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Mirrors > Home > MPE Home > Th. List > ressply1mul | Structured version Visualization version GIF version |
Description: A restricted polynomial algebra has the same 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 |
---|---|
ressply1mul | β’ ((π β§ (π β π΅ β§ π β π΅)) β (π(.rβπ)π) = (π(.rβπ)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 | ressmplmul 21946 | . 2 β’ ((π β§ (π β π΅ β§ π β π΅)) β (π(.rβ(1o mPoly π»))π) = (π(.rβ((1o mPoly π ) βΎs π΅))π)) |
13 | eqid 2727 | . . . 4 β’ (.rβπ) = (.rβπ) | |
14 | 4, 3, 13 | ply1mulr 22118 | . . 3 β’ (.rβπ) = (.rβ(1o mPoly π»)) |
15 | 14 | oveqi 7427 | . 2 β’ (π(.rβπ)π) = (π(.rβ(1o mPoly π»))π) |
16 | ressply1.s | . . . . 5 β’ π = (Poly1βπ ) | |
17 | eqid 2727 | . . . . 5 β’ (.rβπ) = (.rβπ) | |
18 | 16, 1, 17 | ply1mulr 22118 | . . . 4 β’ (.rβπ) = (.rβ(1o mPoly π )) |
19 | 6 | fvexi 6905 | . . . . 5 β’ π΅ β V |
20 | ressply1.p | . . . . . 6 β’ π = (π βΎs π΅) | |
21 | 20, 17 | ressmulr 17273 | . . . . 5 β’ (π΅ β V β (.rβπ) = (.rβπ)) |
22 | 19, 21 | ax-mp 5 | . . . 4 β’ (.rβπ) = (.rβπ) |
23 | eqid 2727 | . . . . . 6 β’ (.rβ(1o mPoly π )) = (.rβ(1o mPoly π )) | |
24 | 11, 23 | ressmulr 17273 | . . . . 5 β’ (π΅ β V β (.rβ(1o mPoly π )) = (.rβ((1o mPoly π ) βΎs π΅))) |
25 | 19, 24 | ax-mp 5 | . . . 4 β’ (.rβ(1o mPoly π )) = (.rβ((1o mPoly π ) βΎs π΅)) |
26 | 18, 22, 25 | 3eqtr3i 2763 | . . 3 β’ (.rβπ) = (.rβ((1o mPoly π ) βΎs π΅)) |
27 | 26 | oveqi 7427 | . 2 β’ (π(.rβπ)π) = (π(.rβ((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(.rβπ)π) = (π(.rβπ)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 .rcmulr 17219 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-rmo 3371 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-ofr 7678 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-pm 8837 df-ixp 8906 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-seq 13985 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-0g 17408 df-gsum 17409 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-submnd 18726 df-grp 18878 df-minusg 18879 df-subg 19062 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-ring 20159 df-subrng 20465 df-subrg 20490 df-psr 21822 df-mpl 21824 df-opsr 21826 df-psr1 22073 df-ply1 22075 |
This theorem is referenced by: evls1muld 33173 |
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