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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ig1pmindeg | Structured version Visualization version GIF version |
Description: The polynomial ideal generator is of minimum degree. (Contributed by Thierry Arnoux, 19-Mar-2025.) |
Ref | Expression |
---|---|
ig1pirred.p | ⊢ 𝑃 = (Poly1‘𝑅) |
ig1pirred.g | ⊢ 𝐺 = (idlGen1p‘𝑅) |
ig1pirred.u | ⊢ 𝑈 = (Base‘𝑃) |
ig1pirred.r | ⊢ (𝜑 → 𝑅 ∈ DivRing) |
ig1pirred.1 | ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑃)) |
ig1pmindeg.d | ⊢ 𝐷 = ( deg1 ‘𝑅) |
ig1pmindeg.o | ⊢ 0 = (0g‘𝑃) |
ig1pmindeg.2 | ⊢ (𝜑 → 𝐹 ∈ 𝐼) |
ig1pmindeg.3 | ⊢ (𝜑 → 𝐹 ≠ 0 ) |
Ref | Expression |
---|---|
ig1pmindeg | ⊢ (𝜑 → (𝐷‘(𝐺‘𝐼)) ≤ (𝐷‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ig1pmindeg.2 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝐼) | |
2 | 1 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 = { 0 }) → 𝐹 ∈ 𝐼) |
3 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 = { 0 }) → 𝐼 = { 0 }) | |
4 | 2, 3 | eleqtrd 2830 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 = { 0 }) → 𝐹 ∈ { 0 }) |
5 | elsni 4641 | . . . 4 ⊢ (𝐹 ∈ { 0 } → 𝐹 = 0 ) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝐼 = { 0 }) → 𝐹 = 0 ) |
7 | ig1pmindeg.3 | . . . 4 ⊢ (𝜑 → 𝐹 ≠ 0 ) | |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐼 = { 0 }) → 𝐹 ≠ 0 ) |
9 | 6, 8 | pm2.21ddne 3021 | . 2 ⊢ ((𝜑 ∧ 𝐼 = { 0 }) → (𝐷‘(𝐺‘𝐼)) ≤ (𝐷‘𝐹)) |
10 | ig1pirred.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ DivRing) | |
11 | 10 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → 𝑅 ∈ DivRing) |
12 | ig1pirred.1 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑃)) | |
13 | 12 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → 𝐼 ∈ (LIdeal‘𝑃)) |
14 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → 𝐼 ≠ { 0 }) | |
15 | ig1pirred.p | . . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅) | |
16 | ig1pirred.g | . . . . . 6 ⊢ 𝐺 = (idlGen1p‘𝑅) | |
17 | ig1pmindeg.o | . . . . . 6 ⊢ 0 = (0g‘𝑃) | |
18 | eqid 2727 | . . . . . 6 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃) | |
19 | ig1pmindeg.d | . . . . . 6 ⊢ 𝐷 = ( deg1 ‘𝑅) | |
20 | eqid 2727 | . . . . . 6 ⊢ (Monic1p‘𝑅) = (Monic1p‘𝑅) | |
21 | 15, 16, 17, 18, 19, 20 | ig1pval3 26099 | . . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ (LIdeal‘𝑃) ∧ 𝐼 ≠ { 0 }) → ((𝐺‘𝐼) ∈ 𝐼 ∧ (𝐺‘𝐼) ∈ (Monic1p‘𝑅) ∧ (𝐷‘(𝐺‘𝐼)) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))) |
22 | 11, 13, 14, 21 | syl3anc 1369 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → ((𝐺‘𝐼) ∈ 𝐼 ∧ (𝐺‘𝐼) ∈ (Monic1p‘𝑅) ∧ (𝐷‘(𝐺‘𝐼)) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ))) |
23 | 22 | simp3d 1142 | . . 3 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → (𝐷‘(𝐺‘𝐼)) = inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < )) |
24 | nfv 1910 | . . . . 5 ⊢ Ⅎ𝑓(𝜑 ∧ 𝐼 ≠ { 0 }) | |
25 | ig1pirred.u | . . . . . . . 8 ⊢ 𝑈 = (Base‘𝑃) | |
26 | 19, 15, 25 | deg1xrf 26004 | . . . . . . 7 ⊢ 𝐷:𝑈⟶ℝ* |
27 | 26 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → 𝐷:𝑈⟶ℝ*) |
28 | 27 | ffund 6720 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → Fun 𝐷) |
29 | 11 | drngringd 20621 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → 𝑅 ∈ Ring) |
30 | 29 | adantr 480 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐼 ≠ { 0 }) ∧ 𝑓 ∈ (𝐼 ∖ { 0 })) → 𝑅 ∈ Ring) |
31 | 25, 18 | lidlss 21097 | . . . . . . . . . 10 ⊢ (𝐼 ∈ (LIdeal‘𝑃) → 𝐼 ⊆ 𝑈) |
32 | 13, 31 | syl 17 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → 𝐼 ⊆ 𝑈) |
33 | 32 | ssdifssd 4138 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → (𝐼 ∖ { 0 }) ⊆ 𝑈) |
34 | 33 | sselda 3978 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐼 ≠ { 0 }) ∧ 𝑓 ∈ (𝐼 ∖ { 0 })) → 𝑓 ∈ 𝑈) |
35 | eldifsni 4789 | . . . . . . . 8 ⊢ (𝑓 ∈ (𝐼 ∖ { 0 }) → 𝑓 ≠ 0 ) | |
36 | 35 | adantl 481 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐼 ≠ { 0 }) ∧ 𝑓 ∈ (𝐼 ∖ { 0 })) → 𝑓 ≠ 0 ) |
37 | 19, 15, 17, 25 | deg1nn0cl 26011 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ 𝑈 ∧ 𝑓 ≠ 0 ) → (𝐷‘𝑓) ∈ ℕ0) |
38 | 30, 34, 36, 37 | syl3anc 1369 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐼 ≠ { 0 }) ∧ 𝑓 ∈ (𝐼 ∖ { 0 })) → (𝐷‘𝑓) ∈ ℕ0) |
39 | nn0uz 12886 | . . . . . 6 ⊢ ℕ0 = (ℤ≥‘0) | |
40 | 38, 39 | eleqtrdi 2838 | . . . . 5 ⊢ (((𝜑 ∧ 𝐼 ≠ { 0 }) ∧ 𝑓 ∈ (𝐼 ∖ { 0 })) → (𝐷‘𝑓) ∈ (ℤ≥‘0)) |
41 | 24, 28, 40 | funimassd 6959 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → (𝐷 “ (𝐼 ∖ { 0 })) ⊆ (ℤ≥‘0)) |
42 | 27 | ffnd 6717 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → 𝐷 Fn 𝑈) |
43 | 1 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → 𝐹 ∈ 𝐼) |
44 | 32, 43 | sseldd 3979 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → 𝐹 ∈ 𝑈) |
45 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → 𝐹 ≠ 0 ) |
46 | nelsn 4664 | . . . . . . 7 ⊢ (𝐹 ≠ 0 → ¬ 𝐹 ∈ { 0 }) | |
47 | 45, 46 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → ¬ 𝐹 ∈ { 0 }) |
48 | 43, 47 | eldifd 3955 | . . . . 5 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → 𝐹 ∈ (𝐼 ∖ { 0 })) |
49 | 42, 44, 48 | fnfvimad 7240 | . . . 4 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → (𝐷‘𝐹) ∈ (𝐷 “ (𝐼 ∖ { 0 }))) |
50 | infssuzle 12937 | . . . 4 ⊢ (((𝐷 “ (𝐼 ∖ { 0 })) ⊆ (ℤ≥‘0) ∧ (𝐷‘𝐹) ∈ (𝐷 “ (𝐼 ∖ { 0 }))) → inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ) ≤ (𝐷‘𝐹)) | |
51 | 41, 49, 50 | syl2anc 583 | . . 3 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → inf((𝐷 “ (𝐼 ∖ { 0 })), ℝ, < ) ≤ (𝐷‘𝐹)) |
52 | 23, 51 | eqbrtrd 5164 | . 2 ⊢ ((𝜑 ∧ 𝐼 ≠ { 0 }) → (𝐷‘(𝐺‘𝐼)) ≤ (𝐷‘𝐹)) |
53 | 9, 52 | pm2.61dane 3024 | 1 ⊢ (𝜑 → (𝐷‘(𝐺‘𝐼)) ≤ (𝐷‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∖ cdif 3941 ⊆ wss 3944 {csn 4624 class class class wbr 5142 “ cima 5675 ⟶wf 6538 ‘cfv 6542 infcinf 9456 ℝcr 11129 0cc0 11130 ℝ*cxr 11269 < clt 11270 ≤ cle 11271 ℕ0cn0 12494 ℤ≥cuz 12844 Basecbs 17171 0gc0g 17412 Ringcrg 20164 DivRingcdr 20613 LIdealclidl 21091 Poly1cpl1 22083 deg1 cdg1 25974 Monic1pcmn1 26048 idlGen1pcig1p 26052 |
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 ax-pre-sup 11208 ax-addf 11209 |
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-tpos 8225 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-inf 9458 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-starv 17239 df-sca 17240 df-vsca 17241 df-ip 17242 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 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-ring 20166 df-cring 20167 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-subrng 20472 df-subrg 20497 df-drng 20615 df-lmod 20734 df-lss 20805 df-sra 21047 df-rgmod 21048 df-lidl 21093 df-rlreg 21219 df-cnfld 21267 df-ascl 21776 df-psr 21829 df-mvr 21830 df-mpl 21831 df-opsr 21833 df-psr1 22086 df-vr1 22087 df-ply1 22088 df-coe1 22089 df-mdeg 25975 df-deg1 25976 df-mon1 26053 df-uc1p 26054 df-ig1p 26057 |
This theorem is referenced by: minplyirredlem 33316 irredminply 33320 |
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