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Mirrors > Home > MPE Home > Th. List > vr1cl | Structured version Visualization version GIF version |
Description: The generator of a univariate polynomial algebra is contained in the base set. (Contributed by Stefan O'Rear, 19-Mar-2015.) |
Ref | Expression |
---|---|
vr1cl.x | ⊢ 𝑋 = (var1‘𝑅) |
vr1cl.p | ⊢ 𝑃 = (Poly1‘𝑅) |
vr1cl.b | ⊢ 𝐵 = (Base‘𝑃) |
Ref | Expression |
---|---|
vr1cl | ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vr1cl.x | . . 3 ⊢ 𝑋 = (var1‘𝑅) | |
2 | 1 | vr1val 19881 | . 2 ⊢ 𝑋 = ((1𝑜 mVar 𝑅)‘∅) |
3 | eqid 2798 | . . 3 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅) | |
4 | eqid 2798 | . . 3 ⊢ (1𝑜 mVar 𝑅) = (1𝑜 mVar 𝑅) | |
5 | vr1cl.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
6 | eqid 2798 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
7 | vr1cl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
8 | 5, 6, 7 | ply1bas 19884 | . . 3 ⊢ 𝐵 = (Base‘(1𝑜 mPoly 𝑅)) |
9 | 1onn 7958 | . . . 4 ⊢ 1𝑜 ∈ ω | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑅 ∈ Ring → 1𝑜 ∈ ω) |
11 | id 22 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring) | |
12 | 0lt1o 7823 | . . . 4 ⊢ ∅ ∈ 1𝑜 | |
13 | 12 | a1i 11 | . . 3 ⊢ (𝑅 ∈ Ring → ∅ ∈ 1𝑜) |
14 | 3, 4, 8, 10, 11, 13 | mvrcl 19769 | . 2 ⊢ (𝑅 ∈ Ring → ((1𝑜 mVar 𝑅)‘∅) ∈ 𝐵) |
15 | 2, 14 | syl5eqel 2881 | 1 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ∅c0 4114 ‘cfv 6100 (class class class)co 6877 ωcom 7298 1𝑜c1o 7791 Basecbs 16181 Ringcrg 18860 mVar cmvr 19672 mPoly cmpl 19673 PwSer1cps1 19864 var1cv1 19865 Poly1cpl1 19866 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-rep 4963 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-int 4667 df-iun 4711 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-of 7130 df-om 7299 df-1st 7400 df-2nd 7401 df-supp 7532 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-1o 7798 df-oadd 7802 df-er 7981 df-map 8096 df-en 8195 df-dom 8196 df-sdom 8197 df-fin 8198 df-fsupp 8517 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-nn 11312 df-2 11373 df-3 11374 df-4 11375 df-5 11376 df-6 11377 df-7 11378 df-8 11379 df-9 11380 df-n0 11578 df-z 11664 df-dec 11781 df-uz 11928 df-fz 12578 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-sca 16280 df-vsca 16281 df-tset 16283 df-ple 16284 df-0g 16414 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-grp 17738 df-mgp 18803 df-ur 18815 df-ring 18862 df-psr 19676 df-mvr 19677 df-mpl 19678 df-opsr 19680 df-psr1 19869 df-vr1 19870 df-ply1 19871 |
This theorem is referenced by: ply1moncl 19960 coe1pwmul 19968 ply1scltm 19970 ply1coefsupp 19984 ply1coe 19985 gsummoncoe1 19993 lply1binom 19995 evls1varpw 20010 evl1var 20019 evl1vard 20020 evls1var 20021 pf1id 20030 evl1scvarpw 20046 evl1scvarpwval 20047 evl1gsummon 20048 pmatcollpwscmatlem1 20919 mply1topmatcllem 20933 mply1topmatcl 20935 pm2mpghm 20946 monmat2matmon 20954 pm2mp 20955 chmatcl 20958 chmatval 20959 chpmat0d 20964 chpmat1dlem 20965 chpmat1d 20966 chpdmatlem0 20967 chpdmatlem2 20969 chpdmatlem3 20970 chpscmat 20972 chpscmatgsumbin 20974 chpscmatgsummon 20975 chp0mat 20976 chpidmat 20977 chfacfscmulcl 20987 chfacfscmul0 20988 chfacfscmulgsum 20990 cpmadugsumlemB 21004 cpmadugsumlemC 21005 cpmadugsumlemF 21006 cpmadugsumfi 21007 cpmidgsum2 21009 deg1pw 24218 ply1remlem 24260 fta1blem 24266 plypf1 24306 lgsqrlem2 25421 lgsqrlem3 25422 lgsqrlem4 25423 hbtlem4 38470 idomrootle 38547 ply1vr1smo 42957 ply1mulgsumlem4 42965 ply1mulgsum 42966 linply1 42969 |
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