![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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.) |
Ref | Expression |
---|---|
mhpinvcl.h | ⊢ 𝐻 = (𝐼 mHomP 𝑅) |
mhpinvcl.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mhpinvcl.m | ⊢ 𝑀 = (invg‘𝑃) |
mhpinvcl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
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 2724 | . 2 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
4 | eqid 2724 | . 2 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | eqid 2724 | . 2 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
6 | mhpinvcl.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
7 | mhpinvcl.r | . 2 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
8 | mhpinvcl.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
9 | 2 | mplgrp 21886 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp) |
10 | 6, 7, 9 | syl2anc 583 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Grp) |
11 | mhpinvcl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁)) | |
12 | 1, 2, 3, 6, 7, 8, 11 | mhpmpl 21995 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑃)) |
13 | mhpinvcl.m | . . . 4 ⊢ 𝑀 = (invg‘𝑃) | |
14 | 3, 13 | grpinvcl 18907 | . . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑃)) → (𝑀‘𝑋) ∈ (Base‘𝑃)) |
15 | 10, 12, 14 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (Base‘𝑃)) |
16 | eqid 2724 | . . . . . 6 ⊢ (invg‘𝑅) = (invg‘𝑅) | |
17 | 2, 3, 16, 13, 6, 7, 12 | mplneg 21879 | . . . . 5 ⊢ (𝜑 → (𝑀‘𝑋) = ((invg‘𝑅) ∘ 𝑋)) |
18 | 17 | oveq1d 7416 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝑋) supp (0g‘𝑅)) = (((invg‘𝑅) ∘ 𝑋) supp (0g‘𝑅))) |
19 | eqid 2724 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
20 | 19, 16 | grpinvfn 18901 | . . . . . 6 ⊢ (invg‘𝑅) Fn (Base‘𝑅) |
21 | 20 | a1i 11 | . . . . 5 ⊢ (𝜑 → (invg‘𝑅) Fn (Base‘𝑅)) |
22 | 2, 19, 3, 5, 12 | mplelf 21867 | . . . . 5 ⊢ (𝜑 → 𝑋:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
23 | ovex 7434 | . . . . . . 7 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
24 | 23 | rabex 5322 | . . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V |
25 | 24 | a1i 11 | . . . . 5 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∈ V) |
26 | fvexd 6896 | . . . . 5 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
27 | 4, 16 | grpinvid 18919 | . . . . . 6 ⊢ (𝑅 ∈ Grp → ((invg‘𝑅)‘(0g‘𝑅)) = (0g‘𝑅)) |
28 | 7, 27 | syl 17 | . . . . 5 ⊢ (𝜑 → ((invg‘𝑅)‘(0g‘𝑅)) = (0g‘𝑅)) |
29 | 21, 22, 25, 26, 28 | suppcoss 8187 | . . . 4 ⊢ (𝜑 → (((invg‘𝑅) ∘ 𝑋) supp (0g‘𝑅)) ⊆ (𝑋 supp (0g‘𝑅))) |
30 | 18, 29 | eqsstrd 4012 | . . 3 ⊢ (𝜑 → ((𝑀‘𝑋) supp (0g‘𝑅)) ⊆ (𝑋 supp (0g‘𝑅))) |
31 | 1, 4, 5, 6, 7, 8, 11 | mhpdeg 21996 | . . 3 ⊢ (𝜑 → (𝑋 supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) |
32 | 30, 31 | sstrd 3984 | . 2 ⊢ (𝜑 → ((𝑀‘𝑋) supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) |
33 | 1, 2, 3, 4, 5, 6, 7, 8, 15, 32 | ismhp2 21993 | 1 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (𝐻‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3424 Vcvv 3466 ◡ccnv 5665 “ cima 5669 ∘ ccom 5670 Fn wfn 6528 ‘cfv 6533 (class class class)co 7401 supp csupp 8140 ↑m cmap 8816 Fincfn 8935 ℕcn 12209 ℕ0cn0 12469 Basecbs 17143 ↾s cress 17172 0gc0g 17384 Σg cgsu 17385 Grpcgrp 18853 invgcminusg 18854 ℂfldccnfld 21228 mPoly cmpl 21768 mHomP cmhp 21982 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-prds 17392 df-pws 17394 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-minusg 18857 df-subg 19040 df-psr 21771 df-mpl 21773 df-mhp 21989 |
This theorem is referenced by: mhpsubg 22004 |
Copyright terms: Public domain | W3C validator |