![]() |
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 2726 | . 2 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
4 | eqid 2726 | . 2 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | eqid 2726 | . 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 21918 | . . . 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 22027 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑃)) |
13 | mhpinvcl.m | . . . 4 ⊢ 𝑀 = (invg‘𝑃) | |
14 | 3, 13 | grpinvcl 18917 | . . 3 ⊢ ((𝑃 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑃)) → (𝑀‘𝑋) ∈ (Base‘𝑃)) |
15 | 10, 12, 14 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (Base‘𝑃)) |
16 | eqid 2726 | . . . . . 6 ⊢ (invg‘𝑅) = (invg‘𝑅) | |
17 | 2, 3, 16, 13, 6, 7, 12 | mplneg 21911 | . . . . 5 ⊢ (𝜑 → (𝑀‘𝑋) = ((invg‘𝑅) ∘ 𝑋)) |
18 | 17 | oveq1d 7420 | . . . 4 ⊢ (𝜑 → ((𝑀‘𝑋) supp (0g‘𝑅)) = (((invg‘𝑅) ∘ 𝑋) supp (0g‘𝑅))) |
19 | eqid 2726 | . . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
20 | 19, 16 | grpinvfn 18911 | . . . . . 6 ⊢ (invg‘𝑅) Fn (Base‘𝑅) |
21 | 20 | a1i 11 | . . . . 5 ⊢ (𝜑 → (invg‘𝑅) Fn (Base‘𝑅)) |
22 | 2, 19, 3, 5, 12 | mplelf 21899 | . . . . 5 ⊢ (𝜑 → 𝑋:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
23 | ovex 7438 | . . . . . . 7 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
24 | 23 | rabex 5325 | . . . . . 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 6900 | . . . . 5 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
27 | 4, 16 | grpinvid 18929 | . . . . . 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 8193 | . . . 4 ⊢ (𝜑 → (((invg‘𝑅) ∘ 𝑋) supp (0g‘𝑅)) ⊆ (𝑋 supp (0g‘𝑅))) |
30 | 18, 29 | eqsstrd 4015 | . . 3 ⊢ (𝜑 → ((𝑀‘𝑋) supp (0g‘𝑅)) ⊆ (𝑋 supp (0g‘𝑅))) |
31 | 1, 4, 5, 6, 7, 8, 11 | mhpdeg 22028 | . . 3 ⊢ (𝜑 → (𝑋 supp (0g‘𝑅)) ⊆ {𝑔 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}) |
32 | 30, 31 | sstrd 3987 | . 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 22025 | 1 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (𝐻‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3426 Vcvv 3468 ◡ccnv 5668 “ cima 5672 ∘ ccom 5673 Fn wfn 6532 ‘cfv 6537 (class class class)co 7405 supp csupp 8146 ↑m cmap 8822 Fincfn 8941 ℕcn 12216 ℕ0cn0 12476 Basecbs 17153 ↾s cress 17182 0gc0g 17394 Σg cgsu 17395 Grpcgrp 18863 invgcminusg 18864 ℂfldccnfld 21240 mPoly cmpl 21800 mHomP cmhp 22014 |
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 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
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-iun 4992 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-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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-hom 17230 df-cco 17231 df-0g 17396 df-prds 17402 df-pws 17404 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-subg 19050 df-psr 21803 df-mpl 21805 df-mhp 22021 |
This theorem is referenced by: mhpsubg 22036 |
Copyright terms: Public domain | W3C validator |