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Mirrors > Home > MPE Home > Th. List > Mathboxes > ply1vr1smo | Structured version Visualization version GIF version |
Description: The variable in a polynomial expressed as scaled monomial. (Contributed by AV, 12-Aug-2019.) |
Ref | Expression |
---|---|
ply1vr1smo.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1vr1smo.i | ⊢ 1 = (1r‘𝑅) |
ply1vr1smo.t | ⊢ · = ( ·𝑠 ‘𝑃) |
ply1vr1smo.m | ⊢ 𝐺 = (mulGrp‘𝑃) |
ply1vr1smo.e | ⊢ ↑ = (.g‘𝐺) |
ply1vr1smo.x | ⊢ 𝑋 = (var1‘𝑅) |
Ref | Expression |
---|---|
ply1vr1smo | ⊢ (𝑅 ∈ Ring → ( 1 · (1 ↑ 𝑋)) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1vr1smo.i | . . . 4 ⊢ 1 = (1r‘𝑅) | |
2 | ply1vr1smo.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
3 | 2 | ply1sca 21368 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃)) |
4 | 3 | fveq2d 6765 | . . . 4 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) = (1r‘(Scalar‘𝑃))) |
5 | 1, 4 | eqtrid 2789 | . . 3 ⊢ (𝑅 ∈ Ring → 1 = (1r‘(Scalar‘𝑃))) |
6 | 5 | oveq1d 7275 | . 2 ⊢ (𝑅 ∈ Ring → ( 1 · (1 ↑ 𝑋)) = ((1r‘(Scalar‘𝑃)) · (1 ↑ 𝑋))) |
7 | 2 | ply1lmod 21367 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod) |
8 | ply1vr1smo.x | . . . . . 6 ⊢ 𝑋 = (var1‘𝑅) | |
9 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
10 | 8, 2, 9 | vr1cl 21332 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃)) |
11 | ply1vr1smo.m | . . . . . . 7 ⊢ 𝐺 = (mulGrp‘𝑃) | |
12 | 11, 9 | mgpbas 19670 | . . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝐺) |
13 | ply1vr1smo.e | . . . . . 6 ⊢ ↑ = (.g‘𝐺) | |
14 | 12, 13 | mulg1 18655 | . . . . 5 ⊢ (𝑋 ∈ (Base‘𝑃) → (1 ↑ 𝑋) = 𝑋) |
15 | 10, 14 | syl 17 | . . . 4 ⊢ (𝑅 ∈ Ring → (1 ↑ 𝑋) = 𝑋) |
16 | 15, 10 | eqeltrd 2837 | . . 3 ⊢ (𝑅 ∈ Ring → (1 ↑ 𝑋) ∈ (Base‘𝑃)) |
17 | eqid 2737 | . . . 4 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
18 | ply1vr1smo.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝑃) | |
19 | eqid 2737 | . . . 4 ⊢ (1r‘(Scalar‘𝑃)) = (1r‘(Scalar‘𝑃)) | |
20 | 9, 17, 18, 19 | lmodvs1 20095 | . . 3 ⊢ ((𝑃 ∈ LMod ∧ (1 ↑ 𝑋) ∈ (Base‘𝑃)) → ((1r‘(Scalar‘𝑃)) · (1 ↑ 𝑋)) = (1 ↑ 𝑋)) |
21 | 7, 16, 20 | syl2anc 583 | . 2 ⊢ (𝑅 ∈ Ring → ((1r‘(Scalar‘𝑃)) · (1 ↑ 𝑋)) = (1 ↑ 𝑋)) |
22 | 6, 21, 15 | 3eqtrd 2781 | 1 ⊢ (𝑅 ∈ Ring → ( 1 · (1 ↑ 𝑋)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ‘cfv 6423 (class class class)co 7260 1c1 10819 Basecbs 16856 Scalarcsca 16909 ·𝑠 cvsca 16910 .gcmg 18644 mulGrpcmgp 19664 1rcur 19681 Ringcrg 19727 LModclmod 20067 var1cv1 21291 Poly1cpl1 21292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7571 ax-cnex 10874 ax-resscn 10875 ax-1cn 10876 ax-icn 10877 ax-addcl 10878 ax-addrcl 10879 ax-mulcl 10880 ax-mulrcl 10881 ax-mulcom 10882 ax-addass 10883 ax-mulass 10884 ax-distr 10885 ax-i2m1 10886 ax-1ne0 10887 ax-1rid 10888 ax-rnegex 10889 ax-rrecex 10890 ax-cnre 10891 ax-pre-lttri 10892 ax-pre-lttrn 10893 ax-pre-ltadd 10894 ax-pre-mulgt0 10895 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3429 df-sbc 3717 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6259 df-on 6260 df-lim 6261 df-suc 6262 df-iota 6381 df-fun 6425 df-fn 6426 df-f 6427 df-f1 6428 df-fo 6429 df-f1o 6430 df-fv 6431 df-riota 7217 df-ov 7263 df-oprab 7264 df-mpo 7265 df-of 7516 df-om 7693 df-1st 7809 df-2nd 7810 df-supp 7954 df-frecs 8073 df-wrecs 8104 df-recs 8178 df-rdg 8217 df-1o 8272 df-er 8461 df-map 8580 df-en 8697 df-dom 8698 df-sdom 8699 df-fin 8700 df-fsupp 9075 df-pnf 10958 df-mnf 10959 df-xr 10960 df-ltxr 10961 df-le 10962 df-sub 11153 df-neg 11154 df-nn 11920 df-2 11982 df-3 11983 df-4 11984 df-5 11985 df-6 11986 df-7 11987 df-8 11988 df-9 11989 df-n0 12180 df-z 12266 df-dec 12383 df-uz 12528 df-fz 13185 df-seq 13666 df-struct 16792 df-sets 16809 df-slot 16827 df-ndx 16839 df-base 16857 df-ress 16886 df-plusg 16919 df-mulr 16920 df-sca 16922 df-vsca 16923 df-tset 16925 df-ple 16926 df-0g 17096 df-mgm 18270 df-sgrp 18319 df-mnd 18330 df-grp 18524 df-minusg 18525 df-sbg 18526 df-mulg 18645 df-subg 18696 df-mgp 19665 df-ur 19682 df-ring 19729 df-lmod 20069 df-lss 20138 df-psr 21056 df-mvr 21057 df-mpl 21058 df-opsr 21060 df-psr1 21295 df-vr1 21296 df-ply1 21297 |
This theorem is referenced by: (None) |
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