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Mirrors > Home > MPE Home > Th. List > mdegval | Structured version Visualization version GIF version |
Description: Value of the multivariate degree function at some particular polynomial. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by AV, 25-Jun-2019.) |
Ref | Expression |
---|---|
mdegval.d | ⊢ 𝐷 = (𝐼 mDeg 𝑅) |
mdegval.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mdegval.b | ⊢ 𝐵 = (Base‘𝑃) |
mdegval.z | ⊢ 0 = (0g‘𝑅) |
mdegval.a | ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} |
mdegval.h | ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) |
Ref | Expression |
---|---|
mdegval | ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7368 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 supp 0 ) = (𝐹 supp 0 )) | |
2 | 1 | imaeq2d 6017 | . . 3 ⊢ (𝑓 = 𝐹 → (𝐻 “ (𝑓 supp 0 )) = (𝐻 “ (𝐹 supp 0 ))) |
3 | 2 | supeq1d 9390 | . 2 ⊢ (𝑓 = 𝐹 → sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < ) = sup((𝐻 “ (𝐹 supp 0 )), ℝ*, < )) |
4 | mdegval.d | . . 3 ⊢ 𝐷 = (𝐼 mDeg 𝑅) | |
5 | mdegval.p | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
6 | mdegval.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
7 | mdegval.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
8 | mdegval.a | . . 3 ⊢ 𝐴 = {𝑚 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑚 “ ℕ) ∈ Fin} | |
9 | mdegval.h | . . 3 ⊢ 𝐻 = (ℎ ∈ 𝐴 ↦ (ℂfld Σg ℎ)) | |
10 | 4, 5, 6, 7, 8, 9 | mdegfval 25450 | . 2 ⊢ 𝐷 = (𝑓 ∈ 𝐵 ↦ sup((𝐻 “ (𝑓 supp 0 )), ℝ*, < )) |
11 | xrltso 13069 | . . 3 ⊢ < Or ℝ* | |
12 | 11 | supex 9407 | . 2 ⊢ sup((𝐻 “ (𝐹 supp 0 )), ℝ*, < ) ∈ V |
13 | 3, 10, 12 | fvmpt 6952 | 1 ⊢ (𝐹 ∈ 𝐵 → (𝐷‘𝐹) = sup((𝐻 “ (𝐹 supp 0 )), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {crab 3406 ↦ cmpt 5192 ◡ccnv 5636 “ cima 5640 ‘cfv 6500 (class class class)co 7361 supp csupp 8096 ↑m cmap 8771 Fincfn 8889 supcsup 9384 ℝ*cxr 11196 < clt 11197 ℕcn 12161 ℕ0cn0 12421 Basecbs 17091 0gc0g 17329 Σg cgsu 17330 ℂfldccnfld 20819 mPoly cmpl 21331 mDeg cmdg 25438 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-sup 9386 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-sca 17157 df-vsca 17158 df-tset 17160 df-psr 21334 df-mpl 21336 df-mdeg 25440 |
This theorem is referenced by: mdegleb 25452 mdeglt 25453 mdegldg 25454 mdegxrcl 25455 mdegcl 25457 mdeg0 25458 mdegvsca 25464 deg1val 25484 |
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