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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 2737 | . . . 4 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
| 3 | eqid 2737 | . . . 4 ⊢ (Scalar‘(1o mPoly 𝑅)) = (Scalar‘(1o mPoly 𝑅)) | |
| 4 | ply1ascl.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
| 5 | 4 | ply1sca 22254 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 = (Scalar‘𝑃)) |
| 6 | 5 | fveq2d 6910 | . . . 4 ⊢ (𝑅 ∈ V → (Base‘𝑅) = (Base‘(Scalar‘𝑃))) |
| 7 | eqid 2737 | . . . . . 6 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
| 8 | 1on 8518 | . . . . . . 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 22033 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 = (Scalar‘(1o mPoly 𝑅))) |
| 12 | 11 | fveq2d 6910 | . . . 4 ⊢ (𝑅 ∈ V → (Base‘𝑅) = (Base‘(Scalar‘(1o mPoly 𝑅)))) |
| 13 | eqid 2737 | . . . . . . 7 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
| 14 | 4, 7, 13 | ply1vsca 22226 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘(1o mPoly 𝑅)) |
| 15 | 14 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ V → ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘(1o mPoly 𝑅))) |
| 16 | 15 | oveqdr 7459 | . . . 4 ⊢ ((𝑅 ∈ V ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ V)) → (𝑥( ·𝑠 ‘𝑃)𝑦) = (𝑥( ·𝑠 ‘(1o mPoly 𝑅))𝑦)) |
| 17 | eqid 2737 | . . . . . 6 ⊢ (1r‘𝑃) = (1r‘𝑃) | |
| 18 | 7, 4, 17 | ply1mpl1 22260 | . . . . 5 ⊢ (1r‘𝑃) = (1r‘(1o mPoly 𝑅)) |
| 19 | 18 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ V → (1r‘𝑃) = (1r‘(1o mPoly 𝑅))) |
| 20 | fvexd 6921 | . . . 4 ⊢ (𝑅 ∈ V → (1r‘𝑃) ∈ V) | |
| 21 | 2, 3, 6, 12, 16, 19, 20 | asclpropd 21917 | . . 3 ⊢ (𝑅 ∈ V → (algSc‘𝑃) = (algSc‘(1o mPoly 𝑅))) |
| 22 | fvprc 6898 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅) | |
| 23 | 4, 22 | eqtrid 2789 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑃 = ∅) |
| 24 | reldmmpl 22008 | . . . . . 6 ⊢ Rel dom mPoly | |
| 25 | 24 | ovprc2 7471 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o mPoly 𝑅) = ∅) |
| 26 | 23, 25 | eqtr4d 2780 | . . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑃 = (1o mPoly 𝑅)) |
| 27 | 26 | fveq2d 6910 | . . 3 ⊢ (¬ 𝑅 ∈ V → (algSc‘𝑃) = (algSc‘(1o mPoly 𝑅))) |
| 28 | 21, 27 | pm2.61i 182 | . 2 ⊢ (algSc‘𝑃) = (algSc‘(1o mPoly 𝑅)) |
| 29 | 1, 28 | eqtri 2765 | 1 ⊢ 𝐴 = (algSc‘(1o mPoly 𝑅)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 Oncon0 6384 ‘cfv 6561 (class class class)co 7431 1oc1o 8499 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 1rcur 20178 algSccascl 21872 mPoly cmpl 21926 Poly1cpl1 22178 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-tset 17316 df-ple 17317 df-0g 17486 df-mgp 20138 df-ur 20179 df-ascl 21875 df-psr 21929 df-mpl 21931 df-opsr 21933 df-psr1 22181 df-ply1 22183 |
| This theorem is referenced by: subrg1ascl 22262 subrg1asclcl 22263 evls1sca 22327 evl1sca 22338 pf1ind 22359 deg1le0 26150 |
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