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Mirrors > Home > MPE Home > Th. List > ply1ascl | Structured version Visualization version GIF version |
Description: The univariate polynomial ring inherits the multivariate ring's scalar function. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by Fan Zheng, 26-Jun-2016.) |
Ref | Expression |
---|---|
ply1ascl.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ply1ascl.a | ⊢ 𝐴 = (algSc‘𝑃) |
Ref | Expression |
---|---|
ply1ascl | ⊢ 𝐴 = (algSc‘(1o mPoly 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1ascl.a | . 2 ⊢ 𝐴 = (algSc‘𝑃) | |
2 | eqid 2821 | . . . 4 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
3 | eqid 2821 | . . . 4 ⊢ (Scalar‘(1o mPoly 𝑅)) = (Scalar‘(1o mPoly 𝑅)) | |
4 | ply1ascl.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | 4 | ply1sca 20420 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 = (Scalar‘𝑃)) |
6 | 5 | fveq2d 6673 | . . . 4 ⊢ (𝑅 ∈ V → (Base‘𝑅) = (Base‘(Scalar‘𝑃))) |
7 | eqid 2821 | . . . . . 6 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
8 | 1on 8108 | . . . . . . 7 ⊢ 1o ∈ On | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝑅 ∈ V → 1o ∈ On) |
10 | id 22 | . . . . . 6 ⊢ (𝑅 ∈ V → 𝑅 ∈ V) | |
11 | 7, 9, 10 | mplsca 20224 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 = (Scalar‘(1o mPoly 𝑅))) |
12 | 11 | fveq2d 6673 | . . . 4 ⊢ (𝑅 ∈ V → (Base‘𝑅) = (Base‘(Scalar‘(1o mPoly 𝑅)))) |
13 | eqid 2821 | . . . . . . 7 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
14 | 4, 7, 13 | ply1vsca 20393 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘(1o mPoly 𝑅)) |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ V → ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘(1o mPoly 𝑅))) |
16 | 15 | oveqdr 7183 | . . . 4 ⊢ ((𝑅 ∈ V ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ V)) → (𝑥( ·𝑠 ‘𝑃)𝑦) = (𝑥( ·𝑠 ‘(1o mPoly 𝑅))𝑦)) |
17 | eqid 2821 | . . . . . 6 ⊢ (1r‘𝑃) = (1r‘𝑃) | |
18 | 7, 4, 17 | ply1mpl1 20424 | . . . . 5 ⊢ (1r‘𝑃) = (1r‘(1o mPoly 𝑅)) |
19 | 18 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ V → (1r‘𝑃) = (1r‘(1o mPoly 𝑅))) |
20 | fvexd 6684 | . . . 4 ⊢ (𝑅 ∈ V → (1r‘𝑃) ∈ V) | |
21 | 2, 3, 6, 12, 16, 19, 20 | asclpropd 20125 | . . 3 ⊢ (𝑅 ∈ V → (algSc‘𝑃) = (algSc‘(1o mPoly 𝑅))) |
22 | fvprc 6662 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅) | |
23 | 4, 22 | syl5eq 2868 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑃 = ∅) |
24 | reldmmpl 20206 | . . . . . 6 ⊢ Rel dom mPoly | |
25 | 24 | ovprc2 7195 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o mPoly 𝑅) = ∅) |
26 | 23, 25 | eqtr4d 2859 | . . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑃 = (1o mPoly 𝑅)) |
27 | 26 | fveq2d 6673 | . . 3 ⊢ (¬ 𝑅 ∈ V → (algSc‘𝑃) = (algSc‘(1o mPoly 𝑅))) |
28 | 21, 27 | pm2.61i 184 | . 2 ⊢ (algSc‘𝑃) = (algSc‘(1o mPoly 𝑅)) |
29 | 1, 28 | eqtri 2844 | 1 ⊢ 𝐴 = (algSc‘(1o mPoly 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 Oncon0 6190 ‘cfv 6354 (class class class)co 7155 1oc1o 8094 Basecbs 16482 Scalarcsca 16567 ·𝑠 cvsca 16568 1rcur 19250 algSccascl 20083 mPoly cmpl 20132 Poly1cpl1 20344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-tset 16583 df-ple 16584 df-0g 16714 df-mgp 19239 df-ur 19251 df-ascl 20086 df-psr 20135 df-mpl 20137 df-opsr 20139 df-psr1 20347 df-ply1 20349 |
This theorem is referenced by: subrg1ascl 20426 subrg1asclcl 20427 evls1sca 20485 evl1sca 20496 pf1ind 20517 deg1le0 24704 |
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