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Theorem plycpn 26237

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 26145	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
2	1	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → 𝐹:ℂ⟶ℂ)
3		cnex 11220	. . . . . . 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 26236	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (Poly‘ℂ))
7		plycn 26208	. . . . . . 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 6733	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → dom 𝐹 = ℂ)
10	9	oveq1d 7435	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → (dom 𝐹–cn→ℂ) = (ℂ–cn→ℂ))
11	8, 10	eleqtrrd 2832	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (dom 𝐹–cn→ℂ))
12		ssidd 4003	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → ℂ ⊆ ℂ)
13		elcpn 25877	. . . . . 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 579	. . . . 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 712	. . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑛))
16	15	ralrimiva 3143	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑛 ∈ ℕ₀ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑛))
17		ssid 4002	. . . 4 ⊢ ℂ ⊆ ℂ
18		fncpn 25876	. . . 4 ⊢ (ℂ ⊆ ℂ → (𝓑C^𝑛‘ℂ) Fn ℕ₀)
19		eleq2 2818	. . . . 5 ⊢ (𝑥 = ((𝓑C^𝑛‘ℂ)‘𝑛) → (𝐹 ∈ 𝑥 ↔ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑛)))
20	19	ralrn 7098	. . . 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 233	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑥 ∈ ran (𝓑C^𝑛‘ℂ)𝐹 ∈ 𝑥)
23		elintg 4957	. 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 205 ∧ wa 395 ∈ wcel 2099 ∀wral 3058 ⊆ wss 3947 ∩ cint 4949 dom cdm 5678 ran crn 5679 Fn wfn 6543 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ↑_pm cpm 8846 ℂcc 11137 ℕ₀cn0 12503 –cn→ccncf 24809 D^𝑛 cdvn 25806 𝓑C^𝑛ccpn 25807 Polycply 26131

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 ax-addf 11218

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-fi 9435 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13790 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-rlim 15466 df-sum 15666 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-pt 17426 df-prds 17429 df-xrs 17484 df-qtop 17489 df-imas 17490 df-xps 17492 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-grp 18893 df-minusg 18894 df-mulg 19024 df-subg 19078 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-cring 20176 df-subrng 20483 df-subrg 20508 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lp 23053 df-perf 23054 df-cn 23144 df-cnp 23145 df-haus 23232 df-tx 23479 df-hmeo 23672 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-xms 24239 df-ms 24240 df-tms 24241 df-cncf 24811 df-0p 25612 df-limc 25808 df-dv 25809 df-dvn 25810 df-cpn 25811 df-ply 26135 df-coe 26137 df-dgr 26138

This theorem is referenced by: aalioulem3 26282

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