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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 19421 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
2 | mdeg0.p | . . . . 5 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
3 | 2 | mplgrp 20832 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → 𝑃 ∈ Grp) |
4 | 1, 3 | sylan2 596 | . . 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 18249 | . . 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 24816 | . . 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 486 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝐼 ∈ 𝑉) | |
15 | 1 | adantl 485 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Grp) |
16 | 2, 10, 9, 6, 14, 15 | mpl0 20822 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 0 = ({𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} × {(0g‘𝑅)})) |
17 | fvex 6687 | . . . . . . . . . 10 ⊢ (0g‘𝑅) ∈ V | |
18 | fnconstg 6566 | . . . . . . . . . 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 6431 | . . . . . . . . 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 260 | . . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → 0 Fn {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin}) |
22 | ovex 7203 | . . . . . . . . . 10 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
23 | 22 | rabex 5200 | . . . . . . . . 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 7887 | . . . . . . . 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 1372 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (( 0 supp (0g‘𝑅)) = ∅ ↔ 0 = ({𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} × {(0g‘𝑅)}))) |
28 | 16, 27 | mpbird 260 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ( 0 supp (0g‘𝑅)) = ∅) |
29 | 28 | imaeq2d 5903 | . . . . 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 5919 | . . . . 5 ⊢ ((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ ∅) = ∅ | |
31 | 29, 30 | eqtrdi 2789 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → ((𝑦 ∈ {𝑥 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑥 “ ℕ) ∈ Fin} ↦ (ℂfld Σg 𝑦)) “ ( 0 supp (0g‘𝑅))) = ∅) |
32 | 31 | supeq1d 8983 | . . 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 12799 | . . 3 ⊢ sup(∅, ℝ*, < ) = -∞ | |
34 | 32, 33 | eqtrdi 2789 | . 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 2773 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring) → (𝐷‘ 0 ) = -∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {crab 3057 Vcvv 3398 ∅c0 4211 {csn 4516 ↦ cmpt 5110 × cxp 5523 ◡ccnv 5524 “ cima 5528 Fn wfn 6334 ‘cfv 6339 (class class class)co 7170 supp csupp 7856 ↑m cmap 8437 Fincfn 8555 supcsup 8977 -∞cmnf 10751 ℝ*cxr 10752 < clt 10753 ℕcn 11716 ℕ0cn0 11976 Basecbs 16586 0gc0g 16816 Σg cgsu 16817 Grpcgrp 18219 Ringcrg 19416 ℂfldccnfld 20217 mPoly cmpl 20719 mDeg cmdg 24803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-sup 8979 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-tset 16687 df-0g 16818 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-grp 18222 df-minusg 18223 df-subg 18394 df-ring 19418 df-psr 20722 df-mpl 20724 df-mdeg 24805 |
This theorem is referenced by: deg1z 24840 |
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