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Mirrors > Home > MPE Home > Th. List > mdegxrcl | Structured version Visualization version GIF version |
Description: Closure of polynomial degree in the extended reals. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Proof shortened by AV, 27-Jul-2019.) |
Ref | Expression |
---|---|
mdegxrcl.d | β’ π· = (πΌ mDeg π ) |
mdegxrcl.p | β’ π = (πΌ mPoly π ) |
mdegxrcl.b | β’ π΅ = (Baseβπ) |
Ref | Expression |
---|---|
mdegxrcl | β’ (πΉ β π΅ β (π·βπΉ) β β*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdegxrcl.d | . . 3 β’ π· = (πΌ mDeg π ) | |
2 | mdegxrcl.p | . . 3 β’ π = (πΌ mPoly π ) | |
3 | mdegxrcl.b | . . 3 β’ π΅ = (Baseβπ) | |
4 | eqid 2728 | . . 3 β’ (0gβπ ) = (0gβπ ) | |
5 | eqid 2728 | . . 3 β’ {π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin} = {π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin} | |
6 | eqid 2728 | . . 3 β’ (π¦ β {π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin} β¦ (βfld Ξ£g π¦)) = (π¦ β {π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin} β¦ (βfld Ξ£g π¦)) | |
7 | 1, 2, 3, 4, 5, 6 | mdegval 25998 | . 2 β’ (πΉ β π΅ β (π·βπΉ) = sup(((π¦ β {π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin} β¦ (βfld Ξ£g π¦)) β (πΉ supp (0gβπ ))), β*, < )) |
8 | imassrn 6074 | . . . 4 β’ ((π¦ β {π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin} β¦ (βfld Ξ£g π¦)) β (πΉ supp (0gβπ ))) β ran (π¦ β {π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin} β¦ (βfld Ξ£g π¦)) | |
9 | 5, 6 | tdeglem1 25990 | . . . . . 6 β’ (π¦ β {π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin} β¦ (βfld Ξ£g π¦)):{π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin}βΆβ0 |
10 | frn 6729 | . . . . . 6 β’ ((π¦ β {π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin} β¦ (βfld Ξ£g π¦)):{π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin}βΆβ0 β ran (π¦ β {π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin} β¦ (βfld Ξ£g π¦)) β β0) | |
11 | 9, 10 | mp1i 13 | . . . . 5 β’ (πΉ β π΅ β ran (π¦ β {π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin} β¦ (βfld Ξ£g π¦)) β β0) |
12 | nn0ssre 12506 | . . . . . 6 β’ β0 β β | |
13 | ressxr 11288 | . . . . . 6 β’ β β β* | |
14 | 12, 13 | sstri 3989 | . . . . 5 β’ β0 β β* |
15 | 11, 14 | sstrdi 3992 | . . . 4 β’ (πΉ β π΅ β ran (π¦ β {π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin} β¦ (βfld Ξ£g π¦)) β β*) |
16 | 8, 15 | sstrid 3991 | . . 3 β’ (πΉ β π΅ β ((π¦ β {π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin} β¦ (βfld Ξ£g π¦)) β (πΉ supp (0gβπ ))) β β*) |
17 | supxrcl 13326 | . . 3 β’ (((π¦ β {π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin} β¦ (βfld Ξ£g π¦)) β (πΉ supp (0gβπ ))) β β* β sup(((π¦ β {π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin} β¦ (βfld Ξ£g π¦)) β (πΉ supp (0gβπ ))), β*, < ) β β*) | |
18 | 16, 17 | syl 17 | . 2 β’ (πΉ β π΅ β sup(((π¦ β {π₯ β (β0 βm πΌ) β£ (β‘π₯ β β) β Fin} β¦ (βfld Ξ£g π¦)) β (πΉ supp (0gβπ ))), β*, < ) β β*) |
19 | 7, 18 | eqeltrd 2829 | 1 β’ (πΉ β π΅ β (π·βπΉ) β β*) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 {crab 3429 β wss 3947 β¦ cmpt 5231 β‘ccnv 5677 ran crn 5679 β cima 5681 βΆwf 6544 βcfv 6548 (class class class)co 7420 supp csupp 8165 βm cmap 8844 Fincfn 8963 supcsup 9463 βcr 11137 β*cxr 11277 < clt 11278 βcn 12242 β0cn0 12502 Basecbs 17179 0gc0g 17420 Ξ£g cgsu 17421 βfldccnfld 21278 mPoly cmpl 21838 mDeg cmdg 25985 |
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 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-0g 17422 df-gsum 17423 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-grp 18892 df-minusg 18893 df-cntz 19267 df-cmn 19736 df-abl 19737 df-mgp 20074 df-ur 20121 df-ring 20174 df-cring 20175 df-cnfld 21279 df-psr 21841 df-mpl 21843 df-mdeg 25987 |
This theorem is referenced by: mdegxrf 26003 mdegaddle 26009 mdegvscale 26010 mdegmullem 26013 |
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