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| Mirrors > Home > MPE Home > Th. List > ply1sca2 | Structured version Visualization version GIF version | ||
| Description: Scalars of a univariate polynomial ring. (Contributed by Stefan O'Rear, 26-Mar-2015.) |
| Ref | Expression |
|---|---|
| ply1lmod.p | β’ π = (Poly1βπ ) |
| Ref | Expression |
|---|---|
| ply1sca2 | β’ ( I βπ ) = (Scalarβπ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fvi 6979 | . . 3 β’ (π β V β ( I βπ ) = π ) | |
| 2 | ply1lmod.p | . . . 4 β’ π = (Poly1βπ ) | |
| 3 | 2 | ply1sca 22190 | . . 3 β’ (π β V β π = (Scalarβπ)) |
| 4 | 1, 3 | eqtrd 2768 | . 2 β’ (π β V β ( I βπ ) = (Scalarβπ)) |
| 5 | fvprc 6894 | . . 3 β’ (Β¬ π β V β ( I βπ ) = β ) | |
| 6 | fvprc 6894 | . . . . 5 β’ (Β¬ π β V β (Poly1βπ ) = β ) | |
| 7 | 6 | fveq2d 6906 | . . . 4 β’ (Β¬ π β V β (Scalarβ(Poly1βπ )) = (Scalarββ )) |
| 8 | 2 | fveq2i 6905 | . . . 4 β’ (Scalarβπ) = (Scalarβ(Poly1βπ )) |
| 9 | scaid 17305 | . . . . 5 β’ Scalar = Slot (Scalarβndx) | |
| 10 | 9 | str0 17167 | . . . 4 β’ β = (Scalarββ ) |
| 11 | 7, 8, 10 | 3eqtr4g 2793 | . . 3 β’ (Β¬ π β V β (Scalarβπ) = β ) |
| 12 | 5, 11 | eqtr4d 2771 | . 2 β’ (Β¬ π β V β ( I βπ ) = (Scalarβπ)) |
| 13 | 4, 12 | pm2.61i 182 | 1 β’ ( I βπ ) = (Scalarβπ) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 = wceq 1533 β wcel 2098 Vcvv 3473 β c0 4326 I cid 5579 βcfv 6553 ndxcnx 17171 Scalarcsca 17245 Poly1cpl1 22114 |
| 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 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
| 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-tp 4637 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-of 7692 df-om 7879 df-1st 8001 df-2nd 8002 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-fsupp 9396 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-fz 13527 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-sca 17258 df-vsca 17259 df-tset 17261 df-ple 17262 df-psr 21856 df-opsr 21860 df-psr1 22117 df-ply1 22119 |
| This theorem is referenced by: ply1tmcl 22210 ply1scltm 22219 ply1sclf 22223 ply1scl0OLD 22229 ply1scl1OLD 22232 deg1invg 26070 |
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