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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 2798 | . . . 4 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
3 | eqid 2798 | . . . 4 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
4 | eqid 2798 | . . . 4 ⊢ (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅)) | |
5 | ply1val.2 | . . . . 5 ⊢ 𝑆 = (PwSer1‘𝑅) | |
6 | eqid 2798 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
7 | 5, 6, 3 | psr1bas2 20819 | . . . 4 ⊢ (Base‘𝑆) = (Base‘(1o mPwSer 𝑅)) |
8 | 2, 3, 4, 7 | mplbasss 20670 | . . 3 ⊢ (Base‘(1o mPoly 𝑅)) ⊆ (Base‘𝑆) |
9 | ply1val.1 | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
10 | 9, 5 | ply1val 20823 | . . . 4 ⊢ 𝑃 = (𝑆 ↾s (Base‘(1o mPoly 𝑅))) |
11 | 10, 6 | ressbas2 16547 | . . 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 2824 | 1 ⊢ 𝑈 = (Base‘(1o mPoly 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ⊆ wss 3881 ‘cfv 6324 (class class class)co 7135 1oc1o 8078 Basecbs 16475 mPwSer cmps 20589 mPoly cmpl 20591 PwSer1cps1 20804 Poly1cpl1 20806 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 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-dec 12087 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-ple 16577 df-psr 20594 df-mpl 20596 df-opsr 20598 df-psr1 20809 df-ply1 20811 |
This theorem is referenced by: ply1lss 20825 ply1subrg 20826 ply1crng 20827 ply1assa 20828 ply1basf 20831 ply1bascl2 20833 vr1cl 20846 ressply1bas2 20857 ressply1add 20859 ressply1mul 20860 ressply1vsca 20861 subrgply1 20862 ply1baspropd 20872 ply1ring 20877 ply1lmod 20881 ply1mpl0 20884 ply1mpl1 20886 subrg1asclcl 20889 subrgvr1cl 20891 coe1add 20893 coe1tm 20902 ply1coe 20925 evls1rhm 20946 evls1sca 20947 evl1rhm 20956 evl1sca 20958 evl1var 20960 evls1var 20962 mpfpf1 20975 pf1mpf 20976 deg1xrf 24682 deg1cl 24684 deg1nn0cl 24689 deg1ldg 24693 deg1leb 24696 deg1val 24697 deg1vscale 24705 deg1vsca 24706 deg1mulle2 24710 deg1le0 24712 fply1 30982 |
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