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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 19043 | . . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
2 | ply1val.1 | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | eqid 2772 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
4 | eqid 2772 | . . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
5 | 2, 3, 4 | ply1subrg 20080 | . . 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 20079 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘𝑃) ∈ (LSubSp‘(PwSer1‘𝑅))) |
8 | 1, 7 | syl 17 | . 2 ⊢ (𝑅 ∈ CRing → (Base‘𝑃) ∈ (LSubSp‘(PwSer1‘𝑅))) |
9 | 3 | psr1assa 20071 | . . 3 ⊢ (𝑅 ∈ CRing → (PwSer1‘𝑅) ∈ AssAlg) |
10 | eqid 2772 | . . . . 5 ⊢ (1r‘(PwSer1‘𝑅)) = (1r‘(PwSer1‘𝑅)) | |
11 | 10 | subrg1cl 19278 | . . . 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 2772 | . . . . 5 ⊢ (Base‘(PwSer1‘𝑅)) = (Base‘(PwSer1‘𝑅)) | |
14 | 13 | subrgss 19271 | . . . 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 20077 | . . . . 5 ⊢ 𝑃 = ((PwSer1‘𝑅) ↾s (Base‘(1o mPoly 𝑅))) |
17 | 2, 3, 4 | ply1bas 20078 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘(1o mPoly 𝑅)) |
18 | 17 | oveq2i 6985 | . . . . 5 ⊢ ((PwSer1‘𝑅) ↾s (Base‘𝑃)) = ((PwSer1‘𝑅) ↾s (Base‘(1o mPoly 𝑅))) |
19 | 16, 18 | eqtr4i 2799 | . . . 4 ⊢ 𝑃 = ((PwSer1‘𝑅) ↾s (Base‘𝑃)) |
20 | eqid 2772 | . . . 4 ⊢ (LSubSp‘(PwSer1‘𝑅)) = (LSubSp‘(PwSer1‘𝑅)) | |
21 | 19, 20, 13, 10 | issubassa 19830 | . . 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 1351 | . 2 ⊢ (𝑅 ∈ CRing → (𝑃 ∈ AssAlg ↔ ((Base‘𝑃) ∈ (SubRing‘(PwSer1‘𝑅)) ∧ (Base‘𝑃) ∈ (LSubSp‘(PwSer1‘𝑅))))) |
23 | 6, 8, 22 | mpbir2and 700 | 1 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ⊆ wss 3823 ‘cfv 6185 (class class class)co 6974 1oc1o 7896 Basecbs 16337 ↾s cress 16338 1rcur 18986 Ringcrg 19032 CRingccrg 19033 SubRingcsubrg 19266 LSubSpclss 19437 AssAlgcasa 19815 mPoly cmpl 19859 PwSer1cps1 20058 Poly1cpl1 20060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-ofr 7226 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-oadd 7907 df-er 8087 df-map 8206 df-pm 8207 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-oi 8767 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-fz 12707 df-fzo 12848 df-seq 13183 df-hash 13504 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-sca 16435 df-vsca 16436 df-tset 16438 df-ple 16439 df-0g 16569 df-gsum 16570 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-mhm 17815 df-submnd 17816 df-grp 17906 df-minusg 17907 df-sbg 17908 df-mulg 18024 df-subg 18072 df-ghm 18139 df-cntz 18230 df-cmn 18680 df-abl 18681 df-mgp 18975 df-ur 18987 df-ring 19034 df-cring 19035 df-subrg 19268 df-lmod 19370 df-lss 19438 df-assa 19818 df-psr 19862 df-mpl 19864 df-opsr 19866 df-psr1 20063 df-ply1 20065 |
This theorem is referenced by: lply1binomsc 20190 evl1vsd 20221 pf1subrg 20225 evl1scvarpw 20240 mat2pmatmul 21055 mat2pmatlin 21059 monmatcollpw 21103 pmatcollpwlem 21104 chpscmatgsumbin 21168 fta1blem 24477 |
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