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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 2740 | . 2 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
2 | dvmptid.1 | . 2 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
3 | 1 | cnfldtopon 23944 | . . 3 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
4 | toponmax 22073 | . . 3 ⊢ ((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) → ℂ ∈ (TopOpen‘ℂfld)) | |
5 | 3, 4 | mp1i 13 | . 2 ⊢ (𝜑 → ℂ ∈ (TopOpen‘ℂfld)) |
6 | recnprss 25066 | . . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
7 | 2, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
8 | df-ss 3909 | . . 3 ⊢ (𝑆 ⊆ ℂ ↔ (𝑆 ∩ ℂ) = 𝑆) | |
9 | 7, 8 | sylib 217 | . 2 ⊢ (𝜑 → (𝑆 ∩ ℂ) = 𝑆) |
10 | simpr 485 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ) | |
11 | 1cnd 10971 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 1 ∈ ℂ) | |
12 | mptresid 5957 | . . . . . 6 ⊢ ( I ↾ ℂ) = (𝑥 ∈ ℂ ↦ 𝑥) | |
13 | 12 | eqcomi 2749 | . . . . 5 ⊢ (𝑥 ∈ ℂ ↦ 𝑥) = ( I ↾ ℂ) |
14 | 13 | oveq2i 7282 | . . . 4 ⊢ (ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (ℂ D ( I ↾ ℂ)) |
15 | dvid 25080 | . . . 4 ⊢ (ℂ D ( I ↾ ℂ)) = (ℂ × {1}) | |
16 | fconstmpt 5650 | . . . 4 ⊢ (ℂ × {1}) = (𝑥 ∈ ℂ ↦ 1) | |
17 | 14, 15, 16 | 3eqtri 2772 | . . 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 25118 | 1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑆 ↦ 𝑥)) = (𝑥 ∈ 𝑆 ↦ 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∩ cin 3891 ⊆ wss 3892 {csn 4567 {cpr 4569 ↦ cmpt 5162 I cid 5489 × cxp 5588 ↾ cres 5592 ‘cfv 6432 (class class class)co 7271 ℂcc 10870 ℝcr 10871 1c1 10873 TopOpenctopn 17130 ℂfldccnfld 20595 TopOnctopon 22057 D cdv 25025 |
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 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 |
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 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-pm 8601 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fi 9148 df-sup 9179 df-inf 9180 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-icc 13085 df-fz 13239 df-seq 13720 df-exp 13781 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-struct 16846 df-slot 16881 df-ndx 16893 df-base 16911 df-plusg 16973 df-mulr 16974 df-starv 16975 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-rest 17131 df-topn 17132 df-topgen 17152 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-fbas 20592 df-fg 20593 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cld 22168 df-ntr 22169 df-cls 22170 df-nei 22247 df-lp 22285 df-perf 22286 df-cn 22376 df-cnp 22377 df-haus 22464 df-fil 22995 df-fm 23087 df-flim 23088 df-flf 23089 df-xms 23471 df-ms 23472 df-cncf 24039 df-limc 25028 df-dv 25029 |
This theorem is referenced by: dvef 25142 dvsincos 25143 mvth 25154 dvlipcn 25156 dvivthlem1 25170 lhop2 25177 dvfsumle 25183 dvfsumabs 25185 dvfsumlem2 25189 dvtaylp 25527 taylthlem2 25531 pige3ALT 25674 advlog 25807 advlogexp 25808 logtayl 25813 dvcxp1 25891 dvcxp2 25892 dvcncxp1 25894 loglesqrt 25909 dvatan 26083 lgamgulmlem2 26177 log2sumbnd 26690 itgexpif 32582 dvasin 35857 areacirclem1 35861 lcmineqlem7 40040 lcmineqlem12 40045 lhe4.4ex1a 41917 expgrowthi 41921 expgrowth 41923 binomcxplemdvbinom 41941 dvsinax 43425 dvmptidg 43429 dvcosax 43438 itgiccshift 43492 itgperiod 43493 itgsbtaddcnst 43494 dirkeritg 43614 fourierdlem39 43658 fourierdlem56 43674 fourierdlem60 43678 fourierdlem61 43679 fourierdlem62 43680 etransclem46 43792 |
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