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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 𝑋 of a field 𝑅 to be integral over a subring 𝑆. (Contributed by Thierry Arnoux, 28-Jan-2025.) |
Ref | Expression |
---|---|
irngval.o | ⊢ 𝑂 = (𝑅 evalSub1 𝑆) |
irngval.u | ⊢ 𝑈 = (𝑅 ↾s 𝑆) |
irngval.b | ⊢ 𝐵 = (Base‘𝑅) |
irngval.0 | ⊢ 0 = (0g‘𝑅) |
elirng.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
elirng.s | ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) |
Ref | Expression |
---|---|
elirng | ⊢ (𝜑 → (𝑋 ∈ (𝑅 IntgRing 𝑆) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic1p‘𝑈)((𝑂‘𝑓)‘𝑋) = 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | irngval.o | . . . . . 6 ⊢ 𝑂 = (𝑅 evalSub1 𝑆) | |
2 | irngval.u | . . . . . 6 ⊢ 𝑈 = (𝑅 ↾s 𝑆) | |
3 | irngval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
4 | irngval.0 | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
5 | elirng.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
6 | 5 | crngringd 20029 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring) |
7 | elirng.s | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅)) | |
8 | 3 | subrgss 20315 | . . . . . . 7 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ 𝐵) |
9 | 7, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
10 | 1, 2, 3, 4, 6, 9 | irngval 32651 | . . . . 5 ⊢ (𝜑 → (𝑅 IntgRing 𝑆) = ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 })) |
11 | 10 | eleq2d 2819 | . . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝑅 IntgRing 𝑆) ↔ 𝑋 ∈ ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }))) |
12 | eliun 4995 | . . . 4 ⊢ (𝑋 ∈ ∪ 𝑓 ∈ (Monic1p‘𝑈)(◡(𝑂‘𝑓) “ { 0 }) ↔ ∃𝑓 ∈ (Monic1p‘𝑈)𝑋 ∈ (◡(𝑂‘𝑓) “ { 0 })) | |
13 | 11, 12 | bitrdi 286 | . . 3 ⊢ (𝜑 → (𝑋 ∈ (𝑅 IntgRing 𝑆) ↔ ∃𝑓 ∈ (Monic1p‘𝑈)𝑋 ∈ (◡(𝑂‘𝑓) “ { 0 }))) |
14 | eqid 2732 | . . . . . 6 ⊢ (𝑅 ↑s 𝐵) = (𝑅 ↑s 𝐵) | |
15 | eqid 2732 | . . . . . 6 ⊢ (Base‘(𝑅 ↑s 𝐵)) = (Base‘(𝑅 ↑s 𝐵)) | |
16 | 6 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (Monic1p‘𝑈)) → 𝑅 ∈ Ring) |
17 | 3 | fvexi 6893 | . . . . . . 7 ⊢ 𝐵 ∈ V |
18 | 17 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (Monic1p‘𝑈)) → 𝐵 ∈ V) |
19 | eqid 2732 | . . . . . . . . . . 11 ⊢ (Poly1‘𝑈) = (Poly1‘𝑈) | |
20 | 1, 3, 14, 2, 19 | evls1rhm 21772 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑅)) → 𝑂 ∈ ((Poly1‘𝑈) RingHom (𝑅 ↑s 𝐵))) |
21 | 5, 7, 20 | syl2anc 584 | . . . . . . . . 9 ⊢ (𝜑 → 𝑂 ∈ ((Poly1‘𝑈) RingHom (𝑅 ↑s 𝐵))) |
22 | eqid 2732 | . . . . . . . . . 10 ⊢ (Base‘(Poly1‘𝑈)) = (Base‘(Poly1‘𝑈)) | |
23 | 22, 15 | rhmf 20215 | . . . . . . . . 9 ⊢ (𝑂 ∈ ((Poly1‘𝑈) RingHom (𝑅 ↑s 𝐵)) → 𝑂:(Base‘(Poly1‘𝑈))⟶(Base‘(𝑅 ↑s 𝐵))) |
24 | 21, 23 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑂:(Base‘(Poly1‘𝑈))⟶(Base‘(𝑅 ↑s 𝐵))) |
25 | 24 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Monic1p‘𝑈)) → 𝑂:(Base‘(Poly1‘𝑈))⟶(Base‘(𝑅 ↑s 𝐵))) |
26 | eqid 2732 | . . . . . . . . 9 ⊢ (Monic1p‘𝑈) = (Monic1p‘𝑈) | |
27 | 19, 22, 26 | mon1pcl 25593 | . . . . . . . 8 ⊢ (𝑓 ∈ (Monic1p‘𝑈) → 𝑓 ∈ (Base‘(Poly1‘𝑈))) |
28 | 27 | adantl 482 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (Monic1p‘𝑈)) → 𝑓 ∈ (Base‘(Poly1‘𝑈))) |
29 | 25, 28 | ffvelcdmd 7073 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (Monic1p‘𝑈)) → (𝑂‘𝑓) ∈ (Base‘(𝑅 ↑s 𝐵))) |
30 | 14, 3, 15, 16, 18, 29 | pwselbas 17419 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (Monic1p‘𝑈)) → (𝑂‘𝑓):𝐵⟶𝐵) |
31 | ffn 6705 | . . . . 5 ⊢ ((𝑂‘𝑓):𝐵⟶𝐵 → (𝑂‘𝑓) Fn 𝐵) | |
32 | fniniseg 7047 | . . . . 5 ⊢ ((𝑂‘𝑓) Fn 𝐵 → (𝑋 ∈ (◡(𝑂‘𝑓) “ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑓)‘𝑋) = 0 ))) | |
33 | 30, 31, 32 | 3syl 18 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (Monic1p‘𝑈)) → (𝑋 ∈ (◡(𝑂‘𝑓) “ { 0 }) ↔ (𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑓)‘𝑋) = 0 ))) |
34 | 33 | rexbidva 3176 | . . 3 ⊢ (𝜑 → (∃𝑓 ∈ (Monic1p‘𝑈)𝑋 ∈ (◡(𝑂‘𝑓) “ { 0 }) ↔ ∃𝑓 ∈ (Monic1p‘𝑈)(𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑓)‘𝑋) = 0 ))) |
35 | 13, 34 | bitrd 278 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑅 IntgRing 𝑆) ↔ ∃𝑓 ∈ (Monic1p‘𝑈)(𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑓)‘𝑋) = 0 ))) |
36 | r19.42v 3190 | . 2 ⊢ (∃𝑓 ∈ (Monic1p‘𝑈)(𝑋 ∈ 𝐵 ∧ ((𝑂‘𝑓)‘𝑋) = 0 ) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic1p‘𝑈)((𝑂‘𝑓)‘𝑋) = 0 )) | |
37 | 35, 36 | bitrdi 286 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝑅 IntgRing 𝑆) ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑓 ∈ (Monic1p‘𝑈)((𝑂‘𝑓)‘𝑋) = 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 Vcvv 3474 ⊆ wss 3945 {csn 4623 ∪ ciun 4991 ◡ccnv 5669 “ cima 5673 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7394 Basecbs 17128 ↾s cress 17157 0gc0g 17369 ↑s cpws 17376 Ringcrg 20016 CRingccrg 20017 RingHom crh 20200 SubRingcsubrg 20310 Poly1cpl1 21632 evalSub1 ces1 21763 Monic1pcmn1 25574 IntgRing cirng 32649 |
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 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 |
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 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-se 5626 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-of 7654 df-ofr 7655 df-om 7840 df-1st 7959 df-2nd 7960 df-supp 8131 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-1o 8450 df-er 8688 df-map 8807 df-pm 8808 df-ixp 8877 df-en 8925 df-dom 8926 df-sdom 8927 df-fin 8928 df-fsupp 9347 df-sup 9421 df-oi 9489 df-card 9918 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-2 12259 df-3 12260 df-4 12261 df-5 12262 df-6 12263 df-7 12264 df-8 12265 df-9 12266 df-n0 12457 df-z 12543 df-dec 12662 df-uz 12807 df-fz 13469 df-fzo 13612 df-seq 13951 df-hash 14275 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17129 df-ress 17158 df-plusg 17194 df-mulr 17195 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-hom 17205 df-cco 17206 df-0g 17371 df-gsum 17372 df-prds 17377 df-pws 17379 df-mre 17514 df-mrc 17515 df-acs 17517 df-mgm 18545 df-sgrp 18594 df-mnd 18605 df-mhm 18649 df-submnd 18650 df-grp 18799 df-minusg 18800 df-sbg 18801 df-mulg 18925 df-subg 18977 df-ghm 19058 df-cntz 19149 df-cmn 19616 df-abl 19617 df-mgp 19949 df-ur 19966 df-srg 19970 df-ring 20018 df-cring 20019 df-rnghom 20203 df-subrg 20312 df-lmod 20424 df-lss 20494 df-lsp 20534 df-assa 21343 df-asp 21344 df-ascl 21345 df-psr 21395 df-mvr 21396 df-mpl 21397 df-opsr 21399 df-evls 21566 df-psr1 21635 df-ply1 21637 df-evls1 21765 df-mon1 25579 df-irng 32650 |
This theorem is referenced by: irngss 32653 irngssv 32654 0ringirng 32655 irngnzply1lem 32656 irngnzply1 32657 |
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