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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 22193 | . 2 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅) |
| 3 | eqid 2737 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
| 4 | eqid 2737 | . . 3 ⊢ (1o mVar 𝑅) = (1o mVar 𝑅) | |
| 5 | vr1cl.p | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
| 6 | vr1cl.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
| 7 | 5, 6 | ply1bas 22196 | . . 3 ⊢ 𝐵 = (Base‘(1o mPoly 𝑅)) |
| 8 | 1onn 8678 | . . . 4 ⊢ 1o ∈ ω | |
| 9 | 8 | a1i 11 | . . 3 ⊢ (𝑅 ∈ Ring → 1o ∈ ω) |
| 10 | id 22 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring) | |
| 11 | 0lt1o 8542 | . . . 4 ⊢ ∅ ∈ 1o | |
| 12 | 11 | a1i 11 | . . 3 ⊢ (𝑅 ∈ Ring → ∅ ∈ 1o) |
| 13 | 3, 4, 7, 9, 10, 12 | mvrcl 22012 | . 2 ⊢ (𝑅 ∈ Ring → ((1o mVar 𝑅)‘∅) ∈ 𝐵) |
| 14 | 2, 13 | eqeltrid 2845 | 1 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 ωcom 7887 1oc1o 8499 Basecbs 17247 Ringcrg 20230 mVar cmvr 21925 mPoly cmpl 21926 var1cv1 22177 Poly1cpl1 22178 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-tset 17316 df-ple 17317 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-mgp 20138 df-ur 20179 df-ring 20232 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-psr1 22181 df-vr1 22182 df-ply1 22183 |
| This theorem is referenced by: ply1moncl 22274 coe1pwmul 22282 ply1scltm 22284 ply1idvr1 22298 ply1coefsupp 22301 ply1coe 22302 gsummoncoe1 22312 lply1binom 22314 ply1fermltlchr 22316 evls1varpw 22331 evl1var 22340 evl1vard 22341 evls1var 22342 pf1id 22351 evl1scvarpw 22367 evl1scvarpwval 22368 evl1gsummon 22369 evls1varpwval 22372 evls1fpws 22373 rhmply1vr1 22391 rhmply1mon 22393 pmatcollpwscmatlem1 22795 mply1topmatcllem 22809 mply1topmatcl 22811 pm2mpghm 22822 monmat2matmon 22830 pm2mp 22831 chmatcl 22834 chmatval 22835 chpmat0d 22840 chpmat1dlem 22841 chpmat1d 22842 chpdmatlem0 22843 chpdmatlem2 22845 chpdmatlem3 22846 chpscmat 22848 chpscmatgsumbin 22850 chpscmatgsummon 22851 chp0mat 22852 chpidmat 22853 chfacfscmulcl 22863 chfacfscmul0 22864 chfacfscmulgsum 22866 cpmadugsumlemB 22880 cpmadugsumlemC 22881 cpmadugsumlemF 22882 cpmadugsumfi 22883 cpmidgsum2 22885 deg1pw 26160 ply1remlem 26204 fta1blem 26210 idomrootle 26212 plypf1 26251 lgsqrlem2 27391 lgsqrlem3 27392 lgsqrlem4 27393 coe1vr1 33613 deg1vr 33614 gsummoncoe1fzo 33618 ply1degltdimlem 33673 ply1degltdim 33674 algextdeglem4 33761 rtelextdg2lem 33767 2sqr3minply 33791 aks6d1c1p2 42110 aks6d1c1p3 42111 aks6d1c1p7 42114 aks6d1c1 42117 aks6d1c2lem4 42128 aks6d1c5lem0 42136 aks6d1c5lem3 42138 aks6d1c5 42140 aks6d1c6lem1 42171 aks5lem2 42188 aks5lem3a 42190 aks5lem5a 42192 hbtlem4 43138 ply1vr1smo 48299 ply1mulgsumlem4 48306 ply1mulgsum 48307 linply1 48310 |
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