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Mirrors > Home > MPE Home > Th. List > Mathboxes > mhphf3 | Structured version Visualization version GIF version |
Description: A homogeneous polynomial defines a homogeneous function; this is mhphf2 41708 with the finite support restriction (frlmpws 21641, frlmbas 21646) on the assignments π΄ from variables to values. See comment of mhphf2 41708. (Contributed by SN, 23-Nov-2024.) |
Ref | Expression |
---|---|
mhphf3.q | β’ π = ((πΌ evalSub π)βπ ) |
mhphf3.h | β’ π» = (πΌ mHomP π) |
mhphf3.u | β’ π = (π βΎs π ) |
mhphf3.k | β’ πΎ = (Baseβπ) |
mhphf3.f | β’ πΉ = (π freeLMod πΌ) |
mhphf3.m | β’ π = (BaseβπΉ) |
mhphf3.b | β’ β = ( Β·π βπΉ) |
mhphf3.x | β’ Β· = (.rβπ) |
mhphf3.e | β’ β = (.gβ(mulGrpβπ)) |
mhphf3.i | β’ (π β πΌ β π) |
mhphf3.s | β’ (π β π β CRing) |
mhphf3.r | β’ (π β π β (SubRingβπ)) |
mhphf3.l | β’ (π β πΏ β π ) |
mhphf3.n | β’ (π β π β β0) |
mhphf3.p | β’ (π β π β (π»βπ)) |
mhphf3.a | β’ (π β π΄ β π) |
Ref | Expression |
---|---|
mhphf3 | β’ (π β ((πβπ)β(πΏ β π΄)) = ((π β πΏ) Β· ((πβπ)βπ΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mhphf3.f | . . . 4 β’ πΉ = (π freeLMod πΌ) | |
2 | mhphf3.m | . . . 4 β’ π = (BaseβπΉ) | |
3 | mhphf3.k | . . . 4 β’ πΎ = (Baseβπ) | |
4 | mhphf3.i | . . . 4 β’ (π β πΌ β π) | |
5 | mhphf3.r | . . . . . 6 β’ (π β π β (SubRingβπ)) | |
6 | 3 | subrgss 20472 | . . . . . 6 β’ (π β (SubRingβπ) β π β πΎ) |
7 | 5, 6 | syl 17 | . . . . 5 β’ (π β π β πΎ) |
8 | mhphf3.l | . . . . 5 β’ (π β πΏ β π ) | |
9 | 7, 8 | sseldd 3978 | . . . 4 β’ (π β πΏ β πΎ) |
10 | mhphf3.a | . . . 4 β’ (π β π΄ β π) | |
11 | mhphf3.b | . . . 4 β’ β = ( Β·π βπΉ) | |
12 | mhphf3.x | . . . 4 β’ Β· = (.rβπ) | |
13 | 1, 2, 3, 4, 9, 10, 11, 12 | frlmvscafval 21657 | . . 3 β’ (π β (πΏ β π΄) = ((πΌ Γ {πΏ}) βf Β· π΄)) |
14 | 13 | fveq2d 6888 | . 2 β’ (π β ((πβπ)β(πΏ β π΄)) = ((πβπ)β((πΌ Γ {πΏ}) βf Β· π΄))) |
15 | mhphf3.q | . . 3 β’ π = ((πΌ evalSub π)βπ ) | |
16 | mhphf3.h | . . 3 β’ π» = (πΌ mHomP π) | |
17 | mhphf3.u | . . 3 β’ π = (π βΎs π ) | |
18 | mhphf3.e | . . 3 β’ β = (.gβ(mulGrpβπ)) | |
19 | mhphf3.s | . . 3 β’ (π β π β CRing) | |
20 | mhphf3.n | . . 3 β’ (π β π β β0) | |
21 | mhphf3.p | . . 3 β’ (π β π β (π»βπ)) | |
22 | 1, 3, 2 | frlmbasmap 21650 | . . . 4 β’ ((πΌ β π β§ π΄ β π) β π΄ β (πΎ βm πΌ)) |
23 | 4, 10, 22 | syl2anc 583 | . . 3 β’ (π β π΄ β (πΎ βm πΌ)) |
24 | 15, 16, 17, 3, 12, 18, 4, 19, 5, 8, 20, 21, 23 | mhphf 41707 | . 2 β’ (π β ((πβπ)β((πΌ Γ {πΏ}) βf Β· π΄)) = ((π β πΏ) Β· ((πβπ)βπ΄))) |
25 | 14, 24 | eqtrd 2766 | 1 β’ (π β ((πβπ)β(πΏ β π΄)) = ((π β πΏ) Β· ((πβπ)βπ΄))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wss 3943 {csn 4623 Γ cxp 5667 βcfv 6536 (class class class)co 7404 βf cof 7664 βm cmap 8819 β0cn0 12473 Basecbs 17151 βΎs cress 17180 .rcmulr 17205 Β·π cvsca 17208 .gcmg 18993 mulGrpcmgp 20037 CRingccrg 20137 SubRingcsubrg 20467 freeLMod cfrlm 21637 evalSub ces 21971 mHomP cmhp 22010 |
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 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-addf 11188 |
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-rmo 3370 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-int 4944 df-iun 4992 df-iin 4993 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-se 5625 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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-fz 13488 df-fzo 13631 df-seq 13970 df-hash 14294 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-0g 17394 df-gsum 17395 df-prds 17400 df-pws 17402 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-subg 19048 df-ghm 19137 df-cntz 19231 df-cmn 19700 df-abl 19701 df-mgp 20038 df-rng 20056 df-ur 20085 df-srg 20090 df-ring 20138 df-cring 20139 df-rhm 20372 df-subrng 20444 df-subrg 20469 df-lmod 20706 df-lss 20777 df-lsp 20817 df-sra 21019 df-rgmod 21020 df-cnfld 21237 df-dsmm 21623 df-frlm 21638 df-assa 21744 df-asp 21745 df-ascl 21746 df-psr 21799 df-mvr 21800 df-mpl 21801 df-evls 21973 df-mhp 22017 |
This theorem is referenced by: mhphf4 41710 |
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