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Mirrors > Home > MPE Home > Th. List > ply1assa | Structured version Visualization version GIF version |
Description: The ring of univariate polynomials is an associative algebra. (Contributed by Mario Carneiro, 9-Feb-2015.) |
Ref | Expression |
---|---|
ply1val.1 | ⊢ 𝑃 = (Poly1‘𝑅) |
Ref | Expression |
---|---|
ply1assa | ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | crngring 19302 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | ply1val.1 | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | eqid 2821 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
4 | eqid 2821 | . . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
5 | 2, 3, 4 | ply1subrg 20359 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘𝑃) ∈ (SubRing‘(PwSer1‘𝑅))) |
6 | 1, 5 | syl 17 | . 2 ⊢ (𝑅 ∈ CRing → (Base‘𝑃) ∈ (SubRing‘(PwSer1‘𝑅))) |
7 | 2, 3, 4 | ply1lss 20358 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘𝑃) ∈ (LSubSp‘(PwSer1‘𝑅))) |
8 | 1, 7 | syl 17 | . 2 ⊢ (𝑅 ∈ CRing → (Base‘𝑃) ∈ (LSubSp‘(PwSer1‘𝑅))) |
9 | 3 | psr1assa 20350 | . . 3 ⊢ (𝑅 ∈ CRing → (PwSer1‘𝑅) ∈ AssAlg) |
10 | eqid 2821 | . . . . 5 ⊢ (1r‘(PwSer1‘𝑅)) = (1r‘(PwSer1‘𝑅)) | |
11 | 10 | subrg1cl 19537 | . . . 4 ⊢ ((Base‘𝑃) ∈ (SubRing‘(PwSer1‘𝑅)) → (1r‘(PwSer1‘𝑅)) ∈ (Base‘𝑃)) |
12 | 6, 11 | syl 17 | . . 3 ⊢ (𝑅 ∈ CRing → (1r‘(PwSer1‘𝑅)) ∈ (Base‘𝑃)) |
13 | eqid 2821 | . . . . 5 ⊢ (Base‘(PwSer1‘𝑅)) = (Base‘(PwSer1‘𝑅)) | |
14 | 13 | subrgss 19530 | . . . 4 ⊢ ((Base‘𝑃) ∈ (SubRing‘(PwSer1‘𝑅)) → (Base‘𝑃) ⊆ (Base‘(PwSer1‘𝑅))) |
15 | 6, 14 | syl 17 | . . 3 ⊢ (𝑅 ∈ CRing → (Base‘𝑃) ⊆ (Base‘(PwSer1‘𝑅))) |
16 | 2, 3 | ply1val 20356 | . . . . 5 ⊢ 𝑃 = ((PwSer1‘𝑅) ↾s (Base‘(1o mPoly 𝑅))) |
17 | 2, 3, 4 | ply1bas 20357 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘(1o mPoly 𝑅)) |
18 | 17 | oveq2i 7161 | . . . . 5 ⊢ ((PwSer1‘𝑅) ↾s (Base‘𝑃)) = ((PwSer1‘𝑅) ↾s (Base‘(1o mPoly 𝑅))) |
19 | 16, 18 | eqtr4i 2847 | . . . 4 ⊢ 𝑃 = ((PwSer1‘𝑅) ↾s (Base‘𝑃)) |
20 | eqid 2821 | . . . 4 ⊢ (LSubSp‘(PwSer1‘𝑅)) = (LSubSp‘(PwSer1‘𝑅)) | |
21 | 19, 20, 13, 10 | issubassa 20092 | . . 3 ⊢ (((PwSer1‘𝑅) ∈ AssAlg ∧ (1r‘(PwSer1‘𝑅)) ∈ (Base‘𝑃) ∧ (Base‘𝑃) ⊆ (Base‘(PwSer1‘𝑅))) → (𝑃 ∈ AssAlg ↔ ((Base‘𝑃) ∈ (SubRing‘(PwSer1‘𝑅)) ∧ (Base‘𝑃) ∈ (LSubSp‘(PwSer1‘𝑅))))) |
22 | 9, 12, 15, 21 | syl3anc 1367 | . 2 ⊢ (𝑅 ∈ CRing → (𝑃 ∈ AssAlg ↔ ((Base‘𝑃) ∈ (SubRing‘(PwSer1‘𝑅)) ∧ (Base‘𝑃) ∈ (LSubSp‘(PwSer1‘𝑅))))) |
23 | 6, 8, 22 | mpbir2and 711 | 1 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ‘cfv 6349 (class class class)co 7150 1oc1o 8089 Basecbs 16477 ↾s cress 16478 1rcur 19245 Ringcrg 19291 CRingccrg 19292 SubRingcsubrg 19525 LSubSpclss 19697 AssAlgcasa 20076 mPoly cmpl 20127 PwSer1cps1 20337 Poly1cpl1 20339 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 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-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-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 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-sca 16575 df-vsca 16576 df-tset 16578 df-ple 16579 df-0g 16709 df-gsum 16710 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-ghm 18350 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-subrg 19527 df-lmod 19630 df-lss 19698 df-assa 20079 df-psr 20130 df-mpl 20132 df-opsr 20134 df-psr1 20342 df-ply1 20344 |
This theorem is referenced by: lply1binomsc 20469 evl1vsd 20501 pf1subrg 20505 evl1scvarpw 20520 mat2pmatmul 21333 mat2pmatlin 21337 monmatcollpw 21381 pmatcollpwlem 21382 chpscmatgsumbin 21446 fta1blem 24756 |
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