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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 2794 | . 2 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
2 | dvmptid.1 | . 2 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
3 | 1 | cnfldtopon 23074 | . . 3 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
4 | toponmax 21218 | . . 3 ⊢ ((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) → ℂ ∈ (TopOpen‘ℂfld)) | |
5 | 3, 4 | mp1i 13 | . 2 ⊢ (𝜑 → ℂ ∈ (TopOpen‘ℂfld)) |
6 | recnprss 24185 | . . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
7 | 2, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
8 | df-ss 3876 | . . 3 ⊢ (𝑆 ⊆ ℂ ↔ (𝑆 ∩ ℂ) = 𝑆) | |
9 | 7, 8 | sylib 219 | . 2 ⊢ (𝜑 → (𝑆 ∩ ℂ) = 𝑆) |
10 | dvmptc.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
11 | 10 | adantr 481 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ) |
12 | 0cnd 10483 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 0 ∈ ℂ) | |
13 | dvconst 24197 | . . . 4 ⊢ (𝐴 ∈ ℂ → (ℂ D (ℂ × {𝐴})) = (ℂ × {0})) | |
14 | 10, 13 | syl 17 | . . 3 ⊢ (𝜑 → (ℂ D (ℂ × {𝐴})) = (ℂ × {0})) |
15 | fconstmpt 5503 | . . . 4 ⊢ (ℂ × {𝐴}) = (𝑥 ∈ ℂ ↦ 𝐴) | |
16 | 15 | oveq2i 7030 | . . 3 ⊢ (ℂ D (ℂ × {𝐴})) = (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) |
17 | fconstmpt 5503 | . . 3 ⊢ (ℂ × {0}) = (𝑥 ∈ ℂ ↦ 0) | |
18 | 14, 16, 17 | 3eqtr3g 2853 | . 2 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 0)) |
19 | 1, 2, 5, 9, 11, 12, 18 | dvmptres3 24236 | 1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑆 ↦ 𝐴)) = (𝑥 ∈ 𝑆 ↦ 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2080 ∩ cin 3860 ⊆ wss 3861 {csn 4474 {cpr 4476 ↦ cmpt 5043 × cxp 5444 ‘cfv 6228 (class class class)co 7019 ℂcc 10384 ℝcr 10385 0cc0 10386 TopOpenctopn 16524 ℂfldccnfld 20227 TopOnctopon 21202 D cdv 24144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 ax-pre-sup 10464 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-int 4785 df-iun 4829 df-iin 4830 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-1st 7548 df-2nd 7549 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-1o 7956 df-oadd 7960 df-er 8142 df-map 8261 df-pm 8262 df-en 8361 df-dom 8362 df-sdom 8363 df-fin 8364 df-fi 8724 df-sup 8755 df-inf 8756 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-div 11148 df-nn 11489 df-2 11550 df-3 11551 df-4 11552 df-5 11553 df-6 11554 df-7 11555 df-8 11556 df-9 11557 df-n0 11748 df-z 11832 df-dec 11949 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-icc 12595 df-fz 12743 df-seq 13220 df-exp 13280 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-plusg 16407 df-mulr 16408 df-starv 16409 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-rest 16525 df-topn 16526 df-topgen 16546 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-fbas 20224 df-fg 20225 df-cnfld 20228 df-top 21186 df-topon 21203 df-topsp 21225 df-bases 21238 df-cld 21311 df-ntr 21312 df-cls 21313 df-nei 21390 df-lp 21428 df-perf 21429 df-cn 21519 df-cnp 21520 df-haus 21607 df-fil 22138 df-fm 22230 df-flim 22231 df-flf 22232 df-xms 22613 df-ms 22614 df-cncf 23169 df-limc 24147 df-dv 24148 |
This theorem is referenced by: dvmptcmul 24244 dvmptfsum 24255 dvef 24260 rolle 24270 dvlipcn 24274 dvtaylp 24641 taylthlem2 24645 advlog 24918 advlogexp 24919 logtayl 24924 loglesqrt 25020 dvatan 25194 lgamgulmlem2 25289 log2sumbnd 25802 dvasin 34522 dvacos 34523 areacirclem1 34526 lhe4.4ex1a 40212 binomcxplemdvbinom 40236 dvsinax 41752 dvmptconst 41754 dvasinbx 41760 dvcosax 41766 itgiccshift 41820 itgperiod 41821 itgsbtaddcnst 41822 fourierdlem60 42007 fourierdlem61 42008 |
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