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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ressply1mon1p | Structured version Visualization version GIF version |
Description: The monic polynomials of a restricted polynomial algebra. (Contributed by Thierry Arnoux, 21-Jan-2025.) |
Ref | Expression |
---|---|
ressply.1 | ⊢ 𝑆 = (Poly1‘𝑅) |
ressply.2 | ⊢ 𝐻 = (𝑅 ↾s 𝑇) |
ressply.3 | ⊢ 𝑈 = (Poly1‘𝐻) |
ressply.4 | ⊢ 𝐵 = (Base‘𝑈) |
ressply.5 | ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) |
ressply1mon1p.m | ⊢ 𝑀 = (Monic1p‘𝑅) |
ressply1mon1p.n | ⊢ 𝑁 = (Monic1p‘𝐻) |
Ref | Expression |
---|---|
ressply1mon1p | ⊢ (𝜑 → 𝑁 = (𝐵 ∩ 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressply.1 | . . . . . 6 ⊢ 𝑆 = (Poly1‘𝑅) | |
2 | eqid 2724 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
3 | eqid 2724 | . . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
4 | eqid 2724 | . . . . . 6 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅) | |
5 | ressply1mon1p.m | . . . . . 6 ⊢ 𝑀 = (Monic1p‘𝑅) | |
6 | eqid 2724 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | ismon1p 26000 | . . . . 5 ⊢ (𝑝 ∈ 𝑀 ↔ (𝑝 ∈ (Base‘𝑆) ∧ 𝑝 ≠ (0g‘𝑆) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝑅)‘𝑝)) = (1r‘𝑅))) |
8 | 7 | anbi2i 622 | . . . 4 ⊢ ((𝑝 ∈ 𝐵 ∧ 𝑝 ∈ 𝑀) ↔ (𝑝 ∈ 𝐵 ∧ (𝑝 ∈ (Base‘𝑆) ∧ 𝑝 ≠ (0g‘𝑆) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝑅)‘𝑝)) = (1r‘𝑅)))) |
9 | ressply.2 | . . . . . . . . . . 11 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
10 | ressply.3 | . . . . . . . . . . 11 ⊢ 𝑈 = (Poly1‘𝐻) | |
11 | ressply.4 | . . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑈) | |
12 | ressply.5 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
13 | eqid 2724 | . . . . . . . . . . 11 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵) | |
14 | 1, 9, 10, 11, 12, 13 | ressply1bas 22070 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆 ↾s 𝐵))) |
15 | 13, 2 | ressbasss 17182 | . . . . . . . . . 10 ⊢ (Base‘(𝑆 ↾s 𝐵)) ⊆ (Base‘𝑆) |
16 | 14, 15 | eqsstrdi 4028 | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑆)) |
17 | 16 | sseld 3973 | . . . . . . . 8 ⊢ (𝜑 → (𝑝 ∈ 𝐵 → 𝑝 ∈ (Base‘𝑆))) |
18 | 17 | pm4.71d 561 | . . . . . . 7 ⊢ (𝜑 → (𝑝 ∈ 𝐵 ↔ (𝑝 ∈ 𝐵 ∧ 𝑝 ∈ (Base‘𝑆)))) |
19 | 18 | anbi1d 629 | . . . . . 6 ⊢ (𝜑 → ((𝑝 ∈ 𝐵 ∧ (𝑝 ≠ (0g‘𝑆) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝑅)‘𝑝)) = (1r‘𝑅))) ↔ ((𝑝 ∈ 𝐵 ∧ 𝑝 ∈ (Base‘𝑆)) ∧ (𝑝 ≠ (0g‘𝑆) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝑅)‘𝑝)) = (1r‘𝑅))))) |
20 | 13an22anass 32175 | . . . . . 6 ⊢ ((𝑝 ∈ 𝐵 ∧ (𝑝 ∈ (Base‘𝑆) ∧ 𝑝 ≠ (0g‘𝑆) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝑅)‘𝑝)) = (1r‘𝑅))) ↔ ((𝑝 ∈ 𝐵 ∧ 𝑝 ∈ (Base‘𝑆)) ∧ (𝑝 ≠ (0g‘𝑆) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝑅)‘𝑝)) = (1r‘𝑅)))) | |
21 | 19, 20 | bitr4di 289 | . . . . 5 ⊢ (𝜑 → ((𝑝 ∈ 𝐵 ∧ (𝑝 ≠ (0g‘𝑆) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝑅)‘𝑝)) = (1r‘𝑅))) ↔ (𝑝 ∈ 𝐵 ∧ (𝑝 ∈ (Base‘𝑆) ∧ 𝑝 ≠ (0g‘𝑆) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝑅)‘𝑝)) = (1r‘𝑅))))) |
22 | 1, 9, 10, 11, 12, 3 | ressply10g 33123 | . . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝑆) = (0g‘𝑈)) |
23 | 22 | neeq2d 2993 | . . . . . . . . 9 ⊢ (𝜑 → (𝑝 ≠ (0g‘𝑆) ↔ 𝑝 ≠ (0g‘𝑈))) |
24 | 23 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (𝑝 ≠ (0g‘𝑆) ↔ 𝑝 ≠ (0g‘𝑈))) |
25 | simpr 484 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝑝 ∈ 𝐵) | |
26 | 12 | adantr 480 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → 𝑇 ∈ (SubRing‘𝑅)) |
27 | 9, 4, 10, 11, 25, 26 | ressdeg1 33118 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (( deg1 ‘𝑅)‘𝑝) = (( deg1 ‘𝐻)‘𝑝)) |
28 | 27 | fveq2d 6885 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → ((coe1‘𝑝)‘(( deg1 ‘𝑅)‘𝑝)) = ((coe1‘𝑝)‘(( deg1 ‘𝐻)‘𝑝))) |
29 | 9, 6 | subrg1 20474 | . . . . . . . . . . 11 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (1r‘𝑅) = (1r‘𝐻)) |
30 | 12, 29 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → (1r‘𝑅) = (1r‘𝐻)) |
31 | 30 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (1r‘𝑅) = (1r‘𝐻)) |
32 | 28, 31 | eqeq12d 2740 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → (((coe1‘𝑝)‘(( deg1 ‘𝑅)‘𝑝)) = (1r‘𝑅) ↔ ((coe1‘𝑝)‘(( deg1 ‘𝐻)‘𝑝)) = (1r‘𝐻))) |
33 | 24, 32 | anbi12d 630 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐵) → ((𝑝 ≠ (0g‘𝑆) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝑅)‘𝑝)) = (1r‘𝑅)) ↔ (𝑝 ≠ (0g‘𝑈) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝐻)‘𝑝)) = (1r‘𝐻)))) |
34 | 33 | pm5.32da 578 | . . . . . 6 ⊢ (𝜑 → ((𝑝 ∈ 𝐵 ∧ (𝑝 ≠ (0g‘𝑆) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝑅)‘𝑝)) = (1r‘𝑅))) ↔ (𝑝 ∈ 𝐵 ∧ (𝑝 ≠ (0g‘𝑈) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝐻)‘𝑝)) = (1r‘𝐻))))) |
35 | 3anass 1092 | . . . . . 6 ⊢ ((𝑝 ∈ 𝐵 ∧ 𝑝 ≠ (0g‘𝑈) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝐻)‘𝑝)) = (1r‘𝐻)) ↔ (𝑝 ∈ 𝐵 ∧ (𝑝 ≠ (0g‘𝑈) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝐻)‘𝑝)) = (1r‘𝐻)))) | |
36 | 34, 35 | bitr4di 289 | . . . . 5 ⊢ (𝜑 → ((𝑝 ∈ 𝐵 ∧ (𝑝 ≠ (0g‘𝑆) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝑅)‘𝑝)) = (1r‘𝑅))) ↔ (𝑝 ∈ 𝐵 ∧ 𝑝 ≠ (0g‘𝑈) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝐻)‘𝑝)) = (1r‘𝐻)))) |
37 | 21, 36 | bitr3d 281 | . . . 4 ⊢ (𝜑 → ((𝑝 ∈ 𝐵 ∧ (𝑝 ∈ (Base‘𝑆) ∧ 𝑝 ≠ (0g‘𝑆) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝑅)‘𝑝)) = (1r‘𝑅))) ↔ (𝑝 ∈ 𝐵 ∧ 𝑝 ≠ (0g‘𝑈) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝐻)‘𝑝)) = (1r‘𝐻)))) |
38 | 8, 37 | bitr2id 284 | . . 3 ⊢ (𝜑 → ((𝑝 ∈ 𝐵 ∧ 𝑝 ≠ (0g‘𝑈) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝐻)‘𝑝)) = (1r‘𝐻)) ↔ (𝑝 ∈ 𝐵 ∧ 𝑝 ∈ 𝑀))) |
39 | eqid 2724 | . . . 4 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
40 | eqid 2724 | . . . 4 ⊢ ( deg1 ‘𝐻) = ( deg1 ‘𝐻) | |
41 | ressply1mon1p.n | . . . 4 ⊢ 𝑁 = (Monic1p‘𝐻) | |
42 | eqid 2724 | . . . 4 ⊢ (1r‘𝐻) = (1r‘𝐻) | |
43 | 10, 11, 39, 40, 41, 42 | ismon1p 26000 | . . 3 ⊢ (𝑝 ∈ 𝑁 ↔ (𝑝 ∈ 𝐵 ∧ 𝑝 ≠ (0g‘𝑈) ∧ ((coe1‘𝑝)‘(( deg1 ‘𝐻)‘𝑝)) = (1r‘𝐻))) |
44 | elin 3956 | . . 3 ⊢ (𝑝 ∈ (𝐵 ∩ 𝑀) ↔ (𝑝 ∈ 𝐵 ∧ 𝑝 ∈ 𝑀)) | |
45 | 38, 43, 44 | 3bitr4g 314 | . 2 ⊢ (𝜑 → (𝑝 ∈ 𝑁 ↔ 𝑝 ∈ (𝐵 ∩ 𝑀))) |
46 | 45 | eqrdv 2722 | 1 ⊢ (𝜑 → 𝑁 = (𝐵 ∩ 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∩ cin 3939 ‘cfv 6533 (class class class)co 7401 Basecbs 17143 ↾s cress 17172 0gc0g 17384 1rcur 20076 SubRingcsubrg 20459 Poly1cpl1 22019 coe1cco1 22020 deg1 cdg1 25909 Monic1pcmn1 25983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 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 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-ofr 7664 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-mhm 18703 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-mulg 18986 df-subg 19040 df-ghm 19129 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-subrng 20436 df-subrg 20461 df-lmod 20698 df-lss 20769 df-cnfld 21229 df-ascl 21718 df-psr 21771 df-mpl 21773 df-opsr 21775 df-psr1 22022 df-ply1 22024 df-coe1 22025 df-mdeg 25910 df-deg1 25911 df-mon1 25988 |
This theorem is referenced by: irngss 33231 |
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