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Theorem plycpn 25182

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 25092	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
2	1	adantr 484	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → 𝐹:ℂ⟶ℂ)
3		cnex 10810	. . . . . . 7 ⊢ ℂ ∈ V
4	3, 3	fpm 8556	. . . . . 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 25181	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (Poly‘ℂ))
7		plycn 25155	. . . . . . 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 6556	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → dom 𝐹 = ℂ)
10	9	oveq1d 7228	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → (dom 𝐹–cn→ℂ) = (ℂ–cn→ℂ))
11	8, 10	eleqtrrd 2841	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑛 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (dom 𝐹–cn→ℂ))
12		ssidd 3924	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → ℂ ⊆ ℂ)
13		elcpn 24831	. . . . . 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 583	. . . . 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 3105	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑛 ∈ ℕ₀ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑛))
17		ssid 3923	. . . 4 ⊢ ℂ ⊆ ℂ
18		fncpn 24830	. . . 4 ⊢ (ℂ ⊆ ℂ → (𝓑C^𝑛‘ℂ) Fn ℕ₀)
19		eleq2 2826	. . . . 5 ⊢ (𝑥 = ((𝓑C^𝑛‘ℂ)‘𝑛) → (𝐹 ∈ 𝑥 ↔ 𝐹 ∈ ((𝓑C^𝑛‘ℂ)‘𝑛)))
20	19	ralrn 6907	. . . 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 237	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑥 ∈ ran (𝓑C^𝑛‘ℂ)𝐹 ∈ 𝑥)
23		elintg 4867	. 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 ∈ ∩ ran (𝓑C^𝑛‘ℂ) ↔ ∀𝑥 ∈ ran (𝓑C^𝑛‘ℂ)𝐹 ∈ 𝑥))
24	22, 23	mpbird 260	1 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ ∩ ran (𝓑C^𝑛‘ℂ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2110 ∀wral 3061 ⊆ wss 3866 ∩ cint 4859 dom cdm 5551 ran crn 5552 Fn wfn 6375 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ↑_pm cpm 8509 ℂcc 10727 ℕ₀cn0 12090 –cn→ccncf 23773 D^𝑛 cdvn 24761 𝓑C^𝑛ccpn 24762 Polycply 25078

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-map 8510 df-pm 8511 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-rlim 15050 df-sum 15250 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-starv 16817 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-unif 16825 df-hom 16826 df-cco 16827 df-rest 16927 df-topn 16928 df-0g 16946 df-gsum 16947 df-topgen 16948 df-pt 16949 df-prds 16952 df-xrs 17007 df-qtop 17012 df-imas 17013 df-xps 17015 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-minusg 18369 df-mulg 18489 df-subg 18540 df-cntz 18711 df-cmn 19172 df-mgp 19505 df-ur 19517 df-ring 19564 df-cring 19565 df-subrg 19798 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-cnfld 20364 df-top 21791 df-topon 21808 df-topsp 21830 df-bases 21843 df-cld 21916 df-ntr 21917 df-cls 21918 df-nei 21995 df-lp 22033 df-perf 22034 df-cn 22124 df-cnp 22125 df-haus 22212 df-tx 22459 df-hmeo 22652 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-xms 23218 df-ms 23219 df-tms 23220 df-cncf 23775 df-0p 24567 df-limc 24763 df-dv 24764 df-dvn 24765 df-cpn 24766 df-ply 25082 df-coe 25084 df-dgr 25085

This theorem is referenced by: aalioulem3 25227

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