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Theorem plycpn 26255

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 26161	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
2	1	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → 𝐹:ℂ⟶ℂ)
3		cnex 11109	. . . . . . 7 ⊢ ℂ ∈ V
4	3, 3	fpm 8815	. . . . . 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 26254	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (Poly‘ℂ))
7		plycn 26224	. . . . . . 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 6672	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → dom 𝐹 = ℂ)
10	9	oveq1d 7373	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → (dom 𝐹–cn→ℂ) = (ℂ–cn→ℂ))
11	8, 10	eleqtrrd 2839	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (dom 𝐹–cn→ℂ))
12		ssidd 3957	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → ℂ ⊆ ℂ)
13		elcpn 25894	. . . . . 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 3128	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑛 ∈ ℕ₀ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑛))
17		ssid 3956	. . . 4 ⊢ ℂ ⊆ ℂ
18		fncpn 25893	. . . 4 ⊢ (ℂ ⊆ ℂ → (𝓑C^𝑛‘ℂ) Fn ℕ₀)
19		eleq2 2825	. . . . 5 ⊢ (𝑥 = ((𝓑C^𝑛‘ℂ)‘𝑛) → (𝐹 ∈ 𝑥 ↔ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑛)))
20	19	ralrn 7033	. . . 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 4910	. 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 2113 ∀wral 3051 ⊆ wss 3901 ∩ cint 4902 dom cdm 5624 ran crn 5625 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ↑_pm cpm 8766 ℂcc 11026 ℕ₀cn0 12403 –cn→ccncf 24827 D^𝑛 cdvn 25823 𝓑C^𝑛ccpn 25824 Polycply 26147

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9552 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8767 df-pm 8768 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fsupp 9267 df-fi 9316 df-sup 9347 df-inf 9348 df-oi 9417 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-icc 13270 df-fz 13426 df-fzo 13573 df-fl 13714 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-rlim 15414 df-sum 15612 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17425 df-qtop 17430 df-imas 17431 df-xps 17433 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-mulg 19000 df-subg 19055 df-cntz 19248 df-cmn 19713 df-abl 19714 df-mgp 20078 df-rng 20090 df-ur 20119 df-ring 20172 df-cring 20173 df-subrng 20481 df-subrg 20505 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-fbas 21308 df-fg 21309 df-cnfld 21312 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22892 df-cld 22965 df-ntr 22966 df-cls 22967 df-nei 23044 df-lp 23082 df-perf 23083 df-cn 23173 df-cnp 23174 df-haus 23261 df-tx 23508 df-hmeo 23701 df-fil 23792 df-fm 23884 df-flim 23885 df-flf 23886 df-xms 24266 df-ms 24267 df-tms 24268 df-cncf 24829 df-0p 25629 df-limc 25825 df-dv 25826 df-dvn 25827 df-cpn 25828 df-ply 26151 df-coe 26153 df-dgr 26154

This theorem is referenced by: aalioulem3 26300

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