![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 24818 | . . . . . . . . . 10 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
4 | dvmptntr.s | . . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ ℂ) | |
5 | resttopon 23184 | . . . . . . . . . 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 2842 | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑆)) |
8 | topontop 22934 | . . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝐽 ∈ Top) | |
9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top) |
10 | dvmptntr.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
11 | toponuni 22935 | . . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐽) | |
12 | 7, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 = ∪ 𝐽) |
13 | 10, 12 | sseqtrd 4035 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ ∪ 𝐽) |
14 | eqid 2734 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
15 | 14 | ntridm 23091 | . . . . . . 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 6910 | . . . . . 6 ⊢ (𝜑 → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑌)) |
19 | 16, 18 | eqtr3d 2776 | . . . . 5 ⊢ (𝜑 → ((int‘𝐽)‘𝑋) = ((int‘𝐽)‘𝑌)) |
20 | 19 | reseq2d 5999 | . . . 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 7134 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) |
23 | 2, 1 | dvres 25960 | . . . . 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 839 | . . . 4 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑋))) |
25 | 14 | ntrss2 23080 | . . . . . . . 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 4034 | . . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
28 | 27, 10 | sstrd 4005 | . . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑆) |
29 | 2, 1 | dvres 25960 | . . . . 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 839 | . . . 4 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑌))) |
31 | 20, 24, 30 | 3eqtr4d 2784 | . . 3 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋)) = (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌))) |
32 | ssid 4017 | . . . . 5 ⊢ 𝑋 ⊆ 𝑋 | |
33 | resmpt 6056 | . . . . 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 7446 | . . 3 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴))) |
36 | 31, 35 | eqtr3d 2776 | . 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴))) |
37 | 27 | resmptd 6059 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐴)) |
38 | 37 | oveq2d 7446 | . 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴))) |
39 | 36, 38 | eqtr3d 2776 | 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 1536 ∈ wcel 2105 ⊆ wss 3962 ∪ cuni 4911 ↦ cmpt 5230 ↾ cres 5690 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ↾t crest 17466 TopOpenctopn 17467 ℂfldccnfld 21381 Topctop 22914 TopOnctopon 22931 intcnt 23040 D cdv 25912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fi 9448 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-fz 13544 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17468 df-topn 17469 df-topgen 17489 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-cnp 23251 df-xms 24345 df-ms 24346 df-limc 25915 df-dv 25916 |
This theorem is referenced by: rolle 26042 cmvth 26043 cmvthOLD 26044 dvlip 26046 dvlipcn 26047 dvle 26060 dvfsumabs 26077 ftc2 26099 itgparts 26102 itgsubstlem 26103 lgamgulmlem2 27087 ftc2nc 37688 areacirc 37699 itgsin0pilem1 45905 itgsbtaddcnst 45937 |
Copyright terms: Public domain | W3C validator |