![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > prjcrv0 | Structured version Visualization version GIF version |
Description: The "curve" (zero set) corresponding to the zero polynomial contains all coordinates. (Contributed by SN, 23-Nov-2024.) |
Ref | Expression |
---|---|
prjcrv0.y | ⊢ 𝑌 = ((0...𝑁) mPoly 𝐾) |
prjcrv0.0 | ⊢ 0 = (0g‘𝑌) |
prjcrv0.p | ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾) |
prjcrv0.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
prjcrv0.k | ⊢ (𝜑 → 𝐾 ∈ Field) |
Ref | Expression |
---|---|
prjcrv0 | ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘ 0 ) = 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . . 3 ⊢ ((0...𝑁) mHomP 𝐾) = ((0...𝑁) mHomP 𝐾) | |
2 | eqid 2732 | . . 3 ⊢ ((0...𝑁) eval 𝐾) = ((0...𝑁) eval 𝐾) | |
3 | prjcrv0.p | . . 3 ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾) | |
4 | eqid 2732 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
5 | prjcrv0.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
6 | prjcrv0.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ Field) | |
7 | fvssunirn 6924 | . . . 4 ⊢ (((0...𝑁) mHomP 𝐾)‘𝑁) ⊆ ∪ ran ((0...𝑁) mHomP 𝐾) | |
8 | prjcrv0.y | . . . . . 6 ⊢ 𝑌 = ((0...𝑁) mPoly 𝐾) | |
9 | eqid 2732 | . . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m (0...𝑁)) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m (0...𝑁)) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
10 | prjcrv0.0 | . . . . . 6 ⊢ 0 = (0g‘𝑌) | |
11 | ovexd 7446 | . . . . . 6 ⊢ (𝜑 → (0...𝑁) ∈ V) | |
12 | 6 | fldcrngd 20513 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ CRing) |
13 | 12 | crnggrpd 20141 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Grp) |
14 | 8, 9, 4, 10, 11, 13 | mpl0 21784 | . . . . 5 ⊢ (𝜑 → 0 = ({ℎ ∈ (ℕ0 ↑m (0...𝑁)) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝐾)})) |
15 | 1, 4, 9, 11, 13, 5 | mhp0cl 21908 | . . . . 5 ⊢ (𝜑 → ({ℎ ∈ (ℕ0 ↑m (0...𝑁)) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝐾)}) ∈ (((0...𝑁) mHomP 𝐾)‘𝑁)) |
16 | 14, 15 | eqeltrd 2833 | . . . 4 ⊢ (𝜑 → 0 ∈ (((0...𝑁) mHomP 𝐾)‘𝑁)) |
17 | 7, 16 | sselid 3980 | . . 3 ⊢ (𝜑 → 0 ∈ ∪ ran ((0...𝑁) mHomP 𝐾)) |
18 | 1, 2, 3, 4, 5, 6, 17 | prjcrvval 41676 | . 2 ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘ 0 ) = {𝑝 ∈ 𝑃 ∣ ((((0...𝑁) eval 𝐾)‘ 0 ) “ 𝑝) = {(0g‘𝐾)}}) |
19 | eqid 2732 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
20 | ovexd 7446 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (0...𝑁) ∈ V) | |
21 | 12 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝐾 ∈ CRing) |
22 | 2, 19, 8, 4, 10, 20, 21 | evl0 41431 | . . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (((0...𝑁) eval 𝐾)‘ 0 ) = (((Base‘𝐾) ↑m (0...𝑁)) × {(0g‘𝐾)})) |
23 | 22 | imaeq1d 6058 | . . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ((((0...𝑁) eval 𝐾)‘ 0 ) “ 𝑝) = ((((Base‘𝐾) ↑m (0...𝑁)) × {(0g‘𝐾)}) “ 𝑝)) |
24 | eqid 2732 | . . . . . . . . . . 11 ⊢ (𝐾 freeLMod (0...𝑁)) = (𝐾 freeLMod (0...𝑁)) | |
25 | eqid 2732 | . . . . . . . . . . 11 ⊢ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) = ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) | |
26 | 6 | flddrngd 20512 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ DivRing) |
27 | 3, 24, 25, 5, 26 | prjspnssbas 41665 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ⊆ 𝒫 ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})) |
28 | eqid 2732 | . . . . . . . . . . . . . . 15 ⊢ {𝑘 ∈ ((Base‘𝐾) ↑m (0...𝑁)) ∣ 𝑘 finSupp (0g‘𝐾)} = {𝑘 ∈ ((Base‘𝐾) ↑m (0...𝑁)) ∣ 𝑘 finSupp (0g‘𝐾)} | |
29 | 24, 19, 4, 28 | frlmbas 21529 | . . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Field ∧ (0...𝑁) ∈ V) → {𝑘 ∈ ((Base‘𝐾) ↑m (0...𝑁)) ∣ 𝑘 finSupp (0g‘𝐾)} = (Base‘(𝐾 freeLMod (0...𝑁)))) |
30 | 6, 11, 29 | syl2anc 584 | . . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑘 ∈ ((Base‘𝐾) ↑m (0...𝑁)) ∣ 𝑘 finSupp (0g‘𝐾)} = (Base‘(𝐾 freeLMod (0...𝑁)))) |
31 | ssrab2 4077 | . . . . . . . . . . . . 13 ⊢ {𝑘 ∈ ((Base‘𝐾) ↑m (0...𝑁)) ∣ 𝑘 finSupp (0g‘𝐾)} ⊆ ((Base‘𝐾) ↑m (0...𝑁)) | |
32 | 30, 31 | eqsstrrdi 4037 | . . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘(𝐾 freeLMod (0...𝑁))) ⊆ ((Base‘𝐾) ↑m (0...𝑁))) |
33 | 32 | ssdifssd 4142 | . . . . . . . . . . 11 ⊢ (𝜑 → ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) ⊆ ((Base‘𝐾) ↑m (0...𝑁))) |
34 | 33 | sspwd 4615 | . . . . . . . . . 10 ⊢ (𝜑 → 𝒫 ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) ⊆ 𝒫 ((Base‘𝐾) ↑m (0...𝑁))) |
35 | 27, 34 | sstrd 3992 | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ⊆ 𝒫 ((Base‘𝐾) ↑m (0...𝑁))) |
36 | 35 | sselda 3982 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝑝 ∈ 𝒫 ((Base‘𝐾) ↑m (0...𝑁))) |
37 | 36 | elpwid 4611 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝑝 ⊆ ((Base‘𝐾) ↑m (0...𝑁))) |
38 | sseqin2 4215 | . . . . . . 7 ⊢ (𝑝 ⊆ ((Base‘𝐾) ↑m (0...𝑁)) ↔ (((Base‘𝐾) ↑m (0...𝑁)) ∩ 𝑝) = 𝑝) | |
39 | 37, 38 | sylib 217 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (((Base‘𝐾) ↑m (0...𝑁)) ∩ 𝑝) = 𝑝) |
40 | 5 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝑁 ∈ ℕ0) |
41 | 26 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝐾 ∈ DivRing) |
42 | simpr 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝑝 ∈ 𝑃) | |
43 | 3, 24, 25, 40, 41, 42 | prjspnn0 41666 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝑝 ≠ ∅) |
44 | 39, 43 | eqnetrd 3008 | . . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (((Base‘𝐾) ↑m (0...𝑁)) ∩ 𝑝) ≠ ∅) |
45 | xpima2 6183 | . . . . 5 ⊢ ((((Base‘𝐾) ↑m (0...𝑁)) ∩ 𝑝) ≠ ∅ → ((((Base‘𝐾) ↑m (0...𝑁)) × {(0g‘𝐾)}) “ 𝑝) = {(0g‘𝐾)}) | |
46 | 44, 45 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ((((Base‘𝐾) ↑m (0...𝑁)) × {(0g‘𝐾)}) “ 𝑝) = {(0g‘𝐾)}) |
47 | 23, 46 | eqtrd 2772 | . . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ((((0...𝑁) eval 𝐾)‘ 0 ) “ 𝑝) = {(0g‘𝐾)}) |
48 | 47 | rabeqcda 3443 | . 2 ⊢ (𝜑 → {𝑝 ∈ 𝑃 ∣ ((((0...𝑁) eval 𝐾)‘ 0 ) “ 𝑝) = {(0g‘𝐾)}} = 𝑃) |
49 | 18, 48 | eqtrd 2772 | 1 ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘ 0 ) = 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 {crab 3432 Vcvv 3474 ∖ cdif 3945 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 {csn 4628 ∪ cuni 4908 class class class wbr 5148 × cxp 5674 ◡ccnv 5675 ran crn 5677 “ cima 5679 ‘cfv 6543 (class class class)co 7411 ↑m cmap 8822 Fincfn 8941 finSupp cfsupp 9363 0cc0 11112 ℕcn 12216 ℕ0cn0 12476 ...cfz 13488 Basecbs 17148 0gc0g 17389 CRingccrg 20128 DivRingcdr 20500 Fieldcfield 20501 freeLMod cfrlm 21520 mPoly cmpl 21678 eval cevl 21853 mHomP cmhp 21891 ℙ𝕣𝕠𝕛ncprjspn 41658 ℙ𝕣𝕠𝕛Crvcprjcrv 41673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 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 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 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 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-hom 17225 df-cco 17226 df-0g 17391 df-gsum 17392 df-prds 17397 df-pws 17399 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18987 df-subg 19039 df-ghm 19128 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-srg 20081 df-ring 20129 df-cring 20130 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-rhm 20363 df-subrng 20434 df-subrg 20459 df-drng 20502 df-field 20503 df-lmod 20616 df-lss 20687 df-lsp 20727 df-lvec 20858 df-sra 20930 df-rgmod 20931 df-dsmm 21506 df-frlm 21521 df-assa 21627 df-asp 21628 df-ascl 21629 df-psr 21681 df-mvr 21682 df-mpl 21683 df-evls 21854 df-evl 21855 df-mhp 21895 df-prjsp 41646 df-prjspn 41659 df-prjcrv 41674 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |