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Theorem cpnres 25988

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 4018	. . . . . 6 ⊢ ℂ ⊆ ℂ
3		elfvdm 6944	. . . . . . . 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 25984	. . . . . . . . 9 ⊢ (ℂ ⊆ ℂ → (𝓑C^𝑛‘ℂ) Fn ℕ₀)
6	2, 5	ax-mp 5	. . . . . . . 8 ⊢ (𝓑C^𝑛‘ℂ) Fn ℕ₀
7		fndm 6672	. . . . . . . 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 2841	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → 𝑁 ∈ ℕ₀)
10		elcpn 25985	. . . . . 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 8909	. . 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 24933	. . . . . 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 6747	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹)
21		dvnres 25982	. . . 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 1372	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
23		resres 6013	. . . . . . 7 ⊢ ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ↾ dom 𝐹) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹))
24		rescom 6023	. . . . . . 7 ⊢ ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ↾ dom 𝐹) = ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) ↾ 𝑆)
25	23, 24	eqtr3i 2765	. . . . . 6 ⊢ (((ℂ D^𝑛 𝐹)‘𝑁) ↾ (𝑆 ∩ dom 𝐹)) = ((((ℂ D^𝑛 𝐹)‘𝑁) ↾ dom 𝐹) ↾ 𝑆)
26		ffn 6737	. . . . . . . 8 ⊢ (((ℂ D^𝑛 𝐹)‘𝑁):dom 𝐹⟶ℂ → ((ℂ D^𝑛 𝐹)‘𝑁) Fn dom 𝐹)
27		fnresdm 6688	. . . . . . . 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 5999	. . . . . 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 2787	. . . . 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 4246	. . . . . 6 ⊢ (𝑆 ∩ dom 𝐹) ⊆ dom 𝐹
32		rescncf 24937	. . . . . 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 2840	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ∈ ((𝑆 ∩ dom 𝐹)–cn→ℂ))
35		dmres 6032	. . . . 5 ⊢ dom (𝐹 ↾ 𝑆) = (𝑆 ∩ dom 𝐹)
36	35	oveq1i 7441	. . . 4 ⊢ (dom (𝐹 ↾ 𝑆)–cn→ℂ) = ((𝑆 ∩ dom 𝐹)–cn→ℂ)
37	34, 36	eleqtrrdi 2850	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆) ∈ (dom (𝐹 ↾ 𝑆)–cn→ℂ))
38	22, 37	eqeltrd 2839	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑁)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) ∈ (dom (𝐹 ↾ 𝑆)–cn→ℂ))
39		recnprss 25954	. . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
40		elcpn 25985	. . 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 1537 ∈ wcel 2106 ∩ cin 3962 ⊆ wss 3963 {cpr 4633 dom cdm 5689 ↾ cres 5691 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑_pm cpm 8866 ℂcc 11151 ℝcr 11152 ℕ₀cn0 12524 –cn→ccncf 24916 D^𝑛 cdvn 25914 𝓑C^𝑛ccpn 25915

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-icc 13391 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-rest 17469 df-topn 17470 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cnp 23252 df-haus 23339 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-cncf 24918 df-limc 25916 df-dv 25917 df-dvn 25918 df-cpn 25919

This theorem is referenced by: aalioulem3 26391

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