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Mirrors > Home > MPE Home > Th. List > dgrval | Structured version Visualization version GIF version |
Description: Value of the degree function. (Contributed by Mario Carneiro, 22-Jul-2014.) |
Ref | Expression |
---|---|
dgrval.1 | β’ π΄ = (coeffβπΉ) |
Ref | Expression |
---|---|
dgrval | β’ (πΉ β (Polyβπ) β (degβπΉ) = sup((β‘π΄ β (β β {0})), β0, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plyssc 26085 | . . 3 β’ (Polyβπ) β (Polyββ) | |
2 | 1 | sseli 3973 | . 2 β’ (πΉ β (Polyβπ) β πΉ β (Polyββ)) |
3 | fveq2 6884 | . . . . . . 7 β’ (π = πΉ β (coeffβπ) = (coeffβπΉ)) | |
4 | dgrval.1 | . . . . . . 7 β’ π΄ = (coeffβπΉ) | |
5 | 3, 4 | eqtr4di 2784 | . . . . . 6 β’ (π = πΉ β (coeffβπ) = π΄) |
6 | 5 | cnveqd 5868 | . . . . 5 β’ (π = πΉ β β‘(coeffβπ) = β‘π΄) |
7 | 6 | imaeq1d 6051 | . . . 4 β’ (π = πΉ β (β‘(coeffβπ) β (β β {0})) = (β‘π΄ β (β β {0}))) |
8 | 7 | supeq1d 9440 | . . 3 β’ (π = πΉ β sup((β‘(coeffβπ) β (β β {0})), β0, < ) = sup((β‘π΄ β (β β {0})), β0, < )) |
9 | df-dgr 26076 | . . 3 β’ deg = (π β (Polyββ) β¦ sup((β‘(coeffβπ) β (β β {0})), β0, < )) | |
10 | nn0ssre 12477 | . . . . 5 β’ β0 β β | |
11 | ltso 11295 | . . . . 5 β’ < Or β | |
12 | soss 5601 | . . . . 5 β’ (β0 β β β ( < Or β β < Or β0)) | |
13 | 10, 11, 12 | mp2 9 | . . . 4 β’ < Or β0 |
14 | 13 | supex 9457 | . . 3 β’ sup((β‘π΄ β (β β {0})), β0, < ) β V |
15 | 8, 9, 14 | fvmpt 6991 | . 2 β’ (πΉ β (Polyββ) β (degβπΉ) = sup((β‘π΄ β (β β {0})), β0, < )) |
16 | 2, 15 | syl 17 | 1 β’ (πΉ β (Polyβπ) β (degβπΉ) = sup((β‘π΄ β (β β {0})), β0, < )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β cdif 3940 β wss 3943 {csn 4623 Or wor 5580 β‘ccnv 5668 β cima 5672 βcfv 6536 supcsup 9434 βcc 11107 βcr 11108 0cc0 11109 < clt 11249 β0cn0 12473 Polycply 26069 coeffccoe 26071 degcdgr 26072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-i2m1 11177 ax-1ne0 11178 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-pnf 11251 df-mnf 11252 df-ltxr 11254 df-nn 12214 df-n0 12474 df-ply 26073 df-dgr 26076 |
This theorem is referenced by: dgrcl 26118 dgrub 26119 dgrlb 26121 coe11 26138 |
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