Theorem dgraaval 37714

Description: Value of the degree function on an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) (Revised by AV, 29-Sep-2020.)

Assertion

Ref	Expression
dgraaval	|- ( A e. AA -> (degAA ` A ) = inf ( { d e. NN | E. p e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` p ) = d /\ ( p ` A ) = 0 ) } , RR , < ) )

Distinct variable group:   A, d, p

Proof of Theorem dgraaval

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . 7 |- ( a = A -> ( p ` a ) = ( p ` A ) )
2	1	eqeq1d 2624	. . . . . 6 |- ( a = A -> ( ( p ` a ) = 0 <-> ( p ` A ) = 0 ) )
3	2	anbi2d 740	. . . . 5 |- ( a = A -> ( ( (deg ` p ) = d /\ ( p ` a ) = 0 ) <-> ( (deg ` p ) = d /\ ( p ` A ) = 0 ) ) )
4	3	rexbidv 3052	. . . 4 |- ( a = A -> ( E. p e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` p ) = d /\ ( p ` a ) = 0 ) <-> E. p e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` p ) = d /\ ( p ` A ) = 0 ) ) )
5	4	rabbidv 3189	. . 3 |- ( a = A -> { d e. NN | E. p e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` p ) = d /\ ( p ` a ) = 0 ) } = { d e. NN | E. p e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` p ) = d /\ ( p ` A ) = 0 ) } )
6	5	infeq1d 8383	. 2 |- ( a = A -> inf ( { d e. NN | E. p e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` p ) = d /\ ( p ` a ) = 0 ) } , RR , < ) = inf ( { d e. NN | E. p e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` p ) = d /\ ( p ` A ) = 0 ) } , RR , < ) )
7		df-dgraa 37712	. 2 |- degAA = ( a e. AA |-> inf ( { d e. NN | E. p e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` p ) = d /\ ( p ` a ) = 0 ) } , RR , < ) )
8		ltso 10118	. . 3 |- < Or RR
9	8	infex 8399	. 2 |- inf ( { d e. NN | E. p e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` p ) = d /\ ( p ` A ) = 0 ) } , RR , < ) e. _V
10	6, 7, 9	fvmpt 6282	1 |- ( A e. AA -> (degAA ` A ) = inf ( { d e. NN | E. p e. ( (Poly ` QQ ) \ { 0p } ) ( (deg ` p ) = d /\ ( p ` A ) = 0 ) } , RR , < ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   \ cdif 3571   {csn 4177   ` cfv 5888  infcinf 8347   RRcr 9935   0cc0 9936   < clt 10074   NNcn 11020   QQcq 11788   0pc0p 23436  Polycply 23940  degcdgr 23943   AAcaa 24069  deg_AAcdgraa 37710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-dgraa 37712

This theorem is referenced by:  dgraalem  37715  dgraaub  37718