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Mirrors > Home > MPE Home > Th. List > mhpinvcl | Structured version Visualization version GIF version |
Description: Homogeneous polynomials are closed under taking the opposite. (Contributed by SN, 12-Sep-2023.) Remove sethood hypothesis. (Revised by SN, 18-May-2025.) |
Ref | Expression |
---|---|
mhpinvcl.h | ⊢ 𝐻 = (𝐼 mHomP 𝑅) |
mhpinvcl.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mhpinvcl.m | ⊢ 𝑀 = (invg‘𝑃) |
mhpinvcl.r | ⊢ (𝜑 → 𝑅 ∈ Grp) |
mhpinvcl.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
mhpinvcl.x | ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) |
Ref | Expression |
---|---|
mhpinvcl | ⊢ (𝜑 → (𝑀‘𝑋) ∈ (𝐻‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mhpinvcl.h | . 2 ⊢ 𝐻 = (𝐼 mHomP 𝑅) | |
2 | mhpinvcl.p | . 2 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
3 | eqid 2725 | . 2 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
4 | eqid 2725 | . 2 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | eqid 2725 | . 2 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
6 | reldmmhp 22070 | . . 3 ⊢ Rel dom mHomP | |
7 | mhpinvcl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) | |
8 | 6, 1, 7 | elfvov1 7458 | . 2 ⊢ (𝜑 → 𝐼 ∈ V) |
9 | mhpinvcl.r | . 2 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
10 | mhpinvcl.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
11 | 2 | mplgrp 21966 | . . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp) |
12 | 8, 9, 11 | syl2anc 582 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Grp) |
13 | 1, 2, 3, 8, 9, 10, 7 | mhpmpl 22076 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑃)) |
14 | mhpinvcl.m | . . . 4 ⊢ 𝑀 = (invg‘𝑃) | |
15 | 3, 14 | grpinvcl 18948 | . . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑃)) → (𝑀‘𝑋) ∈ (Base‘𝑃)) |
16 | 12, 13, 15 | syl2anc 582 | . 2 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (Base‘𝑃)) |
17 | eqid 2725 | . . . . . 6 ⊢ (invg‘𝑅) = (invg‘𝑅) | |
18 | 2, 3, 17, 14, 8, 9, 13 | mplneg 21959 | . . . . 5 ⊢ (𝜑 → (𝑀‘𝑋) = ((invg‘𝑅) ∘ 𝑋)) |
19 | 18 | oveq1d 7431 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝑋) supp (0g‘𝑅)) = (((invg‘𝑅) ∘ 𝑋) supp (0g‘𝑅))) |
20 | eqid 2725 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
21 | 20, 17 | grpinvfn 18942 | . . . . . 6 ⊢ (invg‘𝑅) Fn (Base‘𝑅) |
22 | 21 | a1i 11 | . . . . 5 ⊢ (𝜑 → (invg‘𝑅) Fn (Base‘𝑅)) |
23 | 2, 20, 3, 5, 13 | mplelf 21947 | . . . . 5 ⊢ (𝜑 → 𝑋:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
24 | ovex 7449 | . . . . . . 7 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
25 | 24 | rabex 5329 | . . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V |
26 | 25 | a1i 11 | . . . . 5 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V) |
27 | fvexd 6907 | . . . . 5 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
28 | 4, 17 | grpinvid 18960 | . . . . . 6 ⊢ (𝑅 ∈ Grp → ((invg‘𝑅)‘(0g‘𝑅)) = (0g‘𝑅)) |
29 | 9, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → ((invg‘𝑅)‘(0g‘𝑅)) = (0g‘𝑅)) |
30 | 22, 23, 26, 27, 29 | suppcoss 8211 | . . . 4 ⊢ (𝜑 → (((invg‘𝑅) ∘ 𝑋) supp (0g‘𝑅)) ⊆ (𝑋 supp (0g‘𝑅))) |
31 | 19, 30 | eqsstrd 4011 | . . 3 ⊢ (𝜑 → ((𝑀‘𝑋) supp (0g‘𝑅)) ⊆ (𝑋 supp (0g‘𝑅))) |
32 | 1, 4, 5, 8, 9, 10, 7 | mhpdeg 22077 | . . 3 ⊢ (𝜑 → (𝑋 supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) |
33 | 31, 32 | sstrd 3983 | . 2 ⊢ (𝜑 → ((𝑀‘𝑋) supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) |
34 | 1, 2, 3, 4, 5, 8, 9, 10, 16, 33 | ismhp2 22074 | 1 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (𝐻‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3419 Vcvv 3463 ◡ccnv 5671 “ cima 5675 ∘ ccom 5676 Fn wfn 6538 ‘cfv 6543 (class class class)co 7416 supp csupp 8163 ↑m cmap 8843 Fincfn 8962 ℕcn 12242 ℕ0cn0 12502 Basecbs 17179 ↾s cress 17208 0gc0g 17420 Σg cgsu 17421 Grpcgrp 18894 invgcminusg 18895 ℂfldccnfld 21283 mPoly cmpl 21843 mHomP cmhp 22062 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 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 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-0g 17422 df-prds 17428 df-pws 17430 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-minusg 18898 df-subg 19082 df-psr 21846 df-mpl 21848 df-mhp 22069 |
This theorem is referenced by: mhpsubg 22085 |
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