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Mirrors > Home > MPE Home > Th. List > ply1lss | Structured version Visualization version GIF version |
Description: Univariate polynomials form a linear subspace of the set of univariate power series. (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 |
---|---|
ply1lss | ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (LSubSp‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
2 | eqid 2818 | . . 3 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
3 | ply1val.1 | . . . 4 ⊢ 𝑃 = (Poly1‘𝑅) | |
4 | ply1val.2 | . . . 4 ⊢ 𝑆 = (PwSer1‘𝑅) | |
5 | ply1bas.u | . . . 4 ⊢ 𝑈 = (Base‘𝑃) | |
6 | 3, 4, 5 | ply1bas 20291 | . . 3 ⊢ 𝑈 = (Base‘(1o mPoly 𝑅)) |
7 | 1on 8098 | . . . 4 ⊢ 1o ∈ On | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝑅 ∈ Ring → 1o ∈ On) |
9 | id 22 | . . 3 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring) | |
10 | 1, 2, 6, 8, 9 | mpllss 20146 | . 2 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (LSubSp‘(1o mPwSer 𝑅))) |
11 | eqidd 2819 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(1o mPwSer 𝑅)) = (Base‘(1o mPwSer 𝑅))) | |
12 | 4 | psr1val 20282 | . . . 4 ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅) |
13 | 0ss 4347 | . . . . 5 ⊢ ∅ ⊆ (1o × 1o) | |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ Ring → ∅ ⊆ (1o × 1o)) |
15 | 1, 12, 14 | opsrbas 20187 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(1o mPwSer 𝑅)) = (Base‘𝑆)) |
16 | ssv 3988 | . . . 4 ⊢ (Base‘(1o mPwSer 𝑅)) ⊆ V | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(1o mPwSer 𝑅)) ⊆ V) |
18 | 1, 12, 14 | opsrplusg 20188 | . . . 4 ⊢ (𝑅 ∈ Ring → (+g‘(1o mPwSer 𝑅)) = (+g‘𝑆)) |
19 | 18 | oveqdr 7173 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g‘(1o mPwSer 𝑅))𝑦) = (𝑥(+g‘𝑆)𝑦)) |
20 | ovexd 7180 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘(1o mPwSer 𝑅)))) → (𝑥( ·𝑠 ‘(1o mPwSer 𝑅))𝑦) ∈ V) | |
21 | 1, 12, 14 | opsrvsca 20190 | . . . 4 ⊢ (𝑅 ∈ Ring → ( ·𝑠 ‘(1o mPwSer 𝑅)) = ( ·𝑠 ‘𝑆)) |
22 | 21 | oveqdr 7173 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘(1o mPwSer 𝑅)))) → (𝑥( ·𝑠 ‘(1o mPwSer 𝑅))𝑦) = (𝑥( ·𝑠 ‘𝑆)𝑦)) |
23 | 1, 8, 9 | psrsca 20097 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘(1o mPwSer 𝑅))) |
24 | 23 | fveq2d 6667 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(Scalar‘(1o mPwSer 𝑅)))) |
25 | 1, 12, 14, 8, 9 | opsrsca 20191 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑆)) |
26 | 25 | fveq2d 6667 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(Scalar‘𝑆))) |
27 | 11, 15, 17, 19, 20, 22, 24, 26 | lsspropd 19718 | . 2 ⊢ (𝑅 ∈ Ring → (LSubSp‘(1o mPwSer 𝑅)) = (LSubSp‘𝑆)) |
28 | 10, 27 | eleqtrd 2912 | 1 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (LSubSp‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 ∅c0 4288 × cxp 5546 Oncon0 6184 ‘cfv 6348 (class class class)co 7145 1oc1o 8084 Basecbs 16471 +gcplusg 16553 Scalarcsca 16556 ·𝑠 cvsca 16557 Ringcrg 19226 LSubSpclss 19632 mPwSer cmps 20059 mPoly cmpl 20061 PwSer1cps1 20271 Poly1cpl1 20273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 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-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-tset 16572 df-ple 16573 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-subg 18214 df-mgp 19169 df-ring 19228 df-lss 19633 df-psr 20064 df-mpl 20066 df-opsr 20068 df-psr1 20276 df-ply1 20278 |
This theorem is referenced by: ply1assa 20295 ply1lmod 20348 |
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