Step |
Hyp |
Ref |
Expression |
1 |
|
deg1addle.y |
|- Y = ( Poly1 ` R ) |
2 |
|
deg1addle.d |
|- D = ( deg1 ` R ) |
3 |
|
deg1addle.r |
|- ( ph -> R e. Ring ) |
4 |
|
deg1addle.b |
|- B = ( Base ` Y ) |
5 |
|
deg1addle.p |
|- .+ = ( +g ` Y ) |
6 |
|
deg1addle.f |
|- ( ph -> F e. B ) |
7 |
|
deg1addle.g |
|- ( ph -> G e. B ) |
8 |
|
deg1add.l |
|- ( ph -> ( D ` G ) < ( D ` F ) ) |
9 |
1
|
ply1ring |
|- ( R e. Ring -> Y e. Ring ) |
10 |
3 9
|
syl |
|- ( ph -> Y e. Ring ) |
11 |
4 5
|
ringacl |
|- ( ( Y e. Ring /\ F e. B /\ G e. B ) -> ( F .+ G ) e. B ) |
12 |
10 6 7 11
|
syl3anc |
|- ( ph -> ( F .+ G ) e. B ) |
13 |
2 1 4
|
deg1xrcl |
|- ( ( F .+ G ) e. B -> ( D ` ( F .+ G ) ) e. RR* ) |
14 |
12 13
|
syl |
|- ( ph -> ( D ` ( F .+ G ) ) e. RR* ) |
15 |
2 1 4
|
deg1xrcl |
|- ( F e. B -> ( D ` F ) e. RR* ) |
16 |
6 15
|
syl |
|- ( ph -> ( D ` F ) e. RR* ) |
17 |
1 2 3 4 5 6 7
|
deg1addle |
|- ( ph -> ( D ` ( F .+ G ) ) <_ if ( ( D ` F ) <_ ( D ` G ) , ( D ` G ) , ( D ` F ) ) ) |
18 |
2 1 4
|
deg1xrcl |
|- ( G e. B -> ( D ` G ) e. RR* ) |
19 |
7 18
|
syl |
|- ( ph -> ( D ` G ) e. RR* ) |
20 |
|
xrltnle |
|- ( ( ( D ` G ) e. RR* /\ ( D ` F ) e. RR* ) -> ( ( D ` G ) < ( D ` F ) <-> -. ( D ` F ) <_ ( D ` G ) ) ) |
21 |
19 16 20
|
syl2anc |
|- ( ph -> ( ( D ` G ) < ( D ` F ) <-> -. ( D ` F ) <_ ( D ` G ) ) ) |
22 |
8 21
|
mpbid |
|- ( ph -> -. ( D ` F ) <_ ( D ` G ) ) |
23 |
22
|
iffalsed |
|- ( ph -> if ( ( D ` F ) <_ ( D ` G ) , ( D ` G ) , ( D ` F ) ) = ( D ` F ) ) |
24 |
17 23
|
breqtrd |
|- ( ph -> ( D ` ( F .+ G ) ) <_ ( D ` F ) ) |
25 |
|
nltmnf |
|- ( ( D ` G ) e. RR* -> -. ( D ` G ) < -oo ) |
26 |
19 25
|
syl |
|- ( ph -> -. ( D ` G ) < -oo ) |
27 |
8
|
adantr |
|- ( ( ph /\ F = ( 0g ` Y ) ) -> ( D ` G ) < ( D ` F ) ) |
28 |
|
fveq2 |
|- ( F = ( 0g ` Y ) -> ( D ` F ) = ( D ` ( 0g ` Y ) ) ) |
29 |
|
eqid |
|- ( 0g ` Y ) = ( 0g ` Y ) |
30 |
2 1 29
|
deg1z |
|- ( R e. Ring -> ( D ` ( 0g ` Y ) ) = -oo ) |
31 |
3 30
|
syl |
|- ( ph -> ( D ` ( 0g ` Y ) ) = -oo ) |
32 |
28 31
|
sylan9eqr |
|- ( ( ph /\ F = ( 0g ` Y ) ) -> ( D ` F ) = -oo ) |
33 |
27 32
|
breqtrd |
|- ( ( ph /\ F = ( 0g ` Y ) ) -> ( D ` G ) < -oo ) |
34 |
33
|
ex |
|- ( ph -> ( F = ( 0g ` Y ) -> ( D ` G ) < -oo ) ) |
35 |
34
|
necon3bd |
|- ( ph -> ( -. ( D ` G ) < -oo -> F =/= ( 0g ` Y ) ) ) |
36 |
26 35
|
mpd |
|- ( ph -> F =/= ( 0g ` Y ) ) |
37 |
2 1 29 4
|
deg1nn0cl |
|- ( ( R e. Ring /\ F e. B /\ F =/= ( 0g ` Y ) ) -> ( D ` F ) e. NN0 ) |
38 |
3 6 36 37
|
syl3anc |
|- ( ph -> ( D ` F ) e. NN0 ) |
39 |
|
eqid |
|- ( +g ` R ) = ( +g ` R ) |
40 |
1 4 5 39
|
coe1addfv |
|- ( ( ( R e. Ring /\ F e. B /\ G e. B ) /\ ( D ` F ) e. NN0 ) -> ( ( coe1 ` ( F .+ G ) ) ` ( D ` F ) ) = ( ( ( coe1 ` F ) ` ( D ` F ) ) ( +g ` R ) ( ( coe1 ` G ) ` ( D ` F ) ) ) ) |
41 |
3 6 7 38 40
|
syl31anc |
|- ( ph -> ( ( coe1 ` ( F .+ G ) ) ` ( D ` F ) ) = ( ( ( coe1 ` F ) ` ( D ` F ) ) ( +g ` R ) ( ( coe1 ` G ) ` ( D ` F ) ) ) ) |
42 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
43 |
|
eqid |
|- ( coe1 ` G ) = ( coe1 ` G ) |
44 |
2 1 4 42 43
|
deg1lt |
|- ( ( G e. B /\ ( D ` F ) e. NN0 /\ ( D ` G ) < ( D ` F ) ) -> ( ( coe1 ` G ) ` ( D ` F ) ) = ( 0g ` R ) ) |
45 |
7 38 8 44
|
syl3anc |
|- ( ph -> ( ( coe1 ` G ) ` ( D ` F ) ) = ( 0g ` R ) ) |
46 |
45
|
oveq2d |
|- ( ph -> ( ( ( coe1 ` F ) ` ( D ` F ) ) ( +g ` R ) ( ( coe1 ` G ) ` ( D ` F ) ) ) = ( ( ( coe1 ` F ) ` ( D ` F ) ) ( +g ` R ) ( 0g ` R ) ) ) |
47 |
|
ringgrp |
|- ( R e. Ring -> R e. Grp ) |
48 |
3 47
|
syl |
|- ( ph -> R e. Grp ) |
49 |
|
eqid |
|- ( coe1 ` F ) = ( coe1 ` F ) |
50 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
51 |
49 4 1 50
|
coe1f |
|- ( F e. B -> ( coe1 ` F ) : NN0 --> ( Base ` R ) ) |
52 |
6 51
|
syl |
|- ( ph -> ( coe1 ` F ) : NN0 --> ( Base ` R ) ) |
53 |
52 38
|
ffvelrnd |
|- ( ph -> ( ( coe1 ` F ) ` ( D ` F ) ) e. ( Base ` R ) ) |
54 |
50 39 42
|
grprid |
|- ( ( R e. Grp /\ ( ( coe1 ` F ) ` ( D ` F ) ) e. ( Base ` R ) ) -> ( ( ( coe1 ` F ) ` ( D ` F ) ) ( +g ` R ) ( 0g ` R ) ) = ( ( coe1 ` F ) ` ( D ` F ) ) ) |
55 |
48 53 54
|
syl2anc |
|- ( ph -> ( ( ( coe1 ` F ) ` ( D ` F ) ) ( +g ` R ) ( 0g ` R ) ) = ( ( coe1 ` F ) ` ( D ` F ) ) ) |
56 |
41 46 55
|
3eqtrd |
|- ( ph -> ( ( coe1 ` ( F .+ G ) ) ` ( D ` F ) ) = ( ( coe1 ` F ) ` ( D ` F ) ) ) |
57 |
2 1 29 4 42 49
|
deg1ldg |
|- ( ( R e. Ring /\ F e. B /\ F =/= ( 0g ` Y ) ) -> ( ( coe1 ` F ) ` ( D ` F ) ) =/= ( 0g ` R ) ) |
58 |
3 6 36 57
|
syl3anc |
|- ( ph -> ( ( coe1 ` F ) ` ( D ` F ) ) =/= ( 0g ` R ) ) |
59 |
56 58
|
eqnetrd |
|- ( ph -> ( ( coe1 ` ( F .+ G ) ) ` ( D ` F ) ) =/= ( 0g ` R ) ) |
60 |
|
eqid |
|- ( coe1 ` ( F .+ G ) ) = ( coe1 ` ( F .+ G ) ) |
61 |
2 1 4 42 60
|
deg1ge |
|- ( ( ( F .+ G ) e. B /\ ( D ` F ) e. NN0 /\ ( ( coe1 ` ( F .+ G ) ) ` ( D ` F ) ) =/= ( 0g ` R ) ) -> ( D ` F ) <_ ( D ` ( F .+ G ) ) ) |
62 |
12 38 59 61
|
syl3anc |
|- ( ph -> ( D ` F ) <_ ( D ` ( F .+ G ) ) ) |
63 |
14 16 24 62
|
xrletrid |
|- ( ph -> ( D ` ( F .+ G ) ) = ( D ` F ) ) |