp 21839 df-xms 22933 df-ms 22934 df-limc 24467 df-dv 24468 |
This theorem is referenced by: rolle 24590 cmvth 24591 dvlip 24593 dvlipcn 24594 dvle 24607 dvfsumabs 24623 ftc2 24644 itgparts 24647 itgsubstlem 24648 lgamgulmlem2 25610 ftc2nc 34980 areacirc 34991 itgsin0pilem1 42241 itgsbtaddcnst 42273 |
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