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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dgraaval | Structured version Visualization version GIF version |
Description: Value of the degree function on an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Revised by AV, 29-Sep-2020.) |
Ref | Expression |
---|---|
dgraaval | β’ (π΄ β πΈ β (degAAβπ΄) = inf({π β β β£ βπ β ((Polyββ) β {0π})((degβπ) = π β§ (πβπ΄) = 0)}, β, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveqeq2 6911 | . . . . . 6 β’ (π = π΄ β ((πβπ) = 0 β (πβπ΄) = 0)) | |
2 | 1 | anbi2d 628 | . . . . 5 β’ (π = π΄ β (((degβπ) = π β§ (πβπ) = 0) β ((degβπ) = π β§ (πβπ΄) = 0))) |
3 | 2 | rexbidv 3176 | . . . 4 β’ (π = π΄ β (βπ β ((Polyββ) β {0π})((degβπ) = π β§ (πβπ) = 0) β βπ β ((Polyββ) β {0π})((degβπ) = π β§ (πβπ΄) = 0))) |
4 | 3 | rabbidv 3438 | . . 3 β’ (π = π΄ β {π β β β£ βπ β ((Polyββ) β {0π})((degβπ) = π β§ (πβπ) = 0)} = {π β β β£ βπ β ((Polyββ) β {0π})((degβπ) = π β§ (πβπ΄) = 0)}) |
5 | 4 | infeq1d 9508 | . 2 β’ (π = π΄ β inf({π β β β£ βπ β ((Polyββ) β {0π})((degβπ) = π β§ (πβπ) = 0)}, β, < ) = inf({π β β β£ βπ β ((Polyββ) β {0π})((degβπ) = π β§ (πβπ΄) = 0)}, β, < )) |
6 | df-dgraa 42597 | . 2 β’ degAA = (π β πΈ β¦ inf({π β β β£ βπ β ((Polyββ) β {0π})((degβπ) = π β§ (πβπ) = 0)}, β, < )) | |
7 | ltso 11332 | . . 3 β’ < Or β | |
8 | 7 | infex 9524 | . 2 β’ inf({π β β β£ βπ β ((Polyββ) β {0π})((degβπ) = π β§ (πβπ΄) = 0)}, β, < ) β V |
9 | 5, 6, 8 | fvmpt 7010 | 1 β’ (π΄ β πΈ β (degAAβπ΄) = inf({π β β β£ βπ β ((Polyββ) β {0π})((degβπ) = π β§ (πβπ΄) = 0)}, β, < )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βwrex 3067 {crab 3430 β cdif 3946 {csn 4632 βcfv 6553 infcinf 9472 βcr 11145 0cc0 11146 < clt 11286 βcn 12250 βcq 12970 0πc0p 25618 Polycply 26138 degcdgr 26141 πΈcaa 26269 degAAcdgraa 42595 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11203 ax-pre-lttri 11220 ax-pre-lttrn 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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 3374 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-ltxr 11291 df-dgraa 42597 |
This theorem is referenced by: dgraalem 42600 dgraaub 42603 |
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