Step |
Hyp |
Ref |
Expression |
1 |
|
dgreq0.1 |
⊢ 𝑁 = ( deg ‘ 𝐹 ) |
2 |
|
dgreq0.2 |
⊢ 𝐴 = ( coeff ‘ 𝐹 ) |
3 |
|
simpr |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ 𝐹 = 0𝑝 ) → 𝐹 = 0𝑝 ) |
4 |
3
|
fveq2d |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ 𝐹 = 0𝑝 ) → ( deg ‘ 𝐹 ) = ( deg ‘ 0𝑝 ) ) |
5 |
|
dgr0 |
⊢ ( deg ‘ 0𝑝 ) = 0 |
6 |
5
|
eqcomi |
⊢ 0 = ( deg ‘ 0𝑝 ) |
7 |
4 1 6
|
3eqtr4g |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ 𝐹 = 0𝑝 ) → 𝑁 = 0 ) |
8 |
|
nn0ge0 |
⊢ ( 𝑀 ∈ ℕ0 → 0 ≤ 𝑀 ) |
9 |
8
|
ad2antlr |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ 𝐹 = 0𝑝 ) → 0 ≤ 𝑀 ) |
10 |
7 9
|
eqbrtrd |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ 𝐹 = 0𝑝 ) → 𝑁 ≤ 𝑀 ) |
11 |
3
|
fveq2d |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ 𝐹 = 0𝑝 ) → ( coeff ‘ 𝐹 ) = ( coeff ‘ 0𝑝 ) ) |
12 |
|
coe0 |
⊢ ( coeff ‘ 0𝑝 ) = ( ℕ0 × { 0 } ) |
13 |
12
|
eqcomi |
⊢ ( ℕ0 × { 0 } ) = ( coeff ‘ 0𝑝 ) |
14 |
11 2 13
|
3eqtr4g |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ 𝐹 = 0𝑝 ) → 𝐴 = ( ℕ0 × { 0 } ) ) |
15 |
14
|
fveq1d |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ 𝐹 = 0𝑝 ) → ( 𝐴 ‘ 𝑀 ) = ( ( ℕ0 × { 0 } ) ‘ 𝑀 ) ) |
16 |
|
c0ex |
⊢ 0 ∈ V |
17 |
16
|
fvconst2 |
⊢ ( 𝑀 ∈ ℕ0 → ( ( ℕ0 × { 0 } ) ‘ 𝑀 ) = 0 ) |
18 |
17
|
ad2antlr |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ 𝐹 = 0𝑝 ) → ( ( ℕ0 × { 0 } ) ‘ 𝑀 ) = 0 ) |
19 |
15 18
|
eqtrd |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ 𝐹 = 0𝑝 ) → ( 𝐴 ‘ 𝑀 ) = 0 ) |
20 |
10 19
|
jca |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ 𝐹 = 0𝑝 ) → ( 𝑁 ≤ 𝑀 ∧ ( 𝐴 ‘ 𝑀 ) = 0 ) ) |
21 |
|
dgrcl |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ( deg ‘ 𝐹 ) ∈ ℕ0 ) |
22 |
1 21
|
eqeltrid |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → 𝑁 ∈ ℕ0 ) |
23 |
22
|
nn0red |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → 𝑁 ∈ ℝ ) |
24 |
|
nn0re |
⊢ ( 𝑀 ∈ ℕ0 → 𝑀 ∈ ℝ ) |
25 |
|
ltle |
⊢ ( ( 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ) → ( 𝑁 < 𝑀 → 𝑁 ≤ 𝑀 ) ) |
26 |
23 24 25
|
syl2an |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) → ( 𝑁 < 𝑀 → 𝑁 ≤ 𝑀 ) ) |
27 |
26
|
imp |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ 𝑁 < 𝑀 ) → 𝑁 ≤ 𝑀 ) |
28 |
2 1
|
dgrub |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ∧ ( 𝐴 ‘ 𝑀 ) ≠ 0 ) → 𝑀 ≤ 𝑁 ) |
29 |
28
|
3expia |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝐴 ‘ 𝑀 ) ≠ 0 → 𝑀 ≤ 𝑁 ) ) |
30 |
|
lenlt |
⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( 𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀 ) ) |
31 |
24 23 30
|
syl2anr |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) → ( 𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀 ) ) |
32 |
29 31
|
sylibd |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝐴 ‘ 𝑀 ) ≠ 0 → ¬ 𝑁 < 𝑀 ) ) |
33 |
32
|
necon4ad |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) → ( 𝑁 < 𝑀 → ( 𝐴 ‘ 𝑀 ) = 0 ) ) |
34 |
33
|
imp |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ 𝑁 < 𝑀 ) → ( 𝐴 ‘ 𝑀 ) = 0 ) |
35 |
27 34
|
jca |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ 𝑁 < 𝑀 ) → ( 𝑁 ≤ 𝑀 ∧ ( 𝐴 ‘ 𝑀 ) = 0 ) ) |
36 |
20 35
|
jaodan |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ ( 𝐹 = 0𝑝 ∨ 𝑁 < 𝑀 ) ) → ( 𝑁 ≤ 𝑀 ∧ ( 𝐴 ‘ 𝑀 ) = 0 ) ) |
37 |
|
leloe |
⊢ ( ( 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ ) → ( 𝑁 ≤ 𝑀 ↔ ( 𝑁 < 𝑀 ∨ 𝑁 = 𝑀 ) ) ) |
38 |
23 24 37
|
syl2an |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) → ( 𝑁 ≤ 𝑀 ↔ ( 𝑁 < 𝑀 ∨ 𝑁 = 𝑀 ) ) ) |
39 |
38
|
biimpa |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ 𝑁 ≤ 𝑀 ) → ( 𝑁 < 𝑀 ∨ 𝑁 = 𝑀 ) ) |
40 |
39
|
adantrr |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ ( 𝑁 ≤ 𝑀 ∧ ( 𝐴 ‘ 𝑀 ) = 0 ) ) → ( 𝑁 < 𝑀 ∨ 𝑁 = 𝑀 ) ) |
41 |
|
fveq2 |
⊢ ( 𝑁 = 𝑀 → ( 𝐴 ‘ 𝑁 ) = ( 𝐴 ‘ 𝑀 ) ) |
42 |
1 2
|
dgreq0 |
⊢ ( 𝐹 ∈ ( Poly ‘ 𝑆 ) → ( 𝐹 = 0𝑝 ↔ ( 𝐴 ‘ 𝑁 ) = 0 ) ) |
43 |
42
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ ( 𝑁 ≤ 𝑀 ∧ ( 𝐴 ‘ 𝑀 ) = 0 ) ) → ( 𝐹 = 0𝑝 ↔ ( 𝐴 ‘ 𝑁 ) = 0 ) ) |
44 |
|
simprr |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ ( 𝑁 ≤ 𝑀 ∧ ( 𝐴 ‘ 𝑀 ) = 0 ) ) → ( 𝐴 ‘ 𝑀 ) = 0 ) |
45 |
44
|
eqeq2d |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ ( 𝑁 ≤ 𝑀 ∧ ( 𝐴 ‘ 𝑀 ) = 0 ) ) → ( ( 𝐴 ‘ 𝑁 ) = ( 𝐴 ‘ 𝑀 ) ↔ ( 𝐴 ‘ 𝑁 ) = 0 ) ) |
46 |
43 45
|
bitr4d |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ ( 𝑁 ≤ 𝑀 ∧ ( 𝐴 ‘ 𝑀 ) = 0 ) ) → ( 𝐹 = 0𝑝 ↔ ( 𝐴 ‘ 𝑁 ) = ( 𝐴 ‘ 𝑀 ) ) ) |
47 |
41 46
|
syl5ibr |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ ( 𝑁 ≤ 𝑀 ∧ ( 𝐴 ‘ 𝑀 ) = 0 ) ) → ( 𝑁 = 𝑀 → 𝐹 = 0𝑝 ) ) |
48 |
47
|
orim2d |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ ( 𝑁 ≤ 𝑀 ∧ ( 𝐴 ‘ 𝑀 ) = 0 ) ) → ( ( 𝑁 < 𝑀 ∨ 𝑁 = 𝑀 ) → ( 𝑁 < 𝑀 ∨ 𝐹 = 0𝑝 ) ) ) |
49 |
40 48
|
mpd |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ ( 𝑁 ≤ 𝑀 ∧ ( 𝐴 ‘ 𝑀 ) = 0 ) ) → ( 𝑁 < 𝑀 ∨ 𝐹 = 0𝑝 ) ) |
50 |
49
|
orcomd |
⊢ ( ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) ∧ ( 𝑁 ≤ 𝑀 ∧ ( 𝐴 ‘ 𝑀 ) = 0 ) ) → ( 𝐹 = 0𝑝 ∨ 𝑁 < 𝑀 ) ) |
51 |
36 50
|
impbida |
⊢ ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝑀 ∈ ℕ0 ) → ( ( 𝐹 = 0𝑝 ∨ 𝑁 < 𝑀 ) ↔ ( 𝑁 ≤ 𝑀 ∧ ( 𝐴 ‘ 𝑀 ) = 0 ) ) ) |