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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 6834 | . . . . . 6 β’ (π = π΄ β ((πβπ) = 0 β (πβπ΄) = 0)) | |
2 | 1 | anbi2d 629 | . . . . 5 β’ (π = π΄ β (((degβπ) = π β§ (πβπ) = 0) β ((degβπ) = π β§ (πβπ΄) = 0))) |
3 | 2 | rexbidv 3171 | . . . 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 3411 | . . 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 9334 | . 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 41230 | . 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 11156 | . . 3 β’ < Or β | |
8 | 7 | infex 9350 | . 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 6931 | 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 396 = wceq 1540 β wcel 2105 βwrex 3070 {crab 3403 β cdif 3895 {csn 4573 βcfv 6479 infcinf 9298 βcr 10971 0cc0 10972 < clt 11110 βcn 12074 βcq 12789 0πc0p 24939 Polycply 25451 degcdgr 25454 πΈcaa 25580 degAAcdgraa 41228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-resscn 11029 ax-pre-lttri 11046 ax-pre-lttrn 11047 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-po 5532 df-so 5533 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-sup 9299 df-inf 9300 df-pnf 11112 df-mnf 11113 df-ltxr 11115 df-dgraa 41230 |
This theorem is referenced by: dgraalem 41233 dgraaub 41236 |
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