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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 21386 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
4 | 3 | subrgss 20589 | . . . . . 6 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ⊆ ℂ) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
6 | plyss 26253 | . . . . 5 ⊢ ((𝐶 ⊆ 𝑆 ∧ 𝑆 ⊆ ℂ) → (Poly‘𝐶) ⊆ (Poly‘𝑆)) | |
7 | 1, 5, 6 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (Poly‘𝐶) ⊆ (Poly‘𝑆)) |
8 | cnsrplycl.p | . . . 4 ⊢ (𝜑 → 𝑃 ∈ (Poly‘𝐶)) | |
9 | 7, 8 | sseldd 3996 | . . 3 ⊢ (𝜑 → 𝑃 ∈ (Poly‘𝑆)) |
10 | cnsrplycl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
11 | 5, 10 | sseldd 3996 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
12 | eqid 2735 | . . . 4 ⊢ (coeff‘𝑃) = (coeff‘𝑃) | |
13 | eqid 2735 | . . . 4 ⊢ (deg‘𝑃) = (deg‘𝑃) | |
14 | 12, 13 | coeid2 26293 | . . 3 ⊢ ((𝑃 ∈ (Poly‘𝑆) ∧ 𝑋 ∈ ℂ) → (𝑃‘𝑋) = Σ𝑘 ∈ (0...(deg‘𝑃))(((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘))) |
15 | 9, 11, 14 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝑃‘𝑋) = Σ𝑘 ∈ (0...(deg‘𝑃))(((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘))) |
16 | fzfid 14011 | . . 3 ⊢ (𝜑 → (0...(deg‘𝑃)) ∈ Fin) | |
17 | 2 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → 𝑆 ∈ (SubRing‘ℂfld)) |
18 | subrgsubg 20594 | . . . . . . . 8 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ∈ (SubGrp‘ℂfld)) | |
19 | cnfld0 21423 | . . . . . . . . 9 ⊢ 0 = (0g‘ℂfld) | |
20 | 19 | subg0cl 19165 | . . . . . . . 8 ⊢ (𝑆 ∈ (SubGrp‘ℂfld) → 0 ∈ 𝑆) |
21 | 2, 18, 20 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑆) |
22 | 12 | coef2 26285 | . . . . . . 7 ⊢ ((𝑃 ∈ (Poly‘𝑆) ∧ 0 ∈ 𝑆) → (coeff‘𝑃):ℕ0⟶𝑆) |
23 | 9, 21, 22 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (coeff‘𝑃):ℕ0⟶𝑆) |
24 | 23 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → (coeff‘𝑃):ℕ0⟶𝑆) |
25 | elfznn0 13657 | . . . . . 6 ⊢ (𝑘 ∈ (0...(deg‘𝑃)) → 𝑘 ∈ ℕ0) | |
26 | 25 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → 𝑘 ∈ ℕ0) |
27 | 24, 26 | ffvelcdmd 7105 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → ((coeff‘𝑃)‘𝑘) ∈ 𝑆) |
28 | 10 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → 𝑋 ∈ 𝑆) |
29 | 17, 28, 26 | cnsrexpcl 43154 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → (𝑋↑𝑘) ∈ 𝑆) |
30 | cnfldmul 21390 | . . . . 5 ⊢ · = (.r‘ℂfld) | |
31 | 30 | subrgmcl 20601 | . . . 4 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ ((coeff‘𝑃)‘𝑘) ∈ 𝑆 ∧ (𝑋↑𝑘) ∈ 𝑆) → (((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘)) ∈ 𝑆) |
32 | 17, 27, 29, 31 | syl3anc 1370 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → (((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘)) ∈ 𝑆) |
33 | 2, 16, 32 | fsumcnsrcl 43155 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...(deg‘𝑃))(((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘)) ∈ 𝑆) |
34 | 15, 33 | eqeltrd 2839 | 1 ⊢ (𝜑 → (𝑃‘𝑋) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 · cmul 11158 ℕ0cn0 12524 ...cfz 13544 ↑cexp 14099 Σcsu 15719 SubGrpcsubg 19151 SubRingcsubrg 20586 ℂfldccnfld 21382 Polycply 26238 coeffccoe 26240 degcdgr 26241 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-rlim 15522 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-cnfld 21383 df-0p 25719 df-ply 26242 df-coe 26244 df-dgr 26245 |
This theorem is referenced by: rngunsnply 43158 |
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