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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 | β’ π = (Poly1βπ ) |
ply1ascl.a | β’ π΄ = (algScβπ) |
Ref | Expression |
---|---|
ply1ascl | β’ π΄ = (algScβ(1o mPoly π )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1ascl.a | . 2 β’ π΄ = (algScβπ) | |
2 | eqid 2733 | . . . 4 β’ (Scalarβπ) = (Scalarβπ) | |
3 | eqid 2733 | . . . 4 β’ (Scalarβ(1o mPoly π )) = (Scalarβ(1o mPoly π )) | |
4 | ply1ascl.p | . . . . . 6 β’ π = (Poly1βπ ) | |
5 | 4 | ply1sca 21775 | . . . . 5 β’ (π β V β π = (Scalarβπ)) |
6 | 5 | fveq2d 6896 | . . . 4 β’ (π β V β (Baseβπ ) = (Baseβ(Scalarβπ))) |
7 | eqid 2733 | . . . . . 6 β’ (1o mPoly π ) = (1o mPoly π ) | |
8 | 1on 8478 | . . . . . . 7 β’ 1o β On | |
9 | 8 | a1i 11 | . . . . . 6 β’ (π β V β 1o β On) |
10 | id 22 | . . . . . 6 β’ (π β V β π β V) | |
11 | 7, 9, 10 | mplsca 21572 | . . . . 5 β’ (π β V β π = (Scalarβ(1o mPoly π ))) |
12 | 11 | fveq2d 6896 | . . . 4 β’ (π β V β (Baseβπ ) = (Baseβ(Scalarβ(1o mPoly π )))) |
13 | eqid 2733 | . . . . . . 7 β’ ( Β·π βπ) = ( Β·π βπ) | |
14 | 4, 7, 13 | ply1vsca 21748 | . . . . . 6 β’ ( Β·π βπ) = ( Β·π β(1o mPoly π )) |
15 | 14 | a1i 11 | . . . . 5 β’ (π β V β ( Β·π βπ) = ( Β·π β(1o mPoly π ))) |
16 | 15 | oveqdr 7437 | . . . 4 β’ ((π β V β§ (π₯ β (Baseβπ ) β§ π¦ β V)) β (π₯( Β·π βπ)π¦) = (π₯( Β·π β(1o mPoly π ))π¦)) |
17 | eqid 2733 | . . . . . 6 β’ (1rβπ) = (1rβπ) | |
18 | 7, 4, 17 | ply1mpl1 21779 | . . . . 5 β’ (1rβπ) = (1rβ(1o mPoly π )) |
19 | 18 | a1i 11 | . . . 4 β’ (π β V β (1rβπ) = (1rβ(1o mPoly π ))) |
20 | fvexd 6907 | . . . 4 β’ (π β V β (1rβπ) β V) | |
21 | 2, 3, 6, 12, 16, 19, 20 | asclpropd 21451 | . . 3 β’ (π β V β (algScβπ) = (algScβ(1o mPoly π ))) |
22 | fvprc 6884 | . . . . . 6 β’ (Β¬ π β V β (Poly1βπ ) = β ) | |
23 | 4, 22 | eqtrid 2785 | . . . . 5 β’ (Β¬ π β V β π = β ) |
24 | reldmmpl 21547 | . . . . . 6 β’ Rel dom mPoly | |
25 | 24 | ovprc2 7449 | . . . . 5 β’ (Β¬ π β V β (1o mPoly π ) = β ) |
26 | 23, 25 | eqtr4d 2776 | . . . 4 β’ (Β¬ π β V β π = (1o mPoly π )) |
27 | 26 | fveq2d 6896 | . . 3 β’ (Β¬ π β V β (algScβπ) = (algScβ(1o mPoly π ))) |
28 | 21, 27 | pm2.61i 182 | . 2 β’ (algScβπ) = (algScβ(1o mPoly π )) |
29 | 1, 28 | eqtri 2761 | 1 β’ π΄ = (algScβ(1o mPoly π )) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3475 β c0 4323 Oncon0 6365 βcfv 6544 (class class class)co 7409 1oc1o 8459 Basecbs 17144 Scalarcsca 17200 Β·π cvsca 17201 1rcur 20004 algSccascl 21407 mPoly cmpl 21459 Poly1cpl1 21701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-sca 17213 df-vsca 17214 df-tset 17216 df-ple 17217 df-0g 17387 df-mgp 19988 df-ur 20005 df-ascl 21410 df-psr 21462 df-mpl 21464 df-opsr 21466 df-psr1 21704 df-ply1 21706 |
This theorem is referenced by: subrg1ascl 21781 subrg1asclcl 21782 evls1sca 21842 evl1sca 21853 pf1ind 21874 deg1le0 25629 |
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