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 20549 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
4 | 3 | subrgss 19536 | . . . . . 6 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ⊆ ℂ) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
6 | plyss 24789 | . . . . 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 24829 | . . 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 13342 | . . 3 ⊢ (𝜑 → (0...(deg‘𝑃)) ∈ Fin) | |
17 | 2 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → 𝑆 ∈ (SubRing‘ℂfld)) |
18 | subrgsubg 19541 | . . . . . . . 8 ⊢ (𝑆 ∈ (SubRing‘ℂfld) → 𝑆 ∈ (SubGrp‘ℂfld)) | |
19 | cnfld0 20569 | . . . . . . . . 9 ⊢ 0 = (0g‘ℂfld) | |
20 | 19 | subg0cl 18287 | . . . . . . . 8 ⊢ (𝑆 ∈ (SubGrp‘ℂfld) → 0 ∈ 𝑆) |
21 | 2, 18, 20 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑆) |
22 | 12 | coef2 24821 | . . . . . . 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 13001 | . . . . . 6 ⊢ (𝑘 ∈ (0...(deg‘𝑃)) → 𝑘 ∈ ℕ0) | |
26 | 25 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → 𝑘 ∈ ℕ0) |
27 | 24, 26 | ffvelrnd 6852 | . . . 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 39785 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(deg‘𝑃))) → (𝑋↑𝑘) ∈ 𝑆) |
30 | cnfldmul 20551 | . . . . 5 ⊢ · = (.r‘ℂfld) | |
31 | 30 | subrgmcl 19547 | . . . 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 39786 | . 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 1537 ∈ wcel 2114 ⊆ wss 3936 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 0cc0 10537 · cmul 10542 ℕ0cn0 11898 ...cfz 12893 ↑cexp 13430 Σcsu 15042 SubGrpcsubg 18273 SubRingcsubrg 19531 ℂfldccnfld 20545 Polycply 24774 coeffccoe 24776 degcdgr 24777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-rlim 14846 df-sum 15043 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-subg 18276 df-cmn 18908 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-subrg 19533 df-cnfld 20546 df-0p 24271 df-ply 24778 df-coe 24780 df-dgr 24781 |
This theorem is referenced by: rngunsnply 39793 |
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