| 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 21368 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
| 4 | 3 | subrgss 20572 | . . . . . 6 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ⊆ ℂ) |
| 5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
| 6 | plyss 26238 | . . . . 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 3984 | . . 3 ⊢ (𝜑 → 𝑃 ∈ (Poly‘𝑆)) |
| 10 | cnsrplycl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
| 11 | 5, 10 | sseldd 3984 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
| 12 | eqid 2737 | . . . 4 ⊢ (coeff‘𝑃) = (coeff‘𝑃) | |
| 13 | eqid 2737 | . . . 4 ⊢ (deg‘𝑃) = (deg‘𝑃) | |
| 14 | 12, 13 | coeid2 26278 | . . 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 14014 | . . 3 ⊢ (𝜑 → (0...(deg‘𝑃)) ∈ Fin) | |
| 17 | 2 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → 𝑆 ∈ (SubRing‘ℂfld)) |
| 18 | subrgsubg 20577 | . . . . . . . 8 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ∈ (SubGrp‘ℂfld)) | |
| 19 | cnfld0 21405 | . . . . . . . . 9 ⊢ 0 = (0g‘ℂfld) | |
| 20 | 19 | subg0cl 19152 | . . . . . . . 8 ⊢ (𝑆 ∈ (SubGrp‘ℂfld) → 0 ∈ 𝑆) |
| 21 | 2, 18, 20 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑆) |
| 22 | 12 | coef2 26270 | . . . . . . 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 13660 | . . . . . 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 43177 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → (𝑋↑𝑘) ∈ 𝑆) |
| 30 | cnfldmul 21372 | . . . . 5 ⊢ · = (.r‘ℂfld) | |
| 31 | 30 | subrgmcl 20584 | . . . 4 ⊢ ((𝑆 ∈ (SubRing‘ℂfld) ∧ ((coeff‘𝑃)‘𝑘) ∈ 𝑆 ∧ (𝑋↑𝑘) ∈ 𝑆) → (((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘)) ∈ 𝑆) |
| 32 | 17, 27, 29, 31 | syl3anc 1373 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → (((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘)) ∈ 𝑆) |
| 33 | 2, 16, 32 | fsumcnsrcl 43178 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...(deg‘𝑃))(((coeff‘𝑃)‘𝑘) · (𝑋↑𝑘)) ∈ 𝑆) |
| 34 | 15, 33 | eqeltrd 2841 | 1 ⊢ (𝜑 → (𝑃‘𝑋) ∈ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 · cmul 11160 ℕ0cn0 12526 ...cfz 13547 ↑cexp 14102 Σcsu 15722 SubGrpcsubg 19138 SubRingcsubrg 20569 ℂfldccnfld 21364 Polycply 26223 coeffccoe 26225 degcdgr 26226 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-subg 19141 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-cnfld 21365 df-0p 25705 df-ply 26227 df-coe 26229 df-dgr 26230 |
| This theorem is referenced by: rngunsnply 43181 |
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