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Mirrors > Home > MPE Home > Th. List > dvmptres | Structured version Visualization version GIF version |
Description: Function-builder for derivative: restrict a derivative to an open subset. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
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)) |
dvmptres.y | ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
dvmptres.j | ⊢ 𝐽 = (𝐾 ↾t 𝑆) |
dvmptres.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
dvmptres.t | ⊢ (𝜑 → 𝑌 ∈ 𝐽) |
Ref | Expression |
---|---|
dvmptres | ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑌 ↦ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvmptadd.s | . 2 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | dvmptadd.a | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) | |
3 | dvmptadd.b | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) | |
4 | dvmptadd.da | . 2 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) | |
5 | dvmptres.y | . 2 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) | |
6 | dvmptres.j | . 2 ⊢ 𝐽 = (𝐾 ↾t 𝑆) | |
7 | dvmptres.k | . 2 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
8 | 7 | cnfldtop 23958 | . . . . 5 ⊢ 𝐾 ∈ Top |
9 | resttop 22322 | . . . . 5 ⊢ ((𝐾 ∈ Top ∧ 𝑆 ∈ {ℝ, ℂ}) → (𝐾 ↾t 𝑆) ∈ Top) | |
10 | 8, 1, 9 | sylancr 587 | . . . 4 ⊢ (𝜑 → (𝐾 ↾t 𝑆) ∈ Top) |
11 | 6, 10 | eqeltrid 2845 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) |
12 | dvmptres.t | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐽) | |
13 | isopn3i 22244 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ 𝐽) → ((int‘𝐽)‘𝑌) = 𝑌) | |
14 | 11, 12, 13 | syl2anc 584 | . 2 ⊢ (𝜑 → ((int‘𝐽)‘𝑌) = 𝑌) |
15 | 1, 2, 3, 4, 5, 6, 7, 14 | dvmptres2 25137 | 1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑌 ↦ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ⊆ wss 3892 {cpr 4569 ↦ cmpt 5162 ‘cfv 6432 (class class class)co 7272 ℂcc 10880 ℝcr 10881 ↾t crest 17142 TopOpenctopn 17143 ℂfldccnfld 20608 Topctop 22053 intcnt 22179 D cdv 25038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 ax-pre-sup 10960 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-1st 7825 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-1o 8289 df-er 8490 df-map 8609 df-pm 8610 df-en 8726 df-dom 8727 df-sdom 8728 df-fin 8729 df-fi 9158 df-sup 9189 df-inf 9190 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-div 11644 df-nn 11985 df-2 12047 df-3 12048 df-4 12049 df-5 12050 df-6 12051 df-7 12052 df-8 12053 df-9 12054 df-n0 12245 df-z 12331 df-dec 12449 df-uz 12594 df-q 12700 df-rp 12742 df-xneg 12859 df-xadd 12860 df-xmul 12861 df-fz 13251 df-seq 13733 df-exp 13794 df-cj 14821 df-re 14822 df-im 14823 df-sqrt 14957 df-abs 14958 df-struct 16859 df-slot 16894 df-ndx 16906 df-base 16924 df-plusg 16986 df-mulr 16987 df-starv 16988 df-tset 16992 df-ple 16993 df-ds 16995 df-unif 16996 df-rest 17144 df-topn 17145 df-topgen 17165 df-psmet 20600 df-xmet 20601 df-met 20602 df-bl 20603 df-mopn 20604 df-cnfld 20609 df-top 22054 df-topon 22071 df-topsp 22093 df-bases 22107 df-cld 22181 df-ntr 22182 df-cls 22183 df-cnp 22390 df-xms 23484 df-ms 23485 df-limc 25041 df-dv 25042 |
This theorem is referenced by: dvmptfsum 25150 dvexp3 25153 dvlipcn 25169 dvivthlem1 25183 lhop2 25190 dvfsumle 25196 dvfsumabs 25198 dvfsumlem2 25202 taylthlem2 25544 pserdvlem2 25598 advlog 25820 advlogexp 25821 logtayl 25826 loglesqrt 25922 dvatan 26096 log2sumbnd 26703 dvtan 35836 dvasin 35870 dvacos 35871 areacirclem1 35874 aks4d1p1p6 40090 dvmptconst 43438 dvmptidg 43440 itgsin0pilem1 43473 itgsbtaddcnst 43505 fourierdlem56 43685 fourierdlem60 43689 fourierdlem61 43690 fourierdlem62 43691 |
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