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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 26147 | . . 3 β’ (Polyβπ) β (Polyββ) | |
2 | 1 | sseli 3976 | . 2 β’ (πΉ β (Polyβπ) β πΉ β (Polyββ)) |
3 | fveq2 6897 | . . . . . . 7 β’ (π = πΉ β (coeffβπ) = (coeffβπΉ)) | |
4 | dgrval.1 | . . . . . . 7 β’ π΄ = (coeffβπΉ) | |
5 | 3, 4 | eqtr4di 2786 | . . . . . 6 β’ (π = πΉ β (coeffβπ) = π΄) |
6 | 5 | cnveqd 5878 | . . . . 5 β’ (π = πΉ β β‘(coeffβπ) = β‘π΄) |
7 | 6 | imaeq1d 6062 | . . . 4 β’ (π = πΉ β (β‘(coeffβπ) β (β β {0})) = (β‘π΄ β (β β {0}))) |
8 | 7 | supeq1d 9470 | . . 3 β’ (π = πΉ β sup((β‘(coeffβπ) β (β β {0})), β0, < ) = sup((β‘π΄ β (β β {0})), β0, < )) |
9 | df-dgr 26138 | . . 3 β’ deg = (π β (Polyββ) β¦ sup((β‘(coeffβπ) β (β β {0})), β0, < )) | |
10 | nn0ssre 12507 | . . . . 5 β’ β0 β β | |
11 | ltso 11325 | . . . . 5 β’ < Or β | |
12 | soss 5610 | . . . . 5 β’ (β0 β β β ( < Or β β < Or β0)) | |
13 | 10, 11, 12 | mp2 9 | . . . 4 β’ < Or β0 |
14 | 13 | supex 9487 | . . 3 β’ sup((β‘π΄ β (β β {0})), β0, < ) β V |
15 | 8, 9, 14 | fvmpt 7005 | . 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 1534 β wcel 2099 β cdif 3944 β wss 3947 {csn 4629 Or wor 5589 β‘ccnv 5677 β cima 5681 βcfv 6548 supcsup 9464 βcc 11137 βcr 11138 0cc0 11139 < clt 11279 β0cn0 12503 Polycply 26131 coeffccoe 26133 degcdgr 26134 |
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 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-i2m1 11207 ax-1ne0 11208 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 |
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-op 4636 df-uni 4909 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-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-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-pnf 11281 df-mnf 11282 df-ltxr 11284 df-nn 12244 df-n0 12504 df-ply 26135 df-dgr 26138 |
This theorem is referenced by: dgrcl 26180 dgrub 26181 dgrlb 26183 coe11 26200 |
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