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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 6241 | . . . . 5 ⊢ ( I ↾ ℂ) Fn ℂ | |
2 | rnresi 5720 | . . . . . 6 ⊢ ran ( I ↾ ℂ) = ℂ | |
3 | 2 | eqimssi 3884 | . . . . 5 ⊢ ran ( I ↾ ℂ) ⊆ ℂ |
4 | df-f 6127 | . . . . 5 ⊢ (( I ↾ ℂ):ℂ⟶ℂ ↔ (( I ↾ ℂ) Fn ℂ ∧ ran ( I ↾ ℂ) ⊆ ℂ)) | |
5 | 1, 3, 4 | mpbir2an 704 | . . . 4 ⊢ ( I ↾ ℂ):ℂ⟶ℂ |
6 | 5 | jctr 522 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 ∈ {ℝ, ℂ} ∧ ( I ↾ ℂ):ℂ⟶ℂ)) |
7 | recnprss 24067 | . . . . 5 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
8 | dvid 24080 | . . . . . . 7 ⊢ (ℂ D ( I ↾ ℂ)) = (ℂ × {1}) | |
9 | 8 | dmeqi 5557 | . . . . . 6 ⊢ dom (ℂ D ( I ↾ ℂ)) = dom (ℂ × {1}) |
10 | 1ex 10352 | . . . . . . . 8 ⊢ 1 ∈ V | |
11 | 10 | fconst 6328 | . . . . . . 7 ⊢ (ℂ × {1}):ℂ⟶{1} |
12 | 11 | fdmi 6288 | . . . . . 6 ⊢ dom (ℂ × {1}) = ℂ |
13 | 9, 12 | eqtri 2849 | . . . . 5 ⊢ dom (ℂ D ( I ↾ ℂ)) = ℂ |
14 | 7, 13 | syl6sseqr 3877 | . . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ dom (ℂ D ( I ↾ ℂ))) |
15 | ssid 3848 | . . . 4 ⊢ ℂ ⊆ ℂ | |
16 | 14, 15 | jctil 517 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → (ℂ ⊆ ℂ ∧ 𝑆 ⊆ dom (ℂ D ( I ↾ ℂ)))) |
17 | dvres3 24076 | . . 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 581 | . 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D (( I ↾ ℂ) ↾ 𝑆)) = ((ℂ D ( I ↾ ℂ)) ↾ 𝑆)) |
19 | 7 | resabs1d 5664 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → (( I ↾ ℂ) ↾ 𝑆) = ( I ↾ 𝑆)) |
20 | 19 | oveq2d 6921 | . 2 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D (( I ↾ ℂ) ↾ 𝑆)) = (𝑆 D ( I ↾ 𝑆))) |
21 | 8 | reseq1i 5625 | . . . 4 ⊢ ((ℂ D ( I ↾ ℂ)) ↾ 𝑆) = ((ℂ × {1}) ↾ 𝑆) |
22 | xpssres 5669 | . . . 4 ⊢ (𝑆 ⊆ ℂ → ((ℂ × {1}) ↾ 𝑆) = (𝑆 × {1})) | |
23 | 21, 22 | syl5eq 2873 | . . 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 2869 | 1 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D ( I ↾ 𝑆)) = (𝑆 × {1})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ⊆ wss 3798 {csn 4397 {cpr 4399 I cid 5249 × cxp 5340 dom cdm 5342 ran crn 5343 ↾ cres 5344 Fn wfn 6118 ⟶wf 6119 (class class class)co 6905 ℂcc 10250 ℝcr 10251 1c1 10253 D cdv 24026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fi 8586 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-xadd 12233 df-xmul 12234 df-icc 12470 df-fz 12620 df-seq 13096 df-exp 13155 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-plusg 16318 df-mulr 16319 df-starv 16320 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-rest 16436 df-topn 16437 df-topgen 16457 df-psmet 20098 df-xmet 20099 df-met 20100 df-bl 20101 df-mopn 20102 df-fbas 20103 df-fg 20104 df-cnfld 20107 df-top 21069 df-topon 21086 df-topsp 21108 df-bases 21121 df-cld 21194 df-ntr 21195 df-cls 21196 df-nei 21273 df-lp 21311 df-perf 21312 df-cn 21402 df-cnp 21403 df-haus 21490 df-fil 22020 df-fm 22112 df-flim 22113 df-flf 22114 df-xms 22495 df-ms 22496 df-cncf 23051 df-limc 24029 df-dv 24030 |
This theorem is referenced by: (None) |
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