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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 2798 | . . . 4 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃) | |
3 | eqid 2798 | . . . 4 ⊢ (Scalar‘(1o mPoly 𝑅)) = (Scalar‘(1o mPoly 𝑅)) | |
4 | ply1ascl.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
5 | 4 | ply1sca 20882 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 = (Scalar‘𝑃)) |
6 | 5 | fveq2d 6649 | . . . 4 ⊢ (𝑅 ∈ V → (Base‘𝑅) = (Base‘(Scalar‘𝑃))) |
7 | eqid 2798 | . . . . . 6 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅) | |
8 | 1on 8092 | . . . . . . 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 20684 | . . . . 5 ⊢ (𝑅 ∈ V → 𝑅 = (Scalar‘(1o mPoly 𝑅))) |
12 | 11 | fveq2d 6649 | . . . 4 ⊢ (𝑅 ∈ V → (Base‘𝑅) = (Base‘(Scalar‘(1o mPoly 𝑅)))) |
13 | eqid 2798 | . . . . . . 7 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃) | |
14 | 4, 7, 13 | ply1vsca 20855 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘(1o mPoly 𝑅)) |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝑅 ∈ V → ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘(1o mPoly 𝑅))) |
16 | 15 | oveqdr 7163 | . . . 4 ⊢ ((𝑅 ∈ V ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ V)) → (𝑥( ·𝑠 ‘𝑃)𝑦) = (𝑥( ·𝑠 ‘(1o mPoly 𝑅))𝑦)) |
17 | eqid 2798 | . . . . . 6 ⊢ (1r‘𝑃) = (1r‘𝑃) | |
18 | 7, 4, 17 | ply1mpl1 20886 | . . . . 5 ⊢ (1r‘𝑃) = (1r‘(1o mPoly 𝑅)) |
19 | 18 | a1i 11 | . . . 4 ⊢ (𝑅 ∈ V → (1r‘𝑃) = (1r‘(1o mPoly 𝑅))) |
20 | fvexd 6660 | . . . 4 ⊢ (𝑅 ∈ V → (1r‘𝑃) ∈ V) | |
21 | 2, 3, 6, 12, 16, 19, 20 | asclpropd 20583 | . . 3 ⊢ (𝑅 ∈ V → (algSc‘𝑃) = (algSc‘(1o mPoly 𝑅))) |
22 | fvprc 6638 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅) | |
23 | 4, 22 | syl5eq 2845 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝑃 = ∅) |
24 | reldmmpl 20665 | . . . . . 6 ⊢ Rel dom mPoly | |
25 | 24 | ovprc2 7175 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o mPoly 𝑅) = ∅) |
26 | 23, 25 | eqtr4d 2836 | . . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑃 = (1o mPoly 𝑅)) |
27 | 26 | fveq2d 6649 | . . 3 ⊢ (¬ 𝑅 ∈ V → (algSc‘𝑃) = (algSc‘(1o mPoly 𝑅))) |
28 | 21, 27 | pm2.61i 185 | . 2 ⊢ (algSc‘𝑃) = (algSc‘(1o mPoly 𝑅)) |
29 | 1, 28 | eqtri 2821 | 1 ⊢ 𝐴 = (algSc‘(1o mPoly 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 Oncon0 6159 ‘cfv 6324 (class class class)co 7135 1oc1o 8078 Basecbs 16475 Scalarcsca 16560 ·𝑠 cvsca 16561 1rcur 19244 algSccascl 20541 mPoly cmpl 20591 Poly1cpl1 20806 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-tset 16576 df-ple 16577 df-0g 16707 df-mgp 19233 df-ur 19245 df-ascl 20544 df-psr 20594 df-mpl 20596 df-opsr 20598 df-psr1 20809 df-ply1 20811 |
This theorem is referenced by: subrg1ascl 20888 subrg1asclcl 20889 evls1sca 20947 evl1sca 20958 pf1ind 20979 deg1le0 24712 |
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