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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 20933 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
4 | 3 | subrgss 20352 | . . . . . 6 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ⊆ ℂ) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
6 | plyss 25695 | . . . . 5 ⊢ ((𝐶 ⊆ 𝑆 ∧ 𝑆 ⊆ ℂ) → (Poly‘𝐶) ⊆ (Poly‘𝑆)) | |
7 | 1, 5, 6 | syl2anc 585 | . . . 4 ⊢ (𝜑 → (Poly‘𝐶) ⊆ (Poly‘𝑆)) |
8 | cnsrplycl.p | . . . 4 ⊢ (𝜑 → 𝑃 ∈ (Poly‘𝐶)) | |
9 | 7, 8 | sseldd 3982 | . . 3 ⊢ (𝜑 → 𝑃 ∈ (Poly‘𝑆)) |
10 | cnsrplycl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
11 | 5, 10 | sseldd 3982 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
12 | eqid 2733 | . . . 4 ⊢ (coeff‘𝑃) = (coeff‘𝑃) | |
13 | eqid 2733 | . . . 4 ⊢ (deg‘𝑃) = (deg‘𝑃) | |
14 | 12, 13 | coeid2 25735 | . . 3 ⊢ ((𝑃 ∈ (Poly‘𝑆) ∧ 𝑋 ∈ ℂ) → (𝑃‘𝑋) = Σ𝑘 ∈ (0...(deg‘𝑃))(((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘))) |
15 | 9, 11, 14 | syl2anc 585 | . 2 ⊢ (𝜑 → (𝑃‘𝑋) = Σ𝑘 ∈ (0...(deg‘𝑃))(((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘))) |
16 | fzfid 13934 | . . 3 ⊢ (𝜑 → (0...(deg‘𝑃)) ∈ Fin) | |
17 | 2 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → 𝑆 ∈ (SubRing‘ℂfld)) |
18 | subrgsubg 20357 | . . . . . . . 8 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ∈ (SubGrp‘ℂfld)) | |
19 | cnfld0 20954 | . . . . . . . . 9 ⊢ 0 = (0g‘ℂfld) | |
20 | 19 | subg0cl 19008 | . . . . . . . 8 ⊢ (𝑆 ∈ (SubGrp‘ℂfld) → 0 ∈ 𝑆) |
21 | 2, 18, 20 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑆) |
22 | 12 | coef2 25727 | . . . . . . 7 ⊢ ((𝑃 ∈ (Poly‘𝑆) ∧ 0 ∈ 𝑆) → (coeff‘𝑃):ℕ0⟶𝑆) |
23 | 9, 21, 22 | syl2anc 585 | . . . . . 6 ⊢ (𝜑 → (coeff‘𝑃):ℕ0⟶𝑆) |
24 | 23 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → (coeff‘𝑃):ℕ0⟶𝑆) |
25 | elfznn0 13590 | . . . . . 6 ⊢ (𝑘 ∈ (0...(deg‘𝑃)) → 𝑘 ∈ ℕ0) | |
26 | 25 | adantl 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → 𝑘 ∈ ℕ0) |
27 | 24, 26 | ffvelcdmd 7083 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → ((coeff‘𝑃)‘𝑘) ∈ 𝑆) |
28 | 10 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → 𝑋 ∈ 𝑆) |
29 | 17, 28, 26 | cnsrexpcl 41840 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → (𝑋↑𝑘) ∈ 𝑆) |
30 | cnfldmul 20935 | . . . . 5 ⊢ · = (.r‘ℂfld) | |
31 | 30 | subrgmcl 20363 | . . . 4 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ ((coeff‘𝑃)‘𝑘) ∈ 𝑆 ∧ (𝑋↑𝑘) ∈ 𝑆) → (((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘)) ∈ 𝑆) |
32 | 17, 27, 29, 31 | syl3anc 1372 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → (((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘)) ∈ 𝑆) |
33 | 2, 16, 32 | fsumcnsrcl 41841 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...(deg‘𝑃))(((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘)) ∈ 𝑆) |
34 | 15, 33 | eqeltrd 2834 | 1 ⊢ (𝜑 → (𝑃‘𝑋) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3947 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 ℂcc 11104 0cc0 11106 · cmul 11111 ℕ0cn0 12468 ...cfz 13480 ↑cexp 14023 Σcsu 15628 SubGrpcsubg 18994 SubRingcsubrg 20347 ℂfldccnfld 20929 Polycply 25680 coeffccoe 25682 degcdgr 25683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-subg 18997 df-cmn 19643 df-mgp 19980 df-ur 19997 df-ring 20049 df-cring 20050 df-subrg 20349 df-cnfld 20930 df-0p 25169 df-ply 25684 df-coe 25686 df-dgr 25687 |
This theorem is referenced by: rngunsnply 41848 |
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