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Mirrors > Home > MPE Home > Th. List > dvmptid | Structured version Visualization version GIF version |
Description: Function-builder for derivative: derivative of the identity. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
dvmptid.1 | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
Ref | Expression |
---|---|
dvmptid | ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑆 ↦ 𝑥)) = (𝑥 ∈ 𝑆 ↦ 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . 2 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
2 | dvmptid.1 | . 2 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
3 | 1 | cnfldtopon 24743 | . . 3 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
4 | toponmax 22872 | . . 3 ⊢ ((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) → ℂ ∈ (TopOpen‘ℂfld)) | |
5 | 3, 4 | mp1i 13 | . 2 ⊢ (𝜑 → ℂ ∈ (TopOpen‘ℂfld)) |
6 | recnprss 25877 | . . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
7 | 2, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
8 | dfss2 3962 | . . 3 ⊢ (𝑆 ⊆ ℂ ↔ (𝑆 ∩ ℂ) = 𝑆) | |
9 | 7, 8 | sylib 217 | . 2 ⊢ (𝜑 → (𝑆 ∩ ℂ) = 𝑆) |
10 | simpr 483 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ) | |
11 | 1cnd 11241 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ) | |
12 | mptresid 6055 | . . . . . 6 ⊢ ( I ↾ ℂ) = (𝑥 ∈ ℂ ↦ 𝑥) | |
13 | 12 | eqcomi 2734 | . . . . 5 ⊢ (𝑥 ∈ ℂ ↦ 𝑥) = ( I ↾ ℂ) |
14 | 13 | oveq2i 7430 | . . . 4 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (ℂ D ( I ↾ ℂ)) |
15 | dvid 25891 | . . . 4 ⊢ (ℂ D ( I ↾ ℂ)) = (ℂ × {1}) | |
16 | fconstmpt 5740 | . . . 4 ⊢ (ℂ × {1}) = (𝑥 ∈ ℂ ↦ 1) | |
17 | 14, 15, 16 | 3eqtri 2757 | . . 3 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (𝑥 ∈ ℂ ↦ 1) |
18 | 17 | a1i 11 | . 2 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (𝑥 ∈ ℂ ↦ 1)) |
19 | 1, 2, 5, 9, 10, 11, 18 | dvmptres3 25932 | 1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑆 ↦ 𝑥)) = (𝑥 ∈ 𝑆 ↦ 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∩ cin 3943 ⊆ wss 3944 {csn 4630 {cpr 4632 ↦ cmpt 5232 I cid 5575 × cxp 5676 ↾ cres 5680 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 ℝcr 11139 1c1 11141 TopOpenctopn 17406 ℂfldccnfld 21296 TopOnctopon 22856 D cdv 25836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9436 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-icc 13366 df-fz 13520 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-struct 17119 df-slot 17154 df-ndx 17166 df-base 17184 df-plusg 17249 df-mulr 17250 df-starv 17251 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-rest 17407 df-topn 17408 df-topgen 17428 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24270 df-ms 24271 df-cncf 24842 df-limc 25839 df-dv 25840 |
This theorem is referenced by: dvef 25956 dvsincos 25957 mvth 25969 dvlipcn 25971 dvivthlem1 25985 lhop2 25992 dvfsumle 25998 dvfsumleOLD 25999 dvfsumabs 26001 dvfsumlem2 26005 dvfsumlem2OLD 26006 dvtaylp 26350 taylthlem2 26354 taylthlem2OLD 26355 pige3ALT 26499 advlog 26633 advlogexp 26634 logtayl 26639 dvcxp1 26719 dvcxp2 26720 dvcncxp1 26722 loglesqrt 26738 dvatan 26912 lgamgulmlem2 27007 log2sumbnd 27522 itgexpif 34369 dvasin 37308 areacirclem1 37312 lcmineqlem7 41638 lcmineqlem12 41643 lhe4.4ex1a 43908 expgrowthi 43912 expgrowth 43914 binomcxplemdvbinom 43932 dvsinax 45439 dvmptidg 45443 dvcosax 45452 itgiccshift 45506 itgperiod 45507 itgsbtaddcnst 45508 dirkeritg 45628 fourierdlem39 45672 fourierdlem56 45688 fourierdlem60 45692 fourierdlem61 45693 fourierdlem62 45694 etransclem46 45806 |
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