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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ply1annidllem | Structured version Visualization version GIF version |
Description: Write the set π of polynomials annihilating an element π΄ as the kernel of the ring homomorphism πΉ mapping polynomials π to their subring evaluation at a given point π΄. (Contributed by Thierry Arnoux, 9-Feb-2025.) |
Ref | Expression |
---|---|
ply1annidl.o | β’ π = (π evalSub1 π) |
ply1annidl.p | β’ π = (Poly1β(π βΎs π)) |
ply1annidl.b | β’ π΅ = (Baseβπ ) |
ply1annidl.r | β’ (π β π β CRing) |
ply1annidl.s | β’ (π β π β (SubRingβπ )) |
ply1annidl.a | β’ (π β π΄ β π΅) |
ply1annidl.0 | β’ 0 = (0gβπ ) |
ply1annidl.q | β’ π = {π β dom π β£ ((πβπ)βπ΄) = 0 } |
ply1annidllem.f | β’ πΉ = (π β (Baseβπ) β¦ ((πβπ)βπ΄)) |
Ref | Expression |
---|---|
ply1annidllem | β’ (π β π = (β‘πΉ β { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1annidl.q | . 2 β’ π = {π β dom π β£ ((πβπ)βπ΄) = 0 } | |
2 | nfv 1910 | . . . . . 6 β’ β²ππ | |
3 | fvexd 6906 | . . . . . 6 β’ ((π β§ π β (Baseβπ)) β ((πβπ)βπ΄) β V) | |
4 | ply1annidllem.f | . . . . . 6 β’ πΉ = (π β (Baseβπ) β¦ ((πβπ)βπ΄)) | |
5 | 2, 3, 4 | fnmptd 6690 | . . . . 5 β’ (π β πΉ Fn (Baseβπ)) |
6 | ply1annidl.o | . . . . . . . 8 β’ π = (π evalSub1 π) | |
7 | ply1annidl.p | . . . . . . . 8 β’ π = (Poly1β(π βΎs π)) | |
8 | eqid 2727 | . . . . . . . 8 β’ (Baseβπ) = (Baseβπ) | |
9 | ply1annidl.r | . . . . . . . 8 β’ (π β π β CRing) | |
10 | ply1annidl.s | . . . . . . . 8 β’ (π β π β (SubRingβπ )) | |
11 | 6, 7, 8, 9, 10 | evls1fn 33171 | . . . . . . 7 β’ (π β π Fn (Baseβπ)) |
12 | 11 | fndmd 6653 | . . . . . 6 β’ (π β dom π = (Baseβπ)) |
13 | 12 | fneq2d 6642 | . . . . 5 β’ (π β (πΉ Fn dom π β πΉ Fn (Baseβπ))) |
14 | 5, 13 | mpbird 257 | . . . 4 β’ (π β πΉ Fn dom π) |
15 | fniniseg2 7065 | . . . 4 β’ (πΉ Fn dom π β (β‘πΉ β { 0 }) = {π β dom π β£ (πΉβπ) = 0 }) | |
16 | 14, 15 | syl 17 | . . 3 β’ (π β (β‘πΉ β { 0 }) = {π β dom π β£ (πΉβπ) = 0 }) |
17 | fveq2 6891 | . . . . . . 7 β’ (π = π β (πβπ) = (πβπ)) | |
18 | 17 | fveq1d 6893 | . . . . . 6 β’ (π = π β ((πβπ)βπ΄) = ((πβπ)βπ΄)) |
19 | 12 | eleq2d 2814 | . . . . . . 7 β’ (π β (π β dom π β π β (Baseβπ))) |
20 | 19 | biimpa 476 | . . . . . 6 β’ ((π β§ π β dom π) β π β (Baseβπ)) |
21 | fvexd 6906 | . . . . . 6 β’ ((π β§ π β dom π) β ((πβπ)βπ΄) β V) | |
22 | 4, 18, 20, 21 | fvmptd3 7022 | . . . . 5 β’ ((π β§ π β dom π) β (πΉβπ) = ((πβπ)βπ΄)) |
23 | 22 | eqeq1d 2729 | . . . 4 β’ ((π β§ π β dom π) β ((πΉβπ) = 0 β ((πβπ)βπ΄) = 0 )) |
24 | 23 | rabbidva 3434 | . . 3 β’ (π β {π β dom π β£ (πΉβπ) = 0 } = {π β dom π β£ ((πβπ)βπ΄) = 0 }) |
25 | 16, 24 | eqtr2d 2768 | . 2 β’ (π β {π β dom π β£ ((πβπ)βπ΄) = 0 } = (β‘πΉ β { 0 })) |
26 | 1, 25 | eqtrid 2779 | 1 β’ (π β π = (β‘πΉ β { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 {crab 3427 Vcvv 3469 {csn 4624 β¦ cmpt 5225 β‘ccnv 5671 dom cdm 5672 β cima 5675 Fn wfn 6537 βcfv 6542 (class class class)co 7414 Basecbs 17171 βΎs cress 17200 0gc0g 17412 CRingccrg 20165 SubRingcsubrg 20495 Poly1cpl1 22083 evalSub1 ces1 22219 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-pm 8839 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9378 df-sup 9457 df-oi 9525 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-fzo 13652 df-seq 13991 df-hash 14314 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-hom 17248 df-cco 17249 df-0g 17414 df-gsum 17415 df-prds 17420 df-pws 17422 df-mre 17557 df-mrc 17558 df-acs 17560 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-submnd 18732 df-grp 18884 df-minusg 18885 df-sbg 18886 df-mulg 19015 df-subg 19069 df-ghm 19159 df-cntz 19259 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-srg 20118 df-ring 20166 df-cring 20167 df-rhm 20400 df-subrng 20472 df-subrg 20497 df-lmod 20734 df-lss 20805 df-lsp 20845 df-assa 21774 df-asp 21775 df-ascl 21776 df-psr 21829 df-mvr 21830 df-mpl 21831 df-opsr 21833 df-evls 22005 df-psr1 22086 df-ply1 22088 df-evls1 22221 |
This theorem is referenced by: ply1annidl 33309 ply1annprmidl 33314 algextdeglem4 33324 algextdeglem5 33325 |
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