Theorem deg1fval 23840

Description: Relate univariate polynomial degree to multivariate. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.)

Hypothesis

Ref	Expression
deg1fval.d	|- D = ( deg1 ` R )

Assertion

Ref	Expression
deg1fval	|- D = ( 1o mDeg R )

Proof of Theorem deg1fval

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		deg1fval.d	. 2 |- D = ( deg1 ` R )
2		oveq2 6658	. . . 4 |- ( r = R -> ( 1o mDeg r ) = ( 1o mDeg R ) )
3		df-deg1 23816	. . . 4 |- deg1 = ( r e. _V |-> ( 1o mDeg r ) )
4		ovex 6678	. . . 4 |- ( 1o mDeg R ) e. _V
5	2, 3, 4	fvmpt 6282	. . 3 |- ( R e. _V -> ( deg1 ` R ) = ( 1o mDeg R ) )
6		fvprc 6185	. . . 4 |- ( -. R e. _V -> ( deg1 ` R ) = (/) )
7		reldmmdeg 23817	. . . . 5 |- Rel dom mDeg
8	7	ovprc2 6685	. . . 4 |- ( -. R e. _V -> ( 1o mDeg R ) = (/) )
9	6, 8	eqtr4d 2659	. . 3 |- ( -. R e. _V -> ( deg1 ` R ) = ( 1o mDeg R ) )
10	5, 9	pm2.61i 176	. 2 |- ( deg1 ` R ) = ( 1o mDeg R )
11	1, 10	eqtri 2644	1 |- D = ( 1o mDeg R )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ` cfv 5888  (class class class)co 6650   1oc1o 7553   mDeg cmdg 23813   deg₁ cdg1 23814

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  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-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-mdeg 23815  df-deg1 23816

This theorem is referenced by:  deg1xrf  23841  deg1cl  23843  deg1propd  23846  deg1z  23847  deg1nn0cl  23848  deg1ldg  23852  deg1leb  23855  deg1val  23856  deg1addle  23861  deg1vscale  23864  deg1vsca  23865  deg1mulle2  23869  deg1le0  23871