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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elirng | Structured version Visualization version GIF version |
Description: Property for an element π of a field π to be integral over a subring π. (Contributed by Thierry Arnoux, 28-Jan-2025.) |
Ref | Expression |
---|---|
irngval.o | β’ π = (π evalSub1 π) |
irngval.u | β’ π = (π βΎs π) |
irngval.b | β’ π΅ = (Baseβπ ) |
irngval.0 | β’ 0 = (0gβπ ) |
elirng.r | β’ (π β π β CRing) |
elirng.s | β’ (π β π β (SubRingβπ )) |
Ref | Expression |
---|---|
elirng | β’ (π β (π β (π IntgRing π) β (π β π΅ β§ βπ β (Monic1pβπ)((πβπ)βπ) = 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | irngval.o | . . . . . 6 β’ π = (π evalSub1 π) | |
2 | irngval.u | . . . . . 6 β’ π = (π βΎs π) | |
3 | irngval.b | . . . . . 6 β’ π΅ = (Baseβπ ) | |
4 | irngval.0 | . . . . . 6 β’ 0 = (0gβπ ) | |
5 | elirng.r | . . . . . . 7 β’ (π β π β CRing) | |
6 | 5 | crngringd 20147 | . . . . . 6 β’ (π β π β Ring) |
7 | elirng.s | . . . . . . 7 β’ (π β π β (SubRingβπ )) | |
8 | 3 | subrgss 20470 | . . . . . . 7 β’ (π β (SubRingβπ ) β π β π΅) |
9 | 7, 8 | syl 17 | . . . . . 6 β’ (π β π β π΅) |
10 | 1, 2, 3, 4, 6, 9 | irngval 33204 | . . . . 5 β’ (π β (π IntgRing π) = βͺ π β (Monic1pβπ)(β‘(πβπ) β { 0 })) |
11 | 10 | eleq2d 2818 | . . . 4 β’ (π β (π β (π IntgRing π) β π β βͺ π β (Monic1pβπ)(β‘(πβπ) β { 0 }))) |
12 | eliun 5001 | . . . 4 β’ (π β βͺ π β (Monic1pβπ)(β‘(πβπ) β { 0 }) β βπ β (Monic1pβπ)π β (β‘(πβπ) β { 0 })) | |
13 | 11, 12 | bitrdi 287 | . . 3 β’ (π β (π β (π IntgRing π) β βπ β (Monic1pβπ)π β (β‘(πβπ) β { 0 }))) |
14 | eqid 2731 | . . . . . 6 β’ (π βs π΅) = (π βs π΅) | |
15 | eqid 2731 | . . . . . 6 β’ (Baseβ(π βs π΅)) = (Baseβ(π βs π΅)) | |
16 | 6 | adantr 480 | . . . . . 6 β’ ((π β§ π β (Monic1pβπ)) β π β Ring) |
17 | 3 | fvexi 6905 | . . . . . . 7 β’ π΅ β V |
18 | 17 | a1i 11 | . . . . . 6 β’ ((π β§ π β (Monic1pβπ)) β π΅ β V) |
19 | eqid 2731 | . . . . . . . . . . 11 β’ (Poly1βπ) = (Poly1βπ) | |
20 | 1, 3, 14, 2, 19 | evls1rhm 22161 | . . . . . . . . . 10 β’ ((π β CRing β§ π β (SubRingβπ )) β π β ((Poly1βπ) RingHom (π βs π΅))) |
21 | 5, 7, 20 | syl2anc 583 | . . . . . . . . 9 β’ (π β π β ((Poly1βπ) RingHom (π βs π΅))) |
22 | eqid 2731 | . . . . . . . . . 10 β’ (Baseβ(Poly1βπ)) = (Baseβ(Poly1βπ)) | |
23 | 22, 15 | rhmf 20383 | . . . . . . . . 9 β’ (π β ((Poly1βπ) RingHom (π βs π΅)) β π:(Baseβ(Poly1βπ))βΆ(Baseβ(π βs π΅))) |
24 | 21, 23 | syl 17 | . . . . . . . 8 β’ (π β π:(Baseβ(Poly1βπ))βΆ(Baseβ(π βs π΅))) |
25 | 24 | adantr 480 | . . . . . . 7 β’ ((π β§ π β (Monic1pβπ)) β π:(Baseβ(Poly1βπ))βΆ(Baseβ(π βs π΅))) |
26 | eqid 2731 | . . . . . . . . 9 β’ (Monic1pβπ) = (Monic1pβπ) | |
27 | 19, 22, 26 | mon1pcl 26000 | . . . . . . . 8 β’ (π β (Monic1pβπ) β π β (Baseβ(Poly1βπ))) |
28 | 27 | adantl 481 | . . . . . . 7 β’ ((π β§ π β (Monic1pβπ)) β π β (Baseβ(Poly1βπ))) |
29 | 25, 28 | ffvelcdmd 7087 | . . . . . 6 β’ ((π β§ π β (Monic1pβπ)) β (πβπ) β (Baseβ(π βs π΅))) |
30 | 14, 3, 15, 16, 18, 29 | pwselbas 17442 | . . . . 5 β’ ((π β§ π β (Monic1pβπ)) β (πβπ):π΅βΆπ΅) |
31 | ffn 6717 | . . . . 5 β’ ((πβπ):π΅βΆπ΅ β (πβπ) Fn π΅) | |
32 | fniniseg 7061 | . . . . 5 β’ ((πβπ) Fn π΅ β (π β (β‘(πβπ) β { 0 }) β (π β π΅ β§ ((πβπ)βπ) = 0 ))) | |
33 | 30, 31, 32 | 3syl 18 | . . . 4 β’ ((π β§ π β (Monic1pβπ)) β (π β (β‘(πβπ) β { 0 }) β (π β π΅ β§ ((πβπ)βπ) = 0 ))) |
34 | 33 | rexbidva 3175 | . . 3 β’ (π β (βπ β (Monic1pβπ)π β (β‘(πβπ) β { 0 }) β βπ β (Monic1pβπ)(π β π΅ β§ ((πβπ)βπ) = 0 ))) |
35 | 13, 34 | bitrd 279 | . 2 β’ (π β (π β (π IntgRing π) β βπ β (Monic1pβπ)(π β π΅ β§ ((πβπ)βπ) = 0 ))) |
36 | r19.42v 3189 | . 2 β’ (βπ β (Monic1pβπ)(π β π΅ β§ ((πβπ)βπ) = 0 ) β (π β π΅ β§ βπ β (Monic1pβπ)((πβπ)βπ) = 0 )) | |
37 | 35, 36 | bitrdi 287 | 1 β’ (π β (π β (π IntgRing π) β (π β π΅ β§ βπ β (Monic1pβπ)((πβπ)βπ) = 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1540 β wcel 2105 βwrex 3069 Vcvv 3473 β wss 3948 {csn 4628 βͺ ciun 4997 β‘ccnv 5675 β cima 5679 Fn wfn 6538 βΆwf 6539 βcfv 6543 (class class class)co 7412 Basecbs 17151 βΎs cress 17180 0gc0g 17392 βs cpws 17399 Ringcrg 20134 CRingccrg 20135 RingHom crh 20367 SubRingcsubrg 20465 Poly1cpl1 22020 evalSub1 ces1 22152 Monic1pcmn1 25981 IntgRing cirng 33202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-ofr 7675 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8152 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9368 df-sup 9443 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-fzo 13635 df-seq 13974 df-hash 14298 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-hom 17228 df-cco 17229 df-0g 17394 df-gsum 17395 df-prds 17400 df-pws 17402 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-mhm 18711 df-submnd 18712 df-grp 18864 df-minusg 18865 df-sbg 18866 df-mulg 18994 df-subg 19046 df-ghm 19135 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-srg 20088 df-ring 20136 df-cring 20137 df-rhm 20370 df-subrng 20442 df-subrg 20467 df-lmod 20704 df-lss 20775 df-lsp 20815 df-assa 21718 df-asp 21719 df-ascl 21720 df-psr 21772 df-mvr 21773 df-mpl 21774 df-opsr 21776 df-evls 21946 df-psr1 22023 df-ply1 22025 df-evls1 22154 df-mon1 25986 df-irng 33203 |
This theorem is referenced by: irngss 33206 irngssv 33207 0ringirng 33208 irngnzply1lem 33209 irngnzply1 33210 |
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