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Mirrors > Home > MPE Home > Th. List > mdeg0 | Structured version Visualization version GIF version |
Description: Degree of the zero polynomial. (Contributed by Stefan O'Rear, 20-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
Ref | Expression |
---|---|
mdeg0.d | ⊢ 𝐷 = (𝐼 mDeg 𝑅) |
mdeg0.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mdeg0.z | ⊢ 0 = (0g‘𝑃) |
Ref | Expression |
---|---|
mdeg0 | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (𝐷‘ 0 ) = -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringgrp 19788 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
2 | mdeg0.p | . . . . 5 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
3 | 2 | mplgrp 21222 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp) |
4 | 1, 3 | sylan2 593 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Grp) |
5 | eqid 2738 | . . . 4 ⊢ (Base‘𝑃) = (Base‘𝑃) | |
6 | mdeg0.z | . . . 4 ⊢ 0 = (0g‘𝑃) | |
7 | 5, 6 | grpidcl 18607 | . . 3 ⊢ (𝑃 ∈ Grp → 0 ∈ (Base‘𝑃)) |
8 | mdeg0.d | . . . 4 ⊢ 𝐷 = (𝐼 mDeg 𝑅) | |
9 | eqid 2738 | . . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
10 | eqid 2738 | . . . 4 ⊢ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} = {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} | |
11 | eqid 2738 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) = (𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) | |
12 | 8, 2, 5, 9, 10, 11 | mdegval 25228 | . . 3 ⊢ ( 0 ∈ (Base‘𝑃) → (𝐷‘ 0 ) = sup(((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ ( 0 supp (0g‘𝑅))), ℝ*, < )) |
13 | 4, 7, 12 | 3syl 18 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (𝐷‘ 0 ) = sup(((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ ( 0 supp (0g‘𝑅))), ℝ*, < )) |
14 | simpl 483 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝐼 ∈ 𝑉) | |
15 | 1 | adantl 482 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Grp) |
16 | 2, 10, 9, 6, 14, 15 | mpl0 21212 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 0 = ({𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} × {(0g‘𝑅)})) |
17 | fvex 6787 | . . . . . . . . . 10 ⊢ (0g‘𝑅) ∈ V | |
18 | fnconstg 6662 | . . . . . . . . . 10 ⊢ ((0g‘𝑅) ∈ V → ({𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} × {(0g‘𝑅)}) Fn {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin}) | |
19 | 17, 18 | mp1i 13 | . . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ({𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} × {(0g‘𝑅)}) Fn {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin}) |
20 | 16 | fneq1d 6526 | . . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ( 0 Fn {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↔ ({𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} × {(0g‘𝑅)}) Fn {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin})) |
21 | 19, 20 | mpbird 256 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 0 Fn {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin}) |
22 | ovex 7308 | . . . . . . . . . 10 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
23 | 22 | rabex 5256 | . . . . . . . . 9 ⊢ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ∈ V |
24 | 23 | a1i 11 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ∈ V) |
25 | 17 | a1i 11 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (0g‘𝑅) ∈ V) |
26 | fnsuppeq0 8008 | . . . . . . . 8 ⊢ (( 0 Fn {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ∧ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ∈ V ∧ (0g‘𝑅) ∈ V) → (( 0 supp (0g‘𝑅)) = ∅ ↔ 0 = ({𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} × {(0g‘𝑅)}))) | |
27 | 21, 24, 25, 26 | syl3anc 1370 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (( 0 supp (0g‘𝑅)) = ∅ ↔ 0 = ({𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} × {(0g‘𝑅)}))) |
28 | 16, 27 | mpbird 256 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ( 0 supp (0g‘𝑅)) = ∅) |
29 | 28 | imaeq2d 5969 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ ( 0 supp (0g‘𝑅))) = ((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ ∅)) |
30 | ima0 5985 | . . . . 5 ⊢ ((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ ∅) = ∅ | |
31 | 29, 30 | eqtrdi 2794 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ ( 0 supp (0g‘𝑅))) = ∅) |
32 | 31 | supeq1d 9205 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → sup(((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ ( 0 supp (0g‘𝑅))), ℝ*, < ) = sup(∅, ℝ*, < )) |
33 | xrsup0 13057 | . . 3 ⊢ sup(∅, ℝ*, < ) = -∞ | |
34 | 32, 33 | eqtrdi 2794 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → sup(((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ ( 0 supp (0g‘𝑅))), ℝ*, < ) = -∞) |
35 | 13, 34 | eqtrd 2778 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (𝐷‘ 0 ) = -∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 {crab 3068 Vcvv 3432 ∅c0 4256 {csn 4561 ↦ cmpt 5157 × cxp 5587 ◡ccnv 5588 “ cima 5592 Fn wfn 6428 ‘cfv 6433 (class class class)co 7275 supp csupp 7977 ↑m cmap 8615 Fincfn 8733 supcsup 9199 -∞cmnf 11007 ℝ*cxr 11008 < clt 11009 ℕcn 11973 ℕ0cn0 12233 Basecbs 16912 0gc0g 17150 Σg cgsu 17151 Grpcgrp 18577 Ringcrg 19783 ℂfldccnfld 20597 mPoly cmpl 21109 mDeg cmdg 25215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-tset 16981 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-subg 18752 df-ring 19785 df-psr 21112 df-mpl 21114 df-mdeg 25217 |
This theorem is referenced by: deg1z 25252 |
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