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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvsid | Structured version Visualization version GIF version |
Description: Derivative of the identity function on the real or complex numbers. (Contributed by Steve Rodriguez, 11-Nov-2015.) |
Ref | Expression |
---|---|
dvsid | ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D ( I ↾ 𝑆)) = (𝑆 × {1})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnresi 6679 | . . . . 5 ⊢ ( I ↾ ℂ) Fn ℂ | |
2 | rnresi 6073 | . . . . . 6 ⊢ ran ( I ↾ ℂ) = ℂ | |
3 | 2 | eqimssi 4039 | . . . . 5 ⊢ ran ( I ↾ ℂ) ⊆ ℂ |
4 | df-f 6547 | . . . . 5 ⊢ (( I ↾ ℂ):ℂ⟶ℂ ↔ (( I ↾ ℂ) Fn ℂ ∧ ran ( I ↾ ℂ) ⊆ ℂ)) | |
5 | 1, 3, 4 | mpbir2an 710 | . . . 4 ⊢ ( I ↾ ℂ):ℂ⟶ℂ |
6 | 5 | jctr 524 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 ∈ {ℝ, ℂ} ∧ ( I ↾ ℂ):ℂ⟶ℂ)) |
7 | recnprss 25827 | . . . . 5 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
8 | dvid 25841 | . . . . . . 7 ⊢ (ℂ D ( I ↾ ℂ)) = (ℂ × {1}) | |
9 | 8 | dmeqi 5902 | . . . . . 6 ⊢ dom (ℂ D ( I ↾ ℂ)) = dom (ℂ × {1}) |
10 | 1ex 11235 | . . . . . . . 8 ⊢ 1 ∈ V | |
11 | 10 | fconst 6778 | . . . . . . 7 ⊢ (ℂ × {1}):ℂ⟶{1} |
12 | 11 | fdmi 6729 | . . . . . 6 ⊢ dom (ℂ × {1}) = ℂ |
13 | 9, 12 | eqtri 2756 | . . . . 5 ⊢ dom (ℂ D ( I ↾ ℂ)) = ℂ |
14 | 7, 13 | sseqtrrdi 4030 | . . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ dom (ℂ D ( I ↾ ℂ))) |
15 | ssid 4001 | . . . 4 ⊢ ℂ ⊆ ℂ | |
16 | 14, 15 | jctil 519 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → (ℂ ⊆ ℂ ∧ 𝑆 ⊆ dom (ℂ D ( I ↾ ℂ)))) |
17 | dvres3 25836 | . . 3 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ ( I ↾ ℂ):ℂ⟶ℂ) ∧ (ℂ ⊆ ℂ ∧ 𝑆 ⊆ dom (ℂ D ( I ↾ ℂ)))) → (𝑆 D (( I ↾ ℂ) ↾ 𝑆)) = ((ℂ D ( I ↾ ℂ)) ↾ 𝑆)) | |
18 | 6, 16, 17 | syl2anc 583 | . 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D (( I ↾ ℂ) ↾ 𝑆)) = ((ℂ D ( I ↾ ℂ)) ↾ 𝑆)) |
19 | 7 | resabs1d 6011 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → (( I ↾ ℂ) ↾ 𝑆) = ( I ↾ 𝑆)) |
20 | 19 | oveq2d 7431 | . 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D (( I ↾ ℂ) ↾ 𝑆)) = (𝑆 D ( I ↾ 𝑆))) |
21 | 8 | reseq1i 5976 | . . . 4 ⊢ ((ℂ D ( I ↾ ℂ)) ↾ 𝑆) = ((ℂ × {1}) ↾ 𝑆) |
22 | xpssres 6017 | . . . 4 ⊢ (𝑆 ⊆ ℂ → ((ℂ × {1}) ↾ 𝑆) = (𝑆 × {1})) | |
23 | 21, 22 | eqtrid 2780 | . . 3 ⊢ (𝑆 ⊆ ℂ → ((ℂ D ( I ↾ ℂ)) ↾ 𝑆) = (𝑆 × {1})) |
24 | 7, 23 | syl 17 | . 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → ((ℂ D ( I ↾ ℂ)) ↾ 𝑆) = (𝑆 × {1})) |
25 | 18, 20, 24 | 3eqtr3d 2776 | 1 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D ( I ↾ 𝑆)) = (𝑆 × {1})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⊆ wss 3945 {csn 4625 {cpr 4627 I cid 5570 × cxp 5671 dom cdm 5673 ran crn 5674 ↾ cres 5675 Fn wfn 6538 ⟶wf 6539 (class class class)co 7415 ℂcc 11131 ℝcr 11132 1c1 11134 D cdv 25786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-map 8841 df-pm 8842 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-fi 9429 df-sup 9460 df-inf 9461 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-icc 13358 df-fz 13512 df-seq 13994 df-exp 14054 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-struct 17110 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-mulr 17241 df-starv 17242 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-rest 17398 df-topn 17399 df-topgen 17419 df-psmet 21265 df-xmet 21266 df-met 21267 df-bl 21268 df-mopn 21269 df-fbas 21270 df-fg 21271 df-cnfld 21274 df-top 22790 df-topon 22807 df-topsp 22829 df-bases 22843 df-cld 22917 df-ntr 22918 df-cls 22919 df-nei 22996 df-lp 23034 df-perf 23035 df-cn 23125 df-cnp 23126 df-haus 23213 df-fil 23744 df-fm 23836 df-flim 23837 df-flf 23838 df-xms 24220 df-ms 24221 df-cncf 24792 df-limc 25789 df-dv 25790 |
This theorem is referenced by: (None) |
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