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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 42586 with the finite support restriction (frlmpws 21762, frlmbas 21767) on the assignments 𝐴 from variables to values. See comment of mhphf2 42586. (Contributed by SN, 23-Nov-2024.) |
Ref | Expression |
---|---|
mhphf3.q | ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) |
mhphf3.h | ⊢ 𝐻 = (𝐼 mHomP 𝑈) |
mhphf3.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
mhphf3.k | ⊢ 𝐾 = (Base‘𝑆) |
mhphf3.f | ⊢ 𝐹 = (𝑆 freeLMod 𝐼) |
mhphf3.m | ⊢ 𝑀 = (Base‘𝐹) |
mhphf3.b | ⊢ ∙ = ( ·𝑠 ‘𝐹) |
mhphf3.x | ⊢ · = (.r‘𝑆) |
mhphf3.e | ⊢ ↑ = (.g‘(mulGrp‘𝑆)) |
mhphf3.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
mhphf3.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
mhphf3.l | ⊢ (𝜑 → 𝐿 ∈ 𝑅) |
mhphf3.p | ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) |
mhphf3.a | ⊢ (𝜑 → 𝐴 ∈ 𝑀) |
Ref | Expression |
---|---|
mhphf3 | ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mhphf3.f | . . . 4 ⊢ 𝐹 = (𝑆 freeLMod 𝐼) | |
2 | mhphf3.m | . . . 4 ⊢ 𝑀 = (Base‘𝐹) | |
3 | mhphf3.k | . . . 4 ⊢ 𝐾 = (Base‘𝑆) | |
4 | reldmmhp 22133 | . . . . 5 ⊢ Rel dom mHomP | |
5 | mhphf3.h | . . . . 5 ⊢ 𝐻 = (𝐼 mHomP 𝑈) | |
6 | mhphf3.p | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) | |
7 | 4, 5, 6 | elfvov1 7471 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ V) |
8 | mhphf3.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
9 | 3 | subrgss 20564 | . . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐾) |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ⊆ 𝐾) |
11 | mhphf3.l | . . . . 5 ⊢ (𝜑 → 𝐿 ∈ 𝑅) | |
12 | 10, 11 | sseldd 3983 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ 𝐾) |
13 | mhphf3.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑀) | |
14 | mhphf3.b | . . . 4 ⊢ ∙ = ( ·𝑠 ‘𝐹) | |
15 | mhphf3.x | . . . 4 ⊢ · = (.r‘𝑆) | |
16 | 1, 2, 3, 7, 12, 13, 14, 15 | frlmvscafval 21778 | . . 3 ⊢ (𝜑 → (𝐿 ∙ 𝐴) = ((𝐼 × {𝐿}) ∘f · 𝐴)) |
17 | 16 | fveq2d 6908 | . 2 ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘f · 𝐴))) |
18 | mhphf3.q | . . 3 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅) | |
19 | mhphf3.u | . . 3 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
20 | mhphf3.e | . . 3 ⊢ ↑ = (.g‘(mulGrp‘𝑆)) | |
21 | mhphf3.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
22 | 1, 3, 2 | frlmbasmap 21771 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝐴 ∈ 𝑀) → 𝐴 ∈ (𝐾 ↑m 𝐼)) |
23 | 7, 13, 22 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) |
24 | 18, 5, 19, 3, 15, 20, 21, 8, 11, 6, 23 | mhphf 42585 | . 2 ⊢ (𝜑 → ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴))) |
25 | 17, 24 | eqtrd 2776 | 1 ⊢ (𝜑 → ((𝑄‘𝑋)‘(𝐿 ∙ 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3479 ⊆ wss 3950 {csn 4624 × cxp 5681 ‘cfv 6559 (class class class)co 7429 ∘f cof 7692 ↑m cmap 8862 Basecbs 17243 ↾s cress 17270 .rcmulr 17294 ·𝑠 cvsca 17297 .gcmg 19081 mulGrpcmgp 20133 CRingccrg 20227 SubRingcsubrg 20561 freeLMod cfrlm 21758 evalSub ces 22088 mHomP cmhp 22125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 ax-addf 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-se 5636 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-isom 6568 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-of 7694 df-ofr 7695 df-om 7884 df-1st 8010 df-2nd 8011 df-supp 8182 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-2o 8503 df-er 8741 df-map 8864 df-pm 8865 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9398 df-sup 9478 df-oi 9546 df-card 9975 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-9 12332 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12875 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-ress 17271 df-plusg 17306 df-mulr 17307 df-starv 17308 df-sca 17309 df-vsca 17310 df-ip 17311 df-tset 17312 df-ple 17313 df-ds 17315 df-unif 17316 df-hom 17317 df-cco 17318 df-0g 17482 df-gsum 17483 df-prds 17488 df-pws 17490 df-mre 17625 df-mrc 17626 df-acs 17628 df-mgm 18649 df-sgrp 18728 df-mnd 18744 df-mhm 18792 df-submnd 18793 df-grp 18950 df-minusg 18951 df-sbg 18952 df-mulg 19082 df-subg 19137 df-ghm 19227 df-cntz 19331 df-cmn 19796 df-abl 19797 df-mgp 20134 df-rng 20146 df-ur 20175 df-srg 20180 df-ring 20228 df-cring 20229 df-rhm 20464 df-subrng 20538 df-subrg 20562 df-lmod 20852 df-lss 20922 df-lsp 20962 df-sra 21164 df-rgmod 21165 df-cnfld 21357 df-dsmm 21744 df-frlm 21759 df-assa 21865 df-asp 21866 df-ascl 21867 df-psr 21921 df-mvr 21922 df-mpl 21923 df-evls 22090 df-mhp 22132 |
This theorem is referenced by: mhphf4 42588 |
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