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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 6008 | . . . . 5 ⊢ ( I ↾ ℂ) Fn ℂ | |
2 | rnresi 5479 | . . . . . 6 ⊢ ran ( I ↾ ℂ) = ℂ | |
3 | 2 | eqimssi 3659 | . . . . 5 ⊢ ran ( I ↾ ℂ) ⊆ ℂ |
4 | df-f 5892 | . . . . 5 ⊢ (( I ↾ ℂ):ℂ⟶ℂ ↔ (( I ↾ ℂ) Fn ℂ ∧ ran ( I ↾ ℂ) ⊆ ℂ)) | |
5 | 1, 3, 4 | mpbir2an 955 | . . . 4 ⊢ ( I ↾ ℂ):ℂ⟶ℂ |
6 | 5 | jctr 565 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 ∈ {ℝ, ℂ} ∧ ( I ↾ ℂ):ℂ⟶ℂ)) |
7 | recnprss 23668 | . . . . 5 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
8 | dvid 23681 | . . . . . . 7 ⊢ (ℂ D ( I ↾ ℂ)) = (ℂ × {1}) | |
9 | 8 | dmeqi 5325 | . . . . . 6 ⊢ dom (ℂ D ( I ↾ ℂ)) = dom (ℂ × {1}) |
10 | 1ex 10035 | . . . . . . . 8 ⊢ 1 ∈ V | |
11 | 10 | fconst 6091 | . . . . . . 7 ⊢ (ℂ × {1}):ℂ⟶{1} |
12 | 11 | fdmi 6052 | . . . . . 6 ⊢ dom (ℂ × {1}) = ℂ |
13 | 9, 12 | eqtri 2644 | . . . . 5 ⊢ dom (ℂ D ( I ↾ ℂ)) = ℂ |
14 | 7, 13 | syl6sseqr 3652 | . . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ dom (ℂ D ( I ↾ ℂ))) |
15 | ssid 3624 | . . . 4 ⊢ ℂ ⊆ ℂ | |
16 | 14, 15 | jctil 560 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → (ℂ ⊆ ℂ ∧ 𝑆 ⊆ dom (ℂ D ( I ↾ ℂ)))) |
17 | dvres3 23677 | . . 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 693 | . 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D (( I ↾ ℂ) ↾ 𝑆)) = ((ℂ D ( I ↾ ℂ)) ↾ 𝑆)) |
19 | 7 | resabs1d 5428 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → (( I ↾ ℂ) ↾ 𝑆) = ( I ↾ 𝑆)) |
20 | 19 | oveq2d 6666 | . 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D (( I ↾ ℂ) ↾ 𝑆)) = (𝑆 D ( I ↾ 𝑆))) |
21 | 8 | reseq1i 5392 | . . . 4 ⊢ ((ℂ D ( I ↾ ℂ)) ↾ 𝑆) = ((ℂ × {1}) ↾ 𝑆) |
22 | xpssres 5434 | . . . 4 ⊢ (𝑆 ⊆ ℂ → ((ℂ × {1}) ↾ 𝑆) = (𝑆 × {1})) | |
23 | 21, 22 | syl5eq 2668 | . . 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 2664 | 1 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D ( I ↾ 𝑆)) = (𝑆 × {1})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 {csn 4177 {cpr 4179 I cid 5023 × cxp 5112 dom cdm 5114 ran crn 5115 ↾ cres 5116 Fn wfn 5883 ⟶wf 5884 (class class class)co 6650 ℂcc 9934 ℝcr 9935 1c1 9937 D cdv 23627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fi 8317 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-rest 16083 df-topn 16084 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-cncf 22681 df-limc 23630 df-dv 23631 |
This theorem is referenced by: (None) |
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