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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 2622 | . . . 4 ⊢ (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅) | |
3 | eqid 2622 | . . . 4 ⊢ (1𝑜 mPwSer 𝑅) = (1𝑜 mPwSer 𝑅) | |
4 | eqid 2622 | . . . 4 ⊢ (Base‘(1𝑜 mPoly 𝑅)) = (Base‘(1𝑜 mPoly 𝑅)) | |
5 | ply1val.2 | . . . . 5 ⊢ 𝑆 = (PwSer1‘𝑅) | |
6 | eqid 2622 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
7 | 5, 6, 3 | psr1bas2 19560 | . . . 4 ⊢ (Base‘𝑆) = (Base‘(1𝑜 mPwSer 𝑅)) |
8 | 2, 3, 4, 7 | mplbasss 19432 | . . 3 ⊢ (Base‘(1𝑜 mPoly 𝑅)) ⊆ (Base‘𝑆) |
9 | ply1val.1 | . . . . 5 ⊢ 𝑃 = (Poly1‘𝑅) | |
10 | 9, 5 | ply1val 19564 | . . . 4 ⊢ 𝑃 = (𝑆 ↾s (Base‘(1𝑜 mPoly 𝑅))) |
11 | 10, 6 | ressbas2 15931 | . . 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 2647 | 1 ⊢ 𝑈 = (Base‘(1𝑜 mPoly 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ⊆ wss 3574 ‘cfv 5888 (class class class)co 6650 1𝑜c1o 7553 Basecbs 15857 mPwSer cmps 19351 mPoly cmpl 19353 PwSer1cps1 19545 Poly1cpl1 19547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-dec 11494 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-ple 15961 df-psr 19356 df-mpl 19358 df-opsr 19360 df-psr1 19550 df-ply1 19552 |
This theorem is referenced by: ply1lss 19566 ply1subrg 19567 ply1crng 19568 ply1assa 19569 ply1basf 19572 ply1bascl2 19574 vr1cl 19587 ressply1bas2 19598 ressply1add 19600 ressply1mul 19601 ressply1vsca 19602 subrgply1 19603 ply1baspropd 19613 ply1ring 19618 ply1lmod 19622 ply1mpl0 19625 ply1mpl1 19627 subrg1asclcl 19630 subrgvr1cl 19632 coe1add 19634 coe1tm 19643 ply1coe 19666 evls1rhm 19687 evls1sca 19688 evl1rhm 19696 evl1sca 19698 evl1var 19700 evls1var 19702 mpfpf1 19715 pf1mpf 19716 deg1xrf 23841 deg1cl 23843 deg1nn0cl 23848 deg1ldg 23852 deg1leb 23855 deg1val 23856 deg1vscale 23864 deg1vsca 23865 deg1mulle2 23869 deg1le0 23871 |
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