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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 6968 | . . 3 β’ (π β V β ( I βπ ) = π ) | |
| 2 | ply1lmod.p | . . . 4 β’ π = (Poly1βπ ) | |
| 3 | 2 | ply1sca 21775 | . . 3 β’ (π β V β π = (Scalarβπ)) |
| 4 | 1, 3 | eqtrd 2773 | . 2 β’ (π β V β ( I βπ ) = (Scalarβπ)) |
| 5 | fvprc 6884 | . . 3 β’ (Β¬ π β V β ( I βπ ) = β ) | |
| 6 | fvprc 6884 | . . . . 5 β’ (Β¬ π β V β (Poly1βπ ) = β ) | |
| 7 | 6 | fveq2d 6896 | . . . 4 β’ (Β¬ π β V β (Scalarβ(Poly1βπ )) = (Scalarββ )) |
| 8 | 2 | fveq2i 6895 | . . . 4 β’ (Scalarβπ) = (Scalarβ(Poly1βπ )) |
| 9 | scaid 17260 | . . . . 5 β’ Scalar = Slot (Scalarβndx) | |
| 10 | 9 | str0 17122 | . . . 4 β’ β = (Scalarββ ) |
| 11 | 7, 8, 10 | 3eqtr4g 2798 | . . 3 β’ (Β¬ π β V β (Scalarβπ) = β ) |
| 12 | 5, 11 | eqtr4d 2776 | . 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 1542 β wcel 2107 Vcvv 3475 β c0 4323 I cid 5574 βcfv 6544 ndxcnx 17126 Scalarcsca 17200 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-psr 21462 df-opsr 21466 df-psr1 21704 df-ply1 21706 |
| This theorem is referenced by: ply1tmcl 21794 ply1scltm 21803 ply1sclf 21807 ply1scl0OLD 21813 ply1scl1OLD 21816 deg1invg 25624 |
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