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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 2824 | . . . 4 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
3 | eqid 2824 | . . . 4 ⊢ (Scalar‘(1o mPoly 𝑅)) = (Scalar‘(1o mPoly 𝑅)) | |
4 | ply1ascl.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | 4 | ply1sca 20424 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 = (Scalar‘𝑃)) |
6 | 5 | fveq2d 6677 | . . . 4 ⊢ (𝑅 ∈ V → (Base‘𝑅) = (Base‘(Scalar‘𝑃))) |
7 | eqid 2824 | . . . . . 6 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
8 | 1on 8112 | . . . . . . 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 20228 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 = (Scalar‘(1o mPoly 𝑅))) |
12 | 11 | fveq2d 6677 | . . . 4 ⊢ (𝑅 ∈ V → (Base‘𝑅) = (Base‘(Scalar‘(1o mPoly 𝑅)))) |
13 | eqid 2824 | . . . . . . 7 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
14 | 4, 7, 13 | ply1vsca 20397 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘(1o mPoly 𝑅)) |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ V → ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘(1o mPoly 𝑅))) |
16 | 15 | oveqdr 7187 | . . . 4 ⊢ ((𝑅 ∈ V ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ V)) → (𝑥( ·𝑠 ‘𝑃)𝑦) = (𝑥( ·𝑠 ‘(1o mPoly 𝑅))𝑦)) |
17 | eqid 2824 | . . . . . 6 ⊢ (1r‘𝑃) = (1r‘𝑃) | |
18 | 7, 4, 17 | ply1mpl1 20428 | . . . . 5 ⊢ (1r‘𝑃) = (1r‘(1o mPoly 𝑅)) |
19 | 18 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ V → (1r‘𝑃) = (1r‘(1o mPoly 𝑅))) |
20 | fvexd 6688 | . . . 4 ⊢ (𝑅 ∈ V → (1r‘𝑃) ∈ V) | |
21 | 2, 3, 6, 12, 16, 19, 20 | asclpropd 20129 | . . 3 ⊢ (𝑅 ∈ V → (algSc‘𝑃) = (algSc‘(1o mPoly 𝑅))) |
22 | fvprc 6666 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅) | |
23 | 4, 22 | syl5eq 2871 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑃 = ∅) |
24 | reldmmpl 20210 | . . . . . 6 ⊢ Rel dom mPoly | |
25 | 24 | ovprc2 7199 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o mPoly 𝑅) = ∅) |
26 | 23, 25 | eqtr4d 2862 | . . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑃 = (1o mPoly 𝑅)) |
27 | 26 | fveq2d 6677 | . . 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 2847 | 1 ⊢ 𝐴 = (algSc‘(1o mPoly 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∅c0 4294 Oncon0 6194 ‘cfv 6358 (class class class)co 7159 1oc1o 8098 Basecbs 16486 Scalarcsca 16571 ·𝑠 cvsca 16572 1rcur 19254 algSccascl 20087 mPoly cmpl 20136 Poly1cpl1 20348 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 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 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-tset 16587 df-ple 16588 df-0g 16718 df-mgp 19243 df-ur 19255 df-ascl 20090 df-psr 20139 df-mpl 20141 df-opsr 20143 df-psr1 20351 df-ply1 20353 |
This theorem is referenced by: subrg1ascl 20430 subrg1asclcl 20431 evls1sca 20489 evl1sca 20500 pf1ind 20521 deg1le0 24708 |
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