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Mirrors > Home > MPE Home > Th. List > dvmptc | Structured version Visualization version GIF version |
Description: Function-builder for derivative: derivative of a constant. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
dvmptid.1 | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
dvmptc.2 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
dvmptc | ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2724 | . 2 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
2 | dvmptid.1 | . 2 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
3 | 1 | cnfldtopon 24623 | . . 3 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
4 | toponmax 22752 | . . 3 ⊢ ((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) → ℂ ∈ (TopOpen‘ℂfld)) | |
5 | 3, 4 | mp1i 13 | . 2 ⊢ (𝜑 → ℂ ∈ (TopOpen‘ℂfld)) |
6 | recnprss 25757 | . . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
7 | 2, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
8 | df-ss 3958 | . . 3 ⊢ (𝑆 ⊆ ℂ ↔ (𝑆 ∩ ℂ) = 𝑆) | |
9 | 7, 8 | sylib 217 | . 2 ⊢ (𝜑 → (𝑆 ∩ ℂ) = 𝑆) |
10 | dvmptc.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
11 | 10 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ) |
12 | 0cnd 11205 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 0 ∈ ℂ) | |
13 | dvconst 25770 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℂ D (ℂ × {𝐴})) = (ℂ × {0})) | |
14 | 10, 13 | syl 17 | . . 3 ⊢ (𝜑 → (ℂ D (ℂ × {𝐴})) = (ℂ × {0})) |
15 | fconstmpt 5729 | . . . 4 ⊢ (ℂ × {𝐴}) = (𝑥 ∈ ℂ ↦ 𝐴) | |
16 | 15 | oveq2i 7413 | . . 3 ⊢ (ℂ D (ℂ × {𝐴})) = (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) |
17 | fconstmpt 5729 | . . 3 ⊢ (ℂ × {0}) = (𝑥 ∈ ℂ ↦ 0) | |
18 | 14, 16, 17 | 3eqtr3g 2787 | . 2 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 0)) |
19 | 1, 2, 5, 9, 11, 12, 18 | dvmptres3 25812 | 1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∩ cin 3940 ⊆ wss 3941 {csn 4621 {cpr 4623 ↦ cmpt 5222 × cxp 5665 ‘cfv 6534 (class class class)co 7402 ℂcc 11105 ℝcr 11106 0cc0 11107 TopOpenctopn 17368 ℂfldccnfld 21230 TopOnctopon 22736 D cdv 25716 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fi 9403 df-sup 9434 df-inf 9435 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-uz 12821 df-q 12931 df-rp 12973 df-xneg 13090 df-xadd 13091 df-xmul 13092 df-icc 13329 df-fz 13483 df-seq 13965 df-exp 14026 df-cj 15044 df-re 15045 df-im 15046 df-sqrt 15180 df-abs 15181 df-struct 17081 df-slot 17116 df-ndx 17128 df-base 17146 df-plusg 17211 df-mulr 17212 df-starv 17213 df-tset 17217 df-ple 17218 df-ds 17220 df-unif 17221 df-rest 17369 df-topn 17370 df-topgen 17390 df-psmet 21222 df-xmet 21223 df-met 21224 df-bl 21225 df-mopn 21226 df-fbas 21227 df-fg 21228 df-cnfld 21231 df-top 22720 df-topon 22737 df-topsp 22759 df-bases 22773 df-cld 22847 df-ntr 22848 df-cls 22849 df-nei 22926 df-lp 22964 df-perf 22965 df-cn 23055 df-cnp 23056 df-haus 23143 df-fil 23674 df-fm 23766 df-flim 23767 df-flf 23768 df-xms 24150 df-ms 24151 df-cncf 24722 df-limc 25719 df-dv 25720 |
This theorem is referenced by: dvmptcmul 25820 dvmptfsum 25831 dvef 25836 rolle 25846 dvlipcn 25851 dvtaylp 26225 taylthlem2 26229 advlog 26507 advlogexp 26508 logtayl 26513 loglesqrt 26612 dvatan 26786 lgamgulmlem2 26881 log2sumbnd 27396 gg-taylthlem2 35658 dvasin 37066 dvacos 37067 areacirclem1 37070 lcmineqlem7 41397 lcmineqlem12 41402 aks4d1p1p6 41435 lhe4.4ex1a 43602 binomcxplemdvbinom 43626 dvsinax 45139 dvmptconst 45141 dvasinbx 45146 dvcosax 45152 itgiccshift 45206 itgperiod 45207 itgsbtaddcnst 45208 fourierdlem60 45392 fourierdlem61 45393 |
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