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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 6894 | . . . . . 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 3172 | . . . 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 3434 | . . 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 9474 | . 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 42459 | . 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 11298 | . . 3 β’ < Or β | |
8 | 7 | infex 9490 | . 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 6992 | 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 395 = wceq 1533 β wcel 2098 βwrex 3064 {crab 3426 β cdif 3940 {csn 4623 βcfv 6537 infcinf 9438 βcr 11111 0cc0 11112 < clt 11252 βcn 12216 βcq 12936 0πc0p 25553 Polycply 26073 degcdgr 26076 πΈcaa 26204 degAAcdgraa 42457 |
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-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
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-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 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-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-ltxr 11257 df-dgraa 42459 |
This theorem is referenced by: dgraalem 42462 dgraaub 42465 |
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