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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 2733 | . . 3 ⊢ ((0...𝑁) mHomP 𝐾) = ((0...𝑁) mHomP 𝐾) | |
2 | eqid 2733 | . . 3 ⊢ ((0...𝑁) eval 𝐾) = ((0...𝑁) eval 𝐾) | |
3 | prjcrv0.p | . . 3 ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛n𝐾) | |
4 | eqid 2733 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
5 | prjcrv0.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
6 | prjcrv0.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ Field) | |
7 | fvssunirn 6925 | . . . 4 ⊢ (((0...𝑁) mHomP 𝐾)‘𝑁) ⊆ ∪ ran ((0...𝑁) mHomP 𝐾) | |
8 | prjcrv0.y | . . . . . 6 ⊢ 𝑌 = ((0...𝑁) mPoly 𝐾) | |
9 | eqid 2733 | . . . . . 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 7444 | . . . . . 6 ⊢ (𝜑 → (0...𝑁) ∈ V) | |
12 | 6 | fldcrngd 20370 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ CRing) |
13 | 12 | crnggrpd 20070 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Grp) |
14 | 8, 9, 4, 10, 11, 13 | mpl0 21565 | . . . . 5 ⊢ (𝜑 → 0 = ({ℎ ∈ (ℕ0 ↑m (0...𝑁)) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝐾)})) |
15 | 1, 4, 9, 11, 13, 5 | mhp0cl 21689 | . . . . 5 ⊢ (𝜑 → ({ℎ ∈ (ℕ0 ↑m (0...𝑁)) ∣ (◡ℎ “ ℕ) ∈ Fin} × {(0g‘𝐾)}) ∈ (((0...𝑁) mHomP 𝐾)‘𝑁)) |
16 | 14, 15 | eqeltrd 2834 | . . . 4 ⊢ (𝜑 → 0 ∈ (((0...𝑁) mHomP 𝐾)‘𝑁)) |
17 | 7, 16 | sselid 3981 | . . 3 ⊢ (𝜑 → 0 ∈ ∪ ran ((0...𝑁) mHomP 𝐾)) |
18 | 1, 2, 3, 4, 5, 6, 17 | prjcrvval 41422 | . 2 ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘ 0 ) = {𝑝 ∈ 𝑃 ∣ ((((0...𝑁) eval 𝐾)‘ 0 ) “ 𝑝) = {(0g‘𝐾)}}) |
19 | eqid 2733 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
20 | ovexd 7444 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (0...𝑁) ∈ V) | |
21 | 12 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝐾 ∈ CRing) |
22 | 2, 19, 8, 4, 10, 20, 21 | evl0 41177 | . . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (((0...𝑁) eval 𝐾)‘ 0 ) = (((Base‘𝐾) ↑m (0...𝑁)) × {(0g‘𝐾)})) |
23 | 22 | imaeq1d 6059 | . . . 4 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ((((0...𝑁) eval 𝐾)‘ 0 ) “ 𝑝) = ((((Base‘𝐾) ↑m (0...𝑁)) × {(0g‘𝐾)}) “ 𝑝)) |
24 | eqid 2733 | . . . . . . . . . . 11 ⊢ (𝐾 freeLMod (0...𝑁)) = (𝐾 freeLMod (0...𝑁)) | |
25 | eqid 2733 | . . . . . . . . . . 11 ⊢ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) = ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) | |
26 | 6 | flddrngd 20369 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ DivRing) |
27 | 3, 24, 25, 5, 26 | prjspnssbas 41411 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ⊆ 𝒫 ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))})) |
28 | eqid 2733 | . . . . . . . . . . . . . . 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 21310 | . . . . . . . . . . . . . 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 585 | . . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑘 ∈ ((Base‘𝐾) ↑m (0...𝑁)) ∣ 𝑘 finSupp (0g‘𝐾)} = (Base‘(𝐾 freeLMod (0...𝑁)))) |
31 | ssrab2 4078 | . . . . . . . . . . . . 13 ⊢ {𝑘 ∈ ((Base‘𝐾) ↑m (0...𝑁)) ∣ 𝑘 finSupp (0g‘𝐾)} ⊆ ((Base‘𝐾) ↑m (0...𝑁)) | |
32 | 30, 31 | eqsstrrdi 4038 | . . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘(𝐾 freeLMod (0...𝑁))) ⊆ ((Base‘𝐾) ↑m (0...𝑁))) |
33 | 32 | ssdifssd 4143 | . . . . . . . . . . 11 ⊢ (𝜑 → ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) ⊆ ((Base‘𝐾) ↑m (0...𝑁))) |
34 | 33 | sspwd 4616 | . . . . . . . . . 10 ⊢ (𝜑 → 𝒫 ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0g‘(𝐾 freeLMod (0...𝑁)))}) ⊆ 𝒫 ((Base‘𝐾) ↑m (0...𝑁))) |
35 | 27, 34 | sstrd 3993 | . . . . . . . . 9 ⊢ (𝜑 → 𝑃 ⊆ 𝒫 ((Base‘𝐾) ↑m (0...𝑁))) |
36 | 35 | sselda 3983 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝑝 ∈ 𝒫 ((Base‘𝐾) ↑m (0...𝑁))) |
37 | 36 | elpwid 4612 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝑝 ⊆ ((Base‘𝐾) ↑m (0...𝑁))) |
38 | sseqin2 4216 | . . . . . . 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 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝑁 ∈ ℕ0) |
41 | 26 | adantr 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝐾 ∈ DivRing) |
42 | simpr 486 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝑝 ∈ 𝑃) | |
43 | 3, 24, 25, 40, 41, 42 | prjspnn0 41412 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → 𝑝 ≠ ∅) |
44 | 39, 43 | eqnetrd 3009 | . . . . 5 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → (((Base‘𝐾) ↑m (0...𝑁)) ∩ 𝑝) ≠ ∅) |
45 | xpima2 6184 | . . . . 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 2773 | . . 3 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝑃) → ((((0...𝑁) eval 𝐾)‘ 0 ) “ 𝑝) = {(0g‘𝐾)}) |
48 | 47 | rabeqcda 3444 | . 2 ⊢ (𝜑 → {𝑝 ∈ 𝑃 ∣ ((((0...𝑁) eval 𝐾)‘ 0 ) “ 𝑝) = {(0g‘𝐾)}} = 𝑃) |
49 | 18, 48 | eqtrd 2773 | 1 ⊢ (𝜑 → ((𝑁ℙ𝕣𝕠𝕛Crv𝐾)‘ 0 ) = 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 {crab 3433 Vcvv 3475 ∖ cdif 3946 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 𝒫 cpw 4603 {csn 4629 ∪ cuni 4909 class class class wbr 5149 × cxp 5675 ◡ccnv 5676 ran crn 5678 “ cima 5680 ‘cfv 6544 (class class class)co 7409 ↑m cmap 8820 Fincfn 8939 finSupp cfsupp 9361 0cc0 11110 ℕcn 12212 ℕ0cn0 12472 ...cfz 13484 Basecbs 17144 0gc0g 17385 CRingccrg 20057 DivRingcdr 20357 Fieldcfield 20358 freeLMod cfrlm 21301 mPoly cmpl 21459 eval cevl 21634 mHomP cmhp 21672 ℙ𝕣𝕠𝕛ncprjspn 41404 ℙ𝕣𝕠𝕛Crvcprjcrv 41419 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-ofr 7671 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-ec 8705 df-qs 8709 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-fzo 13628 df-seq 13967 df-hash 14291 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-hom 17221 df-cco 17222 df-0g 17387 df-gsum 17388 df-prds 17393 df-pws 17395 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-submnd 18672 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mulg 18951 df-subg 19003 df-ghm 19090 df-cntz 19181 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-srg 20010 df-ring 20058 df-cring 20059 df-oppr 20150 df-dvdsr 20171 df-unit 20172 df-invr 20202 df-rnghom 20251 df-subrg 20317 df-drng 20359 df-field 20360 df-lmod 20473 df-lss 20543 df-lsp 20583 df-lvec 20714 df-sra 20785 df-rgmod 20786 df-dsmm 21287 df-frlm 21302 df-assa 21408 df-asp 21409 df-ascl 21410 df-psr 21462 df-mvr 21463 df-mpl 21464 df-evls 21635 df-evl 21636 df-mhp 21676 df-prjsp 41392 df-prjspn 41405 df-prjcrv 41420 |
This theorem is referenced by: (None) |
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