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Mirrors > Home > MPE Home > Th. List > ply1vsca | Structured version Visualization version GIF version |
Description: Value of scalar multiplication in a univariate polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 4-Jul-2015.) |
Ref | Expression |
---|---|
ply1plusg.y | โข ๐ = (Poly1โ๐ ) |
ply1plusg.s | โข ๐ = (1o mPoly ๐ ) |
ply1vscafval.n | โข ยท = ( ยท๐ โ๐) |
Ref | Expression |
---|---|
ply1vsca | โข ยท = ( ยท๐ โ๐) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1vscafval.n | . 2 โข ยท = ( ยท๐ โ๐) | |
2 | ply1plusg.s | . . . 4 โข ๐ = (1o mPoly ๐ ) | |
3 | eqid 2728 | . . . 4 โข (1o mPwSer ๐ ) = (1o mPwSer ๐ ) | |
4 | eqid 2728 | . . . 4 โข ( ยท๐ โ๐) = ( ยท๐ โ๐) | |
5 | 2, 3, 4 | mplvsca2 21963 | . . 3 โข ( ยท๐ โ๐) = ( ยท๐ โ(1o mPwSer ๐ )) |
6 | eqid 2728 | . . . 4 โข (PwSer1โ๐ ) = (PwSer1โ๐ ) | |
7 | eqid 2728 | . . . 4 โข ( ยท๐ โ(PwSer1โ๐ )) = ( ยท๐ โ(PwSer1โ๐ )) | |
8 | 6, 3, 7 | psr1vsca 22147 | . . 3 โข ( ยท๐ โ(PwSer1โ๐ )) = ( ยท๐ โ(1o mPwSer ๐ )) |
9 | fvex 6915 | . . . 4 โข (Baseโ(1o mPoly ๐ )) โ V | |
10 | ply1plusg.y | . . . . . 6 โข ๐ = (Poly1โ๐ ) | |
11 | 10, 6 | ply1val 22120 | . . . . 5 โข ๐ = ((PwSer1โ๐ ) โพs (Baseโ(1o mPoly ๐ ))) |
12 | 11, 7 | ressvsca 17332 | . . . 4 โข ((Baseโ(1o mPoly ๐ )) โ V โ ( ยท๐ โ(PwSer1โ๐ )) = ( ยท๐ โ๐)) |
13 | 9, 12 | ax-mp 5 | . . 3 โข ( ยท๐ โ(PwSer1โ๐ )) = ( ยท๐ โ๐) |
14 | 5, 8, 13 | 3eqtr2i 2762 | . 2 โข ( ยท๐ โ๐) = ( ยท๐ โ๐) |
15 | 1, 14 | eqtr4i 2759 | 1 โข ยท = ( ยท๐ โ๐) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 โ wcel 2098 Vcvv 3473 โcfv 6553 (class class class)co 7426 1oc1o 8486 Basecbs 17187 ยท๐ cvsca 17244 mPwSer cmps 21844 mPoly cmpl 21846 PwSer1cps1 22101 Poly1cpl1 22103 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-dec 12716 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-vsca 17257 df-ple 17260 df-psr 21849 df-mpl 21851 df-opsr 21853 df-psr1 22106 df-ply1 22108 |
This theorem is referenced by: ply1ass23l 22152 ressply1vsca 22157 ply1ascl 22184 coe1tm 22199 ply1coe 22224 deg1vscale 26060 deg1vsca 26061 |
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