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| Mirrors > Home > MPE Home > Th. List > ressply1vsca | Structured version Visualization version GIF version | ||
| Description: A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.) |
| Ref | Expression |
|---|---|
| ressply1.s | β’ π = (Poly1βπ ) |
| ressply1.h | β’ π» = (π βΎs π) |
| ressply1.u | β’ π = (Poly1βπ») |
| ressply1.b | β’ π΅ = (Baseβπ) |
| ressply1.2 | β’ (π β π β (SubRingβπ )) |
| ressply1.p | β’ π = (π βΎs π΅) |
| Ref | Expression |
|---|---|
| ressply1vsca | β’ ((π β§ (π β π β§ π β π΅)) β (π( Β·π βπ)π) = (π( Β·π βπ)π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2727 | . . 3 β’ (1o mPoly π ) = (1o mPoly π ) | |
| 2 | ressply1.h | . . 3 β’ π» = (π βΎs π) | |
| 3 | eqid 2727 | . . 3 β’ (1o mPoly π») = (1o mPoly π») | |
| 4 | ressply1.u | . . . 4 β’ π = (Poly1βπ») | |
| 5 | eqid 2727 | . . . 4 β’ (PwSer1βπ») = (PwSer1βπ») | |
| 6 | ressply1.b | . . . 4 β’ π΅ = (Baseβπ) | |
| 7 | 4, 5, 6 | ply1bas 22088 | . . 3 β’ π΅ = (Baseβ(1o mPoly π»)) |
| 8 | 1on 8490 | . . . 4 β’ 1o β On | |
| 9 | 8 | a1i 11 | . . 3 β’ (π β 1o β On) |
| 10 | ressply1.2 | . . 3 β’ (π β π β (SubRingβπ )) | |
| 11 | eqid 2727 | . . 3 β’ ((1o mPoly π ) βΎs π΅) = ((1o mPoly π ) βΎs π΅) | |
| 12 | 1, 2, 3, 7, 9, 10, 11 | ressmplvsca 21947 | . 2 β’ ((π β§ (π β π β§ π β π΅)) β (π( Β·π β(1o mPoly π»))π) = (π( Β·π β((1o mPoly π ) βΎs π΅))π)) |
| 13 | eqid 2727 | . . . 4 β’ ( Β·π βπ) = ( Β·π βπ) | |
| 14 | 4, 3, 13 | ply1vsca 22117 | . . 3 β’ ( Β·π βπ) = ( Β·π β(1o mPoly π»)) |
| 15 | 14 | oveqi 7427 | . 2 β’ (π( Β·π βπ)π) = (π( Β·π β(1o mPoly π»))π) |
| 16 | ressply1.s | . . . . 5 β’ π = (Poly1βπ ) | |
| 17 | eqid 2727 | . . . . 5 β’ ( Β·π βπ) = ( Β·π βπ) | |
| 18 | 16, 1, 17 | ply1vsca 22117 | . . . 4 β’ ( Β·π βπ) = ( Β·π β(1o mPoly π )) |
| 19 | 6 | fvexi 6905 | . . . . 5 β’ π΅ β V |
| 20 | ressply1.p | . . . . . 6 β’ π = (π βΎs π΅) | |
| 21 | 20, 17 | ressvsca 17310 | . . . . 5 β’ (π΅ β V β ( Β·π βπ) = ( Β·π βπ)) |
| 22 | 19, 21 | ax-mp 5 | . . . 4 β’ ( Β·π βπ) = ( Β·π βπ) |
| 23 | eqid 2727 | . . . . . 6 β’ ( Β·π β(1o mPoly π )) = ( Β·π β(1o mPoly π )) | |
| 24 | 11, 23 | ressvsca 17310 | . . . . 5 β’ (π΅ β V β ( Β·π β(1o mPoly π )) = ( Β·π β((1o mPoly π ) βΎs π΅))) |
| 25 | 19, 24 | ax-mp 5 | . . . 4 β’ ( Β·π β(1o mPoly π )) = ( Β·π β((1o mPoly π ) βΎs π΅)) |
| 26 | 18, 22, 25 | 3eqtr3i 2763 | . . 3 β’ ( Β·π βπ) = ( Β·π β((1o mPoly π ) βΎs π΅)) |
| 27 | 26 | oveqi 7427 | . 2 β’ (π( Β·π βπ)π) = (π( Β·π β((1o mPoly π ) βΎs π΅))π) |
| 28 | 12, 15, 27 | 3eqtr4g 2792 | 1 β’ ((π β§ (π β π β§ π β π΅)) β (π( Β·π βπ)π) = (π( Β·π βπ)π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 Vcvv 3469 Oncon0 6363 βcfv 6542 (class class class)co 7414 1oc1o 8471 Basecbs 17165 βΎs cress 17194 Β·π cvsca 17222 SubRingcsubrg 20488 mPoly cmpl 21819 PwSer1cps1 22068 Poly1cpl1 22070 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-fz 13503 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-tset 17237 df-ple 17238 df-subg 19062 df-ring 20159 df-subrg 20490 df-psr 21822 df-mpl 21824 df-opsr 21826 df-psr1 22073 df-ply1 22075 |
| This theorem is referenced by: evls1vsca 33174 asclply1subcl 33180 |
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