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Metamath Proof Explorer

Theorem deg1add

Description: Exact degree of a sum of two polynomials of unequal degree. (Contributed by Stefan O'Rear, 28-Mar-2015)

Ref Expression
Hypotheses deg1addle.y ⊢ 𝑌 = ( Poly1 ‘ 𝑅 )
deg1addle.d ⊢ 𝐷 = ( deg1 ‘ 𝑅 )
deg1addle.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
deg1addle.b ⊢ 𝐵 = ( Base ‘ 𝑌 )
deg1addle.p ⊢ + = ( +g ‘ 𝑌 )
deg1addle.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
deg1addle.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐵 )
deg1add.l ⊢ ( 𝜑 → ( 𝐷 ‘ 𝐺 ) < ( 𝐷 ‘ 𝐹 ) )
Assertion deg1add ( 𝜑 → ( 𝐷 ‘ ( 𝐹 + 𝐺 ) ) = ( 𝐷 ‘ 𝐹 ) )

Proof

Step Hyp Ref Expression
1 deg1addle.y ⊢ 𝑌 = ( Poly1 ‘ 𝑅 )
2 deg1addle.d ⊢ 𝐷 = ( deg1 ‘ 𝑅 )
3 deg1addle.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
4 deg1addle.b ⊢ 𝐵 = ( Base ‘ 𝑌 )
5 deg1addle.p ⊢ + = ( +g ‘ 𝑌 )
6 deg1addle.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
7 deg1addle.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐵 )
8 deg1add.l ⊢ ( 𝜑 → ( 𝐷 ‘ 𝐺 ) < ( 𝐷 ‘ 𝐹 ) )
9 1 ply1ring ⊢ ( 𝑅 ∈ Ring → 𝑌 ∈ Ring )
10 3 9 syl ⊢ ( 𝜑 → 𝑌 ∈ Ring )
11 4 5 ringacl ⊢ ( ( 𝑌 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → ( 𝐹 + 𝐺 ) ∈ 𝐵 )
12 10 6 7 11 syl3anc ⊢ ( 𝜑 → ( 𝐹 + 𝐺 ) ∈ 𝐵 )
13 2 1 4 deg1xrcl ⊢ ( ( 𝐹 + 𝐺 ) ∈ 𝐵 → ( 𝐷 ‘ ( 𝐹 + 𝐺 ) ) ∈ ℝ* )
14 12 13 syl ⊢ ( 𝜑 → ( 𝐷 ‘ ( 𝐹 + 𝐺 ) ) ∈ ℝ* )
15 2 1 4 deg1xrcl ⊢ ( 𝐹 ∈ 𝐵 → ( 𝐷 ‘ 𝐹 ) ∈ ℝ* )
16 6 15 syl ⊢ ( 𝜑 → ( 𝐷 ‘ 𝐹 ) ∈ ℝ* )
17 1 2 3 4 5 6 7 deg1addle ⊢ ( 𝜑 → ( 𝐷 ‘ ( 𝐹 + 𝐺 ) ) ≤ if ( ( 𝐷 ‘ 𝐹 ) ≤ ( 𝐷 ‘ 𝐺 ) , ( 𝐷 ‘ 𝐺 ) , ( 𝐷 ‘ 𝐹 ) ) )
18 2 1 4 deg1xrcl ⊢ ( 𝐺 ∈ 𝐵 → ( 𝐷 ‘ 𝐺 ) ∈ ℝ* )
19 7 18 syl ⊢ ( 𝜑 → ( 𝐷 ‘ 𝐺 ) ∈ ℝ* )
20 xrltnle ⊢ ( ( ( 𝐷 ‘ 𝐺 ) ∈ ℝ* ∧ ( 𝐷 ‘ 𝐹 ) ∈ ℝ* ) → ( ( 𝐷 ‘ 𝐺 ) < ( 𝐷 ‘ 𝐹 ) ↔ ¬ ( 𝐷 ‘ 𝐹 ) ≤ ( 𝐷 ‘ 𝐺 ) ) )
21 19 16 20 syl2anc ⊢ ( 𝜑 → ( ( 𝐷 ‘ 𝐺 ) < ( 𝐷 ‘ 𝐹 ) ↔ ¬ ( 𝐷 ‘ 𝐹 ) ≤ ( 𝐷 ‘ 𝐺 ) ) )
22 8 21 mpbid ⊢ ( 𝜑 → ¬ ( 𝐷 ‘ 𝐹 ) ≤ ( 𝐷 ‘ 𝐺 ) )
23 22 iffalsed ⊢ ( 𝜑 → if ( ( 𝐷 ‘ 𝐹 ) ≤ ( 𝐷 ‘ 𝐺 ) , ( 𝐷 ‘ 𝐺 ) , ( 𝐷 ‘ 𝐹 ) ) = ( 𝐷 ‘ 𝐹 ) )
24 17 23 breqtrd ⊢ ( 𝜑 → ( 𝐷 ‘ ( 𝐹 + 𝐺 ) ) ≤ ( 𝐷 ‘ 𝐹 ) )
25 nltmnf ⊢ ( ( 𝐷 ‘ 𝐺 ) ∈ ℝ* → ¬ ( 𝐷 ‘ 𝐺 ) < -∞ )
26 19 25 syl ⊢ ( 𝜑 → ¬ ( 𝐷 ‘ 𝐺 ) < -∞ )
27 8 adantr ⊢ ( ( 𝜑 ∧ 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 ∈ Ring → ( 𝐷 ‘ ( 0g ‘ 𝑌 ) ) = -∞ )
31 3 30 syl ⊢ ( 𝜑 → ( 𝐷 ‘ ( 0g ‘ 𝑌 ) ) = -∞ )
32 28 31 sylan9eqr ⊢ ( ( 𝜑 ∧ 𝐹 = ( 0g ‘ 𝑌 ) ) → ( 𝐷 ‘ 𝐹 ) = -∞ )
33 27 32 breqtrd ⊢ ( ( 𝜑 ∧ 𝐹 = ( 0g ‘ 𝑌 ) ) → ( 𝐷 ‘ 𝐺 ) < -∞ )
34 33 ex ⊢ ( 𝜑 → ( 𝐹 = ( 0g ‘ 𝑌 ) → ( 𝐷 ‘ 𝐺 ) < -∞ ) )
35 34 necon3bd ⊢ ( 𝜑 → ( ¬ ( 𝐷 ‘ 𝐺 ) < -∞ → 𝐹 ≠ ( 0g ‘ 𝑌 ) ) )
36 26 35 mpd ⊢ ( 𝜑 → 𝐹 ≠ ( 0g ‘ 𝑌 ) )
37 2 1 29 4 deg1nn0cl ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ ( 0g ‘ 𝑌 ) ) → ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 )
38 3 6 36 37 syl3anc ⊢ ( 𝜑 → ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 )
39 eqid ⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
40 1 4 5 39 coe1addfv ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) ∧ ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 ) → ( ( coe1 ‘ ( 𝐹 + 𝐺 ) ) ‘ ( 𝐷 ‘ 𝐹 ) ) = ( ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ( +g ‘ 𝑅 ) ( ( coe1 ‘ 𝐺 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ) )
41 3 6 7 38 40 syl31anc ⊢ ( 𝜑 → ( ( 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 ∈ 𝐵 ∧ ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 ∧ ( 𝐷 ‘ 𝐺 ) < ( 𝐷 ‘ 𝐹 ) ) → ( ( coe1 ‘ 𝐺 ) ‘ ( 𝐷 ‘ 𝐹 ) ) = ( 0g ‘ 𝑅 ) )
45 7 38 8 44 syl3anc ⊢ ( 𝜑 → ( ( coe1 ‘ 𝐺 ) ‘ ( 𝐷 ‘ 𝐹 ) ) = ( 0g ‘ 𝑅 ) )
46 45 oveq2d ⊢ ( 𝜑 → ( ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ( +g ‘ 𝑅 ) ( ( coe1 ‘ 𝐺 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ) = ( ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ( +g ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) )
47 ringgrp ⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp )
48 3 47 syl ⊢ ( 𝜑 → 𝑅 ∈ Grp )
49 eqid ⊢ ( coe1 ‘ 𝐹 ) = ( coe1 ‘ 𝐹 )
50 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
51 49 4 1 50 coe1f ⊢ ( 𝐹 ∈ 𝐵 → ( coe1 ‘ 𝐹 ) : ℕ0 ⟶ ( Base ‘ 𝑅 ) )
52 6 51 syl ⊢ ( 𝜑 → ( coe1 ‘ 𝐹 ) : ℕ0 ⟶ ( Base ‘ 𝑅 ) )
53 52 38 ffvelcdmd ⊢ ( 𝜑 → ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ∈ ( Base ‘ 𝑅 ) )
54 50 39 42 grprid ⊢ ( ( 𝑅 ∈ Grp ∧ ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ∈ ( Base ‘ 𝑅 ) ) → ( ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ( +g ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) = ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) )
55 48 53 54 syl2anc ⊢ ( 𝜑 → ( ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ( +g ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) = ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) )
56 41 46 55 3eqtrd ⊢ ( 𝜑 → ( ( coe1 ‘ ( 𝐹 + 𝐺 ) ) ‘ ( 𝐷 ‘ 𝐹 ) ) = ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) )
57 2 1 29 4 42 49 deg1ldg ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ ( 0g ‘ 𝑌 ) ) → ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ≠ ( 0g ‘ 𝑅 ) )
58 3 6 36 57 syl3anc ⊢ ( 𝜑 → ( ( coe1 ‘ 𝐹 ) ‘ ( 𝐷 ‘ 𝐹 ) ) ≠ ( 0g ‘ 𝑅 ) )
59 56 58 eqnetrd ⊢ ( 𝜑 → ( ( coe1 ‘ ( 𝐹 + 𝐺 ) ) ‘ ( 𝐷 ‘ 𝐹 ) ) ≠ ( 0g ‘ 𝑅 ) )
60 eqid ⊢ ( coe1 ‘ ( 𝐹 + 𝐺 ) ) = ( coe1 ‘ ( 𝐹 + 𝐺 ) )
61 2 1 4 42 60 deg1ge ⊢ ( ( ( 𝐹 + 𝐺 ) ∈ 𝐵 ∧ ( 𝐷 ‘ 𝐹 ) ∈ ℕ0 ∧ ( ( coe1 ‘ ( 𝐹 + 𝐺 ) ) ‘ ( 𝐷 ‘ 𝐹 ) ) ≠ ( 0g ‘ 𝑅 ) ) → ( 𝐷 ‘ 𝐹 ) ≤ ( 𝐷 ‘ ( 𝐹 + 𝐺 ) ) )
62 12 38 59 61 syl3anc ⊢ ( 𝜑 → ( 𝐷 ‘ 𝐹 ) ≤ ( 𝐷 ‘ ( 𝐹 + 𝐺 ) ) )
63 14 16 24 62 xrletrid ⊢ ( 𝜑 → ( 𝐷 ‘ ( 𝐹 + 𝐺 ) ) = ( 𝐷 ‘ 𝐹 ) )