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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 2610 | . . 3 ⊢ (1𝑜 mPwSer 𝑅) = (1𝑜 mPwSer 𝑅) | |
2 | eqid 2610 | . . 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 19386 | . . 3 ⊢ 𝑈 = (Base‘(1𝑜 mPoly 𝑅)) |
7 | 1on 7454 | . . . 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 19259 | . 2 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (LSubSp‘(1𝑜 mPwSer 𝑅))) |
11 | eqidd 2611 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(1𝑜 mPwSer 𝑅)) = (Base‘(1𝑜 mPwSer 𝑅))) | |
12 | 4 | psr1val 19377 | . . . 4 ⊢ 𝑆 = ((1𝑜 ordPwSer 𝑅)‘∅) |
13 | 0ss 3924 | . . . . 5 ⊢ ∅ ⊆ (1𝑜 × 1𝑜) | |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ Ring → ∅ ⊆ (1𝑜 × 1𝑜)) |
15 | 1, 12, 14 | opsrbas 19300 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(1𝑜 mPwSer 𝑅)) = (Base‘𝑆)) |
16 | ssv 3588 | . . . 4 ⊢ (Base‘(1𝑜 mPwSer 𝑅)) ⊆ V | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(1𝑜 mPwSer 𝑅)) ⊆ V) |
18 | 1, 12, 14 | opsrplusg 19301 | . . . 4 ⊢ (𝑅 ∈ Ring → (+g‘(1𝑜 mPwSer 𝑅)) = (+g‘𝑆)) |
19 | 18 | oveqdr 6573 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g‘(1𝑜 mPwSer 𝑅))𝑦) = (𝑥(+g‘𝑆)𝑦)) |
20 | ovex 6577 | . . . 4 ⊢ (𝑥( ·𝑠 ‘(1𝑜 mPwSer 𝑅))𝑦) ∈ V | |
21 | 20 | a1i 11 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘(1𝑜 mPwSer 𝑅)))) → (𝑥( ·𝑠 ‘(1𝑜 mPwSer 𝑅))𝑦) ∈ V) |
22 | 1, 12, 14 | opsrvsca 19303 | . . . 4 ⊢ (𝑅 ∈ Ring → ( ·𝑠 ‘(1𝑜 mPwSer 𝑅)) = ( ·𝑠 ‘𝑆)) |
23 | 22 | oveqdr 6573 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘(1𝑜 mPwSer 𝑅)))) → (𝑥( ·𝑠 ‘(1𝑜 mPwSer 𝑅))𝑦) = (𝑥( ·𝑠 ‘𝑆)𝑦)) |
24 | 1, 8, 9 | psrsca 19210 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘(1𝑜 mPwSer 𝑅))) |
25 | 24 | fveq2d 6107 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(Scalar‘(1𝑜 mPwSer 𝑅)))) |
26 | 1, 12, 14, 8, 9 | opsrsca 19304 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑆)) |
27 | 26 | fveq2d 6107 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(Scalar‘𝑆))) |
28 | 11, 15, 17, 19, 21, 23, 25, 27 | lsspropd 18838 | . 2 ⊢ (𝑅 ∈ Ring → (LSubSp‘(1𝑜 mPwSer 𝑅)) = (LSubSp‘𝑆)) |
29 | 10, 28 | eleqtrd 2690 | 1 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (LSubSp‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 × cxp 5036 Oncon0 5640 ‘cfv 5804 (class class class)co 6549 1𝑜c1o 7440 Basecbs 15695 +gcplusg 15768 Scalarcsca 15771 ·𝑠 cvsca 15772 Ringcrg 18370 LSubSpclss 18753 mPwSer cmps 19172 mPoly cmpl 19174 PwSer1cps1 19366 Poly1cpl1 19368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-tset 15787 df-ple 15788 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-subg 17414 df-mgp 18313 df-ring 18372 df-lss 18754 df-psr 19177 df-mpl 19179 df-opsr 19181 df-psr1 19371 df-ply1 19373 |
This theorem is referenced by: ply1assa 19390 ply1lmod 19443 |
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