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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 6961 | . . 3 β’ (π β V β ( I βπ ) = π ) | |
| 2 | ply1lmod.p | . . . 4 β’ π = (Poly1βπ ) | |
| 3 | 2 | ply1sca 22126 | . . 3 β’ (π β V β π = (Scalarβπ)) |
| 4 | 1, 3 | eqtrd 2766 | . 2 β’ (π β V β ( I βπ ) = (Scalarβπ)) |
| 5 | fvprc 6877 | . . 3 β’ (Β¬ π β V β ( I βπ ) = β ) | |
| 6 | fvprc 6877 | . . . . 5 β’ (Β¬ π β V β (Poly1βπ ) = β ) | |
| 7 | 6 | fveq2d 6889 | . . . 4 β’ (Β¬ π β V β (Scalarβ(Poly1βπ )) = (Scalarββ )) |
| 8 | 2 | fveq2i 6888 | . . . 4 β’ (Scalarβπ) = (Scalarβ(Poly1βπ )) |
| 9 | scaid 17269 | . . . . 5 β’ Scalar = Slot (Scalarβndx) | |
| 10 | 9 | str0 17131 | . . . 4 β’ β = (Scalarββ ) |
| 11 | 7, 8, 10 | 3eqtr4g 2791 | . . 3 β’ (Β¬ π β V β (Scalarβπ) = β ) |
| 12 | 5, 11 | eqtr4d 2769 | . 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 3468 β c0 4317 I cid 5566 βcfv 6537 ndxcnx 17135 Scalarcsca 17209 Poly1cpl1 22051 |
| 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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-tset 17225 df-ple 17226 df-psr 21803 df-opsr 21807 df-psr1 22054 df-ply1 22056 |
| This theorem is referenced by: ply1tmcl 22146 ply1scltm 22155 ply1sclf 22159 ply1scl0OLD 22165 ply1scl1OLD 22168 deg1invg 25997 |
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