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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 2823 | . . 3 ⊢ (1o mPwSer 𝑅) = (1o mPwSer 𝑅) | |
2 | eqid 2823 | . . 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 20365 | . . 3 ⊢ 𝑈 = (Base‘(1o mPoly 𝑅)) |
7 | 1on 8111 | . . . 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 20220 | . 2 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (LSubSp‘(1o mPwSer 𝑅))) |
11 | eqidd 2824 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(1o mPwSer 𝑅)) = (Base‘(1o mPwSer 𝑅))) | |
12 | 4 | psr1val 20356 | . . . 4 ⊢ 𝑆 = ((1o ordPwSer 𝑅)‘∅) |
13 | 0ss 4352 | . . . . 5 ⊢ ∅ ⊆ (1o × 1o) | |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ Ring → ∅ ⊆ (1o × 1o)) |
15 | 1, 12, 14 | opsrbas 20261 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(1o mPwSer 𝑅)) = (Base‘𝑆)) |
16 | ssv 3993 | . . . 4 ⊢ (Base‘(1o mPwSer 𝑅)) ⊆ V | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘(1o mPwSer 𝑅)) ⊆ V) |
18 | 1, 12, 14 | opsrplusg 20262 | . . . 4 ⊢ (𝑅 ∈ Ring → (+g‘(1o mPwSer 𝑅)) = (+g‘𝑆)) |
19 | 18 | oveqdr 7186 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g‘(1o mPwSer 𝑅))𝑦) = (𝑥(+g‘𝑆)𝑦)) |
20 | ovexd 7193 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘(1o mPwSer 𝑅)))) → (𝑥( ·𝑠 ‘(1o mPwSer 𝑅))𝑦) ∈ V) | |
21 | 1, 12, 14 | opsrvsca 20264 | . . . 4 ⊢ (𝑅 ∈ Ring → ( ·𝑠 ‘(1o mPwSer 𝑅)) = ( ·𝑠 ‘𝑆)) |
22 | 21 | oveqdr 7186 | . . 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 20171 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘(1o mPwSer 𝑅))) |
24 | 23 | fveq2d 6676 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(Scalar‘(1o mPwSer 𝑅)))) |
25 | 1, 12, 14, 8, 9 | opsrsca 20265 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑆)) |
26 | 25 | fveq2d 6676 | . . 3 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘(Scalar‘𝑆))) |
27 | 11, 15, 17, 19, 20, 22, 24, 26 | lsspropd 19791 | . 2 ⊢ (𝑅 ∈ Ring → (LSubSp‘(1o mPwSer 𝑅)) = (LSubSp‘𝑆)) |
28 | 10, 27 | eleqtrd 2917 | 1 ⊢ (𝑅 ∈ Ring → 𝑈 ∈ (LSubSp‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 ∅c0 4293 × cxp 5555 Oncon0 6193 ‘cfv 6357 (class class class)co 7158 1oc1o 8097 Basecbs 16485 +gcplusg 16567 Scalarcsca 16570 ·𝑠 cvsca 16571 Ringcrg 19299 LSubSpclss 19705 mPwSer cmps 20133 mPoly cmpl 20135 PwSer1cps1 20345 Poly1cpl1 20347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-tset 16586 df-ple 16587 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-subg 18278 df-mgp 19242 df-ring 19301 df-lss 19706 df-psr 20138 df-mpl 20140 df-opsr 20142 df-psr1 20350 df-ply1 20352 |
This theorem is referenced by: ply1assa 20369 ply1lmod 20422 |
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