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Theorem cpnres 25891

Description: The restriction of a 𝓑C^𝑛 function is 𝓑C^𝑛. (Contributed by Mario Carneiro, 11-Feb-2015.)

**Assertion**
Ref	Expression
cpnres	⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (𝐹 ↾ 𝑆) ∈ ((𝓑C^𝑛‘𝑆)‘𝑁))

**Proof of Theorem cpnres**
Step	Hyp	Ref	Expression
1		simpr 484	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁))
2		ssid 3981	. . . . . 6 ⊢ ℂ ⊆ ℂ
3		elfvdm 6913	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁) → 𝑁 ∈ dom (𝓑C^𝑛‘ℂ))
4	3	adantl 481	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → 𝑁 ∈ dom (𝓑C^𝑛‘ℂ))
5		fncpn 25887	. . . . . . . . 9 ⊢ (ℂ ⊆ ℂ → (𝓑C^𝑛‘ℂ) Fn ℕ₀)
6	2, 5	ax-mp 5	. . . . . . . 8 ⊢ (𝓑C^𝑛‘ℂ) Fn ℕ₀
7		fndm 6641	. . . . . . . 8 ⊢ ((𝓑C^𝑛‘ℂ) Fn ℕ₀ → dom (𝓑C^𝑛‘ℂ) = ℕ₀)
8	6, 7	mp1i 13	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → dom (𝓑C^𝑛‘ℂ) = ℕ₀)
9	4, 8	eleqtrd 2836	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → 𝑁 ∈ ℕ₀)
10		elcpn 25888	. . . . . 6 ⊢ ((ℂ ⊆ ℂ ∧ 𝑁 ∈ ℕ₀) → (𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁) ↔ (𝐹 ∈ (ℂ ↑_pm ℂ) ∧ ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ))))
11	2, 9, 10	sylancr 587	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁) ↔ (𝐹 ∈ (ℂ ↑_pm ℂ) ∧ ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ))))
12	1, 11	mpbid 232	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (𝐹 ∈ (ℂ ↑_pm ℂ) ∧ ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ)))
13	12	simpld 494	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → 𝐹 ∈ (ℂ ↑_pm ℂ))
14		pmresg 8884	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆))
15	13, 14	syldan 591	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆))
16		simpl 482	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → 𝑆 ∈ {ℝ, ℂ})
17	12	simprd 495	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ))
18		cncff 24837	. . . . . 6 ⊢ (((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ) → ((ℂ D^𝑛 𝐹)‘𝑁):dom 𝐹⟶ℂ)
19	17, 18	syl 17	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((ℂ D^𝑛 𝐹)‘𝑁):dom 𝐹⟶ℂ)
20	19	fdmd 6716	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹)
21		dvnres 25885	. . . 4 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ) ∧ 𝑁 ∈ ℕ₀) ∧ dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
22	16, 13, 9, 20, 21	syl31anc 1375	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
23		resres 5979	. . . . . . 7 ⊢ ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ↾ dom 𝐹) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹))
24		rescom 5989	. . . . . . 7 ⊢ ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ↾ dom 𝐹) = ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) ↾ 𝑆)
25	23, 24	eqtr3i 2760	. . . . . 6 ⊢ (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹)) = ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) ↾ 𝑆)
26		ffn 6706	. . . . . . . 8 ⊢ (((ℂ D^𝑛 𝐹)‘𝑁):dom 𝐹⟶ℂ → ((ℂ D^𝑛 𝐹)‘𝑁) Fn dom 𝐹)
27		fnresdm 6657	. . . . . . . 8 ⊢ (((ℂ D^𝑛 𝐹)‘𝑁) Fn dom 𝐹 → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) = ((ℂ D^𝑛 𝐹)‘𝑁))
28	19, 26, 27	3syl 18	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) = ((ℂ D^𝑛 𝐹)‘𝑁))
29	28	reseq1d 5965	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
30	25, 29	eqtrid 2782	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹)) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
31		inss2 4213	. . . . . 6 ⊢ (𝑆 ∩ dom 𝐹) ⊆ dom 𝐹
32		rescncf 24841	. . . . . 6 ⊢ ((𝑆 ∩ dom 𝐹) ⊆ dom 𝐹 → (((ℂ D^𝑛 𝐹)‘𝑁) ∈ (dom 𝐹–cn→ℂ) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹)) ∈ ((𝑆 ∩ dom 𝐹)–cn→ℂ)))
33	31, 17, 32	mpsyl 68	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹)) ∈ ((𝑆 ∩ dom 𝐹)–cn→ℂ))
34	30, 33	eqeltrrd 2835	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ∈ ((𝑆 ∩ dom 𝐹)–cn→ℂ))
35		dmres 5999	. . . . 5 ⊢ dom (𝐹 ↾ 𝑆) = (𝑆 ∩ dom 𝐹)
36	35	oveq1i 7415	. . . 4 ⊢ (dom (𝐹 ↾ 𝑆)–cn→ℂ) = ((𝑆 ∩ dom 𝐹)–cn→ℂ)
37	34, 36	eleqtrrdi 2845	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ∈ (dom (𝐹 ↾ 𝑆)–cn→ℂ))
38	22, 37	eqeltrd 2834	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) ∈ (dom (𝐹 ↾ 𝑆)–cn→ℂ))
39		recnprss 25857	. . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
40		elcpn 25888	. . 3 ⊢ ((𝑆 ⊆ ℂ ∧ 𝑁 ∈ ℕ₀) → ((𝐹 ↾ 𝑆) ∈ ((𝓑C^𝑛‘𝑆)‘𝑁) ↔ ((𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) ∈ (dom (𝐹 ↾ 𝑆)–cn→ℂ))))
41	39, 9, 40	syl2an2r 685	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((𝐹 ↾ 𝑆) ∈ ((𝓑C^𝑛‘𝑆)‘𝑁) ↔ ((𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆) ∧ ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) ∈ (dom (𝐹 ↾ 𝑆)–cn→ℂ))))
42	15, 38, 41	mpbir2and 713	1 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (𝐹 ↾ 𝑆) ∈ ((𝓑C^𝑛‘𝑆)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3925 ⊆ wss 3926 {cpr 4603 dom cdm 5654 ↾ cres 5656 Fn wfn 6526 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 ↑_pm cpm 8841 ℂcc 11127 ℝcr 11128 ℕ₀cn0 12501 –cn→ccncf 24820 D^𝑛 cdvn 25817 𝓑C^𝑛ccpn 25818

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-map 8842 df-pm 8843 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fi 9423 df-sup 9454 df-inf 9455 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-icc 13369 df-fz 13525 df-seq 14020 df-exp 14080 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-struct 17166 df-slot 17201 df-ndx 17213 df-base 17229 df-plusg 17284 df-mulr 17285 df-starv 17286 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-rest 17436 df-topn 17437 df-topgen 17457 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-fbas 21312 df-fg 21313 df-cnfld 21316 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cld 22957 df-ntr 22958 df-cls 22959 df-nei 23036 df-lp 23074 df-perf 23075 df-cnp 23166 df-haus 23253 df-fil 23784 df-fm 23876 df-flim 23877 df-flf 23878 df-xms 24259 df-ms 24260 df-cncf 24822 df-limc 25819 df-dv 25820 df-dvn 25821 df-cpn 25822

This theorem is referenced by: aalioulem3 26294

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