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Mirrors > Home > MPE Home > Th. List > ply1bas | Structured version Visualization version GIF version |
Description: The value of the base set of univariate polynomials. (Contributed by Mario Carneiro, 9-Feb-2015.) |
Ref | Expression |
---|---|
ply1val.1 | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1val.2 | ⊢ 𝑆 = (PwSer1‘𝑅) |
ply1bas.u | ⊢ 𝑈 = (Base‘𝑃) |
Ref | Expression |
---|---|
ply1bas | ⊢ 𝑈 = (Base‘(1o mPoly 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1bas.u | . 2 ⊢ 𝑈 = (Base‘𝑃) | |
2 | eqid 2825 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
3 | eqid 2825 | . . . 4 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
4 | eqid 2825 | . . . 4 ⊢ (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅)) | |
5 | ply1val.2 | . . . . 5 ⊢ 𝑆 = (PwSer1‘𝑅) | |
6 | eqid 2825 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
7 | 5, 6, 3 | psr1bas2 19920 | . . . 4 ⊢ (Base‘𝑆) = (Base‘(1o mPwSer 𝑅)) |
8 | 2, 3, 4, 7 | mplbasss 19793 | . . 3 ⊢ (Base‘(1o mPoly 𝑅)) ⊆ (Base‘𝑆) |
9 | ply1val.1 | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
10 | 9, 5 | ply1val 19924 | . . . 4 ⊢ 𝑃 = (𝑆 ↾s (Base‘(1o mPoly 𝑅))) |
11 | 10, 6 | ressbas2 16294 | . . 3 ⊢ ((Base‘(1o mPoly 𝑅)) ⊆ (Base‘𝑆) → (Base‘(1o mPoly 𝑅)) = (Base‘𝑃)) |
12 | 8, 11 | ax-mp 5 | . 2 ⊢ (Base‘(1o mPoly 𝑅)) = (Base‘𝑃) |
13 | 1, 12 | eqtr4i 2852 | 1 ⊢ 𝑈 = (Base‘(1o mPoly 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ⊆ wss 3798 ‘cfv 6123 (class class class)co 6905 1oc1o 7819 Basecbs 16222 mPwSer cmps 19712 mPoly cmpl 19714 PwSer1cps1 19905 Poly1cpl1 19907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-dec 11822 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-ple 16325 df-psr 19717 df-mpl 19719 df-opsr 19721 df-psr1 19910 df-ply1 19912 |
This theorem is referenced by: ply1lss 19926 ply1subrg 19927 ply1crng 19928 ply1assa 19929 ply1basf 19932 ply1bascl2 19934 vr1cl 19947 ressply1bas2 19958 ressply1add 19960 ressply1mul 19961 ressply1vsca 19962 subrgply1 19963 ply1baspropd 19973 ply1ring 19978 ply1lmod 19982 ply1mpl0 19985 ply1mpl1 19987 subrg1asclcl 19990 subrgvr1cl 19992 coe1add 19994 coe1tm 20003 ply1coe 20026 evls1rhm 20047 evls1sca 20048 evl1rhm 20056 evl1sca 20058 evl1var 20060 evls1var 20062 mpfpf1 20075 pf1mpf 20076 deg1xrf 24240 deg1cl 24242 deg1nn0cl 24247 deg1ldg 24251 deg1leb 24254 deg1val 24255 deg1vscale 24263 deg1vsca 24264 deg1mulle2 24268 deg1le0 24270 |
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