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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 20288 | . 2 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅) |
3 | eqid 2818 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
4 | eqid 2818 | . . 3 ⊢ (1o mVar 𝑅) = (1o mVar 𝑅) | |
5 | vr1cl.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
6 | eqid 2818 | . . . 4 ⊢ (PwSer1‘𝑅) = (PwSer1‘𝑅) | |
7 | vr1cl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
8 | 5, 6, 7 | ply1bas 20291 | . . 3 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅)) |
9 | 1onn 8254 | . . . 4 ⊢ 1o ∈ ω | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑅 ∈ Ring → 1o ∈ ω) |
11 | id 22 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring) | |
12 | 0lt1o 8118 | . . . 4 ⊢ ∅ ∈ 1o | |
13 | 12 | a1i 11 | . . 3 ⊢ (𝑅 ∈ Ring → ∅ ∈ 1o) |
14 | 3, 4, 8, 10, 11, 13 | mvrcl 20157 | . 2 ⊢ (𝑅 ∈ Ring → ((1o mVar 𝑅)‘∅) ∈ 𝐵) |
15 | 2, 14 | eqeltrid 2914 | 1 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∅c0 4288 ‘cfv 6348 (class class class)co 7145 ωcom 7569 1oc1o 8084 Basecbs 16471 Ringcrg 19226 mVar cmvr 20060 mPoly cmpl 20061 PwSer1cps1 20271 var1cv1 20272 Poly1cpl1 20273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-tset 16572 df-ple 16573 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-mgp 19169 df-ur 19181 df-ring 19228 df-psr 20064 df-mvr 20065 df-mpl 20066 df-opsr 20068 df-psr1 20276 df-vr1 20277 df-ply1 20278 |
This theorem is referenced by: ply1moncl 20367 coe1pwmul 20375 ply1scltm 20377 ply1coefsupp 20391 ply1coe 20392 gsummoncoe1 20400 lply1binom 20402 evls1varpw 20418 evl1var 20427 evl1vard 20428 evls1var 20429 pf1id 20438 evl1scvarpw 20454 evl1scvarpwval 20455 evl1gsummon 20456 pmatcollpwscmatlem1 21325 mply1topmatcllem 21339 mply1topmatcl 21341 pm2mpghm 21352 monmat2matmon 21360 pm2mp 21361 chmatcl 21364 chmatval 21365 chpmat0d 21370 chpmat1dlem 21371 chpmat1d 21372 chpdmatlem0 21373 chpdmatlem2 21375 chpdmatlem3 21376 chpscmat 21378 chpscmatgsumbin 21380 chpscmatgsummon 21381 chp0mat 21382 chpidmat 21383 chfacfscmulcl 21393 chfacfscmul0 21394 chfacfscmulgsum 21396 cpmadugsumlemB 21410 cpmadugsumlemC 21411 cpmadugsumlemF 21412 cpmadugsumfi 21413 cpmidgsum2 21415 deg1pw 24641 ply1remlem 24683 fta1blem 24689 plypf1 24729 lgsqrlem2 25850 lgsqrlem3 25851 lgsqrlem4 25852 hbtlem4 39604 idomrootle 39673 ply1vr1smo 44363 ply1mulgsumlem4 44371 ply1mulgsum 44372 linply1 44375 |
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