Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnsrplycl | Structured version Visualization version GIF version |
Description: Polynomials are closed in number rings. (Contributed by Stefan O'Rear, 30-Nov-2014.) |
Ref | Expression |
---|---|
cnsrplycl.s | ⊢ (𝜑 → 𝑆 ∈ (SubRing‘ℂfld)) |
cnsrplycl.p | ⊢ (𝜑 → 𝑃 ∈ (Poly‘𝐶)) |
cnsrplycl.x | ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
cnsrplycl.c | ⊢ (𝜑 → 𝐶 ⊆ 𝑆) |
Ref | Expression |
---|---|
cnsrplycl | ⊢ (𝜑 → (𝑃‘𝑋) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnsrplycl.c | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝑆) | |
2 | cnsrplycl.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘ℂfld)) | |
3 | cnfldbas 20543 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
4 | 3 | subrgss 19530 | . . . . . 6 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ⊆ ℂ) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
6 | plyss 24783 | . . . . 5 ⊢ ((𝐶 ⊆ 𝑆 ∧ 𝑆 ⊆ ℂ) → (Poly‘𝐶) ⊆ (Poly‘𝑆)) | |
7 | 1, 5, 6 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (Poly‘𝐶) ⊆ (Poly‘𝑆)) |
8 | cnsrplycl.p | . . . 4 ⊢ (𝜑 → 𝑃 ∈ (Poly‘𝐶)) | |
9 | 7, 8 | sseldd 3968 | . . 3 ⊢ (𝜑 → 𝑃 ∈ (Poly‘𝑆)) |
10 | cnsrplycl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
11 | 5, 10 | sseldd 3968 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
12 | eqid 2821 | . . . 4 ⊢ (coeff‘𝑃) = (coeff‘𝑃) | |
13 | eqid 2821 | . . . 4 ⊢ (deg‘𝑃) = (deg‘𝑃) | |
14 | 12, 13 | coeid2 24823 | . . 3 ⊢ ((𝑃 ∈ (Poly‘𝑆) ∧ 𝑋 ∈ ℂ) → (𝑃‘𝑋) = Σ𝑘 ∈ (0...(deg‘𝑃))(((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘))) |
15 | 9, 11, 14 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝑃‘𝑋) = Σ𝑘 ∈ (0...(deg‘𝑃))(((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘))) |
16 | fzfid 13335 | . . 3 ⊢ (𝜑 → (0...(deg‘𝑃)) ∈ Fin) | |
17 | 2 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → 𝑆 ∈ (SubRing‘ℂfld)) |
18 | subrgsubg 19535 | . . . . . . . 8 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ∈ (SubGrp‘ℂfld)) | |
19 | cnfld0 20563 | . . . . . . . . 9 ⊢ 0 = (0g‘ℂfld) | |
20 | 19 | subg0cl 18281 | . . . . . . . 8 ⊢ (𝑆 ∈ (SubGrp‘ℂfld) → 0 ∈ 𝑆) |
21 | 2, 18, 20 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑆) |
22 | 12 | coef2 24815 | . . . . . . 7 ⊢ ((𝑃 ∈ (Poly‘𝑆) ∧ 0 ∈ 𝑆) → (coeff‘𝑃):ℕ0⟶𝑆) |
23 | 9, 21, 22 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (coeff‘𝑃):ℕ0⟶𝑆) |
24 | 23 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → (coeff‘𝑃):ℕ0⟶𝑆) |
25 | elfznn0 12994 | . . . . . 6 ⊢ (𝑘 ∈ (0...(deg‘𝑃)) → 𝑘 ∈ ℕ0) | |
26 | 25 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → 𝑘 ∈ ℕ0) |
27 | 24, 26 | ffvelrnd 6847 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → ((coeff‘𝑃)‘𝑘) ∈ 𝑆) |
28 | 10 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → 𝑋 ∈ 𝑆) |
29 | 17, 28, 26 | cnsrexpcl 39758 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → (𝑋↑𝑘) ∈ 𝑆) |
30 | cnfldmul 20545 | . . . . 5 ⊢ · = (.r‘ℂfld) | |
31 | 30 | subrgmcl 19541 | . . . 4 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ ((coeff‘𝑃)‘𝑘) ∈ 𝑆 ∧ (𝑋↑𝑘) ∈ 𝑆) → (((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘)) ∈ 𝑆) |
32 | 17, 27, 29, 31 | syl3anc 1367 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → (((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘)) ∈ 𝑆) |
33 | 2, 16, 32 | fsumcnsrcl 39759 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...(deg‘𝑃))(((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘)) ∈ 𝑆) |
34 | 15, 33 | eqeltrd 2913 | 1 ⊢ (𝜑 → (𝑃‘𝑋) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3936 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 0cc0 10531 · cmul 10536 ℕ0cn0 11891 ...cfz 12886 ↑cexp 13423 Σcsu 15036 SubGrpcsubg 18267 SubRingcsubrg 19525 ℂfldccnfld 20539 Polycply 24768 coeffccoe 24770 degcdgr 24771 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 df-sum 15037 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-subg 18270 df-cmn 18902 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-subrg 19527 df-cnfld 20540 df-0p 24265 df-ply 24772 df-coe 24774 df-dgr 24775 |
This theorem is referenced by: rngunsnply 39766 |
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