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Theorem plycpn 26254

Description: Polynomials are smooth. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

**Assertion**
Ref	Expression
plycpn	⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ ∩ ran (𝓑C^𝑛‘ℂ))

Proof of Theorem plycpn Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plyf 26160	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
2	1	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → 𝐹:ℂ⟶ℂ)
3		cnex 11215	. . . . . . 7 ⊢ ℂ ∈ V
4	3, 3	fpm 8894	. . . . . 6 ⊢ (𝐹:ℂ⟶ℂ → 𝐹 ∈ (ℂ ↑_pm ℂ))
5	2, 4	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → 𝐹 ∈ (ℂ ↑_pm ℂ))
6		dvnply 26253	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (Poly‘ℂ))
7		plycn 26223	. . . . . . 7 ⊢ (((ℂ D^𝑛 𝐹)‘𝑛) ∈ (Poly‘ℂ) → ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (ℂ–cn→ℂ))
8	6, 7	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (ℂ–cn→ℂ))
9	2	fdmd 6721	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → dom 𝐹 = ℂ)
10	9	oveq1d 7425	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → (dom 𝐹–cn→ℂ) = (ℂ–cn→ℂ))
11	8, 10	eleqtrrd 2838	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (dom 𝐹–cn→ℂ))
12		ssidd 3987	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → ℂ ⊆ ℂ)
13		elcpn 25893	. . . . . 6 ⊢ ((ℂ ⊆ ℂ ∧ 𝑛 ∈ ℕ₀) → (𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑛) ↔ (𝐹 ∈ (ℂ ↑_pm ℂ) ∧ ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (dom 𝐹–cn→ℂ))))
14	12, 13	sylan 580	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → (𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑛) ↔ (𝐹 ∈ (ℂ ↑_pm ℂ) ∧ ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (dom 𝐹–cn→ℂ))))
15	5, 11, 14	mpbir2and 713	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑛))
16	15	ralrimiva 3133	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑛 ∈ ℕ₀ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑛))
17		ssid 3986	. . . 4 ⊢ ℂ ⊆ ℂ
18		fncpn 25892	. . . 4 ⊢ (ℂ ⊆ ℂ → (𝓑C^𝑛‘ℂ) Fn ℕ₀)
19		eleq2 2824	. . . . 5 ⊢ (𝑥 = ((𝓑C^𝑛‘ℂ)‘𝑛) → (𝐹 ∈ 𝑥 ↔ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑛)))
20	19	ralrn 7083	. . . 4 ⊢ ((𝓑C^𝑛‘ℂ) Fn ℕ₀ → (∀𝑥 ∈ ran (𝓑C^𝑛‘ℂ)𝐹 ∈ 𝑥 ↔ ∀𝑛 ∈ ℕ₀ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑛)))
21	17, 18, 20	mp2b 10	. . 3 ⊢ (∀𝑥 ∈ ran (𝓑C^𝑛‘ℂ)𝐹 ∈ 𝑥 ↔ ∀𝑛 ∈ ℕ₀ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑛))
22	16, 21	sylibr 234	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑥 ∈ ran (𝓑C^𝑛‘ℂ)𝐹 ∈ 𝑥)
23		elintg 4935	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 ∈ ∩ ran (𝓑C^𝑛‘ℂ) ↔ ∀𝑥 ∈ ran (𝓑C^𝑛‘ℂ)𝐹 ∈ 𝑥))
24	22, 23	mpbird 257	1 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ ∩ ran (𝓑C^𝑛‘ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2109 ∀wral 3052 ⊆ wss 3931 ∩ cint 4927 dom cdm 5659 ran crn 5660 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ↑_pm cpm 8846 ℂcc 11132 ℕ₀cn0 12506 –cn→ccncf 24825 D^𝑛 cdvn 25822 𝓑C^𝑛ccpn 25823 Polycply 26146

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213

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 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-seq 14025 df-exp 14085 df-hash 14354 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-clim 15509 df-rlim 15510 df-sum 15708 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-grp 18924 df-minusg 18925 df-mulg 19056 df-subg 19111 df-cntz 19305 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-cring 20201 df-subrng 20511 df-subrg 20535 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-0p 25628 df-limc 25824 df-dv 25825 df-dvn 25826 df-cpn 25827 df-ply 26150 df-coe 26152 df-dgr 26153

This theorem is referenced by: aalioulem3 26299

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